Extremal Kähler Poincaré Type Metrics on Toric Varieties

Abstract

We develop a general theory for the existence of extremal Kähler metrics of Poincaré type in the sense of Auvray (J Reine Angew Math 722:1–64, 2017), defined on the complement of a torus invariant divisor of a smooth compact toric variety. In the case when the divisor is smooth, we obtain a list of necessary conditions which must be satisfied for such a metric to exist. Using the explicit methods of Apostolov et al. (Ann Sci Ecole Norm Supp (4) 48:1075–1112, 2015; J Reine Angew Math 721:109–147, 2016, https://doi.org/10.1515/crelle-2014-0060) together with the computational approach of Sektnan (N Y J Math 24:317–354, 2018), we show that on a Hirzebruch complex surface the necessary conditions are also sufficient. In particular, on such a complex surface the complement of the infinity section admits an extremal Kähler metric of Poincaré type, whereas the complement of a fibre fixed by the torus action admits a complete ambitoric extremal Kähler metric which is not of Poincaré type.

1 Introduction

In this article, we are interested in the study of (non-compact) complete extremal Kähler metrics, defined on the complement of a simple normal crossing divisor Z in an n-dimensional Kähler manifold X. Such metrics naturally appear (see e.g. [22]) in attempts to apply continuity methods, or to study global properties of geometric flows, aiming at producing extremal Kähler metrics on X in the framework of the general problem of finding canonical Kähler metrics formulated by Calabi [12].

The main conjecture regarding the Calabi problem is the Yau–Tian–Donaldson conjecture which relates the existence of an extremal Kähler metric in the first Chern class $c_1(L)$ of an ample line bundle L on X to an algebro-geometric notion of stability of the polarized projective variety (X, L). In this context, a key point is to understand what happens when an extremal Kähler metric does not exist in $c_1(L)$. For toric varieties, Donaldson conjectured [22, Conj. 7.2.3.] that there should be a splitting of the corresponding Delzant polytope into sub-polytopes which are semistable when attaching a 0 measure to the facets that are not from the original polytope; furthermore, in the case when a semistable polytope in the splitting is stable, it is conjectured to admit a symplectic potential inducing a (unique) complete extremal Kähler metric on the complement of the divisors corresponding to the facets with 0 measure. Such extremal toric Kähler metrics have a finite volume, and we shall refer to them as Donaldson metrics.

The main motivation for this paper is to study, in the toric case, the precise link between the extremal Donaldson metrics and the class of complete Kähler metrics of finite volume on $X\setminus Z$, called of Poincaré type, early used for instance in [18, 47], and studied by the second named author in [8].

Definition 1.1

Let $Z \subset X$ be a simple normal crossing divisor in a compact complex n-dimensional Kähler manifold $(X, \omega _0)$. A Kähler metric $\omega$ on $X\setminus Z$ is said to be of Poincaré type of class $[\omega _0]$ if

• On any open subset $U\subset X$ with holomorphic coordinates $(z_1, \ldots , z_n)$ such that $Z\cap U$ is given by $z_1\cdots z_k =0$, $\omega$ is quasi-isometric to the (1, 1)-form

  $$\begin{aligned} \omega _\mathrm{mod} =\sqrt{-1} \left( \sum _{j=1}^k \frac{1}{|z_j|^2(\log |z_j|)^2} {\text {d}}z_j \wedge {\text {d}}\bar{z}_j + \sum _{j=k+1}^n {\text {d}}z_j \wedge {\text {d}}\bar{z}_j \right) \end{aligned}$$

  near Z, and

• $\omega = \omega _0 + dd^c \varphi$ where $\varphi$ is a smooth function on $X\setminus Z$ and we have that $\varphi = O\big (\sum _{j=1}^k \log (-\log |z_j|)\big )$ in the coordinates $(z_1, \ldots , z_n)$ as above, with ${\text {d}}\varphi$ having bounded derivatives of any order with respect to the model metric $\omega _{\mathrm{mod}}$ above.

General theory for extremal Poincaré type metrics on $(X\setminus Z, [\omega _0])$ has been developed in [7,8,9,10]. In particular, a differential-geometric obstruction for the existence of a constant scalar curvature Kähler (CSCK) metric of Poincaré type on $X\subset Z$, reminiscent to the usual Futaki invariant, is introduced in [10]. Furthermore, in the special case when the Kähler class $[\omega _0]= c_1(L)$ is associated to a polarization L on X, an algebro-geometric notion of (relative) K-stability of (X, Z, L) is formulated by Székelyhidi [45], who introduced a suitable version of the Donaldson–Futaki invariant of a test configuration associated to the triple (X, Z, L). Székelyhidi also defined a numerical constraint, which we shall refer to in this paper as Székelyhidi’s numerical constraint (see Definition 2.6), which is related to the deformation to the normal cone of $Z\subset X$, and is designed to guarantee the existence of a Poincaré type metric (and not a complete extremal Kähler metric with different asymptotics near Z). It was later shown in [7] that Székelyhidi’s numerical constraint is a necessary condition for the existence of a CSCK Poincaré type metric on $X\setminus Z$ in the class $c_1(L)$. The case when Z is smooth and admits a Kähler metric of non-positive constant scalar curvature has been also studied in [41,42,43], where it is conjectured that (X, Z, L) is then K-semistable and admits a complete Kähler metric of negative constant scalar curvature.

Thus motivated, in Sect. 3 we turn to the case when (X, L) is a smooth toric variety, and Z a divisor invariant under the torus action. Compared to the theory in [43], we are dealing with the case when each component of Z is a toric variety, and therefore can only admit a Kähler metric of positive constant scalar curvature. In terms of the corresponding momentum polytope $\Delta$, Z is the pre-image by the moment map of the union $F= \cup _{i} F_i$ of facets $F_i$ of $\Delta$. In this setting (and taking F to be a single facet), we show that Székelyhidi’s numerical constraint takes a particularly simple form (Lemma 3.2), and matches the necessary numerical condition for the existence of an extremal Kähler metric of Poincaré type on $X\setminus Z$ found in [10].

In Sect. 4, we develop the Abreu–Guillemin formalism of toric Kähler metrics of Poincaré type, thus leading to a natural class of symplectic potentials (see Definition 4.16 and Theorem 4.18) which give rise to Poincaré type metrics in the sense of Definition 1.1. While these conditions are sufficient, they are not necessary in general (but are conjecturally sharp when adding the extremality condition). We show that within this class of Poincaré type metrics on $X\setminus Z$, the extremal ones are unique.

Building on the recent results in [9] and a conjecture from [22] (see Conjecture 4.11), we state a precise conjectural picture concerning the existence of an extremal Poincaré type toric Kähler metric on $X\setminus Z$. When Z is smooth, the conjecture says the following:

Conjecture 1.2

A smooth compact toric Kähler manifold $(X, \omega _0)$ with momentum polytope $\Delta$ and a divisor $Z \subset X$ corresponding to the pre-image of the union of disjoint facets $F=F_1 \cup \cdots \cup F_k$ of $\Delta$ admits an extremal toric Kähler metric of Poincaré type in $[\omega _0]$ if and only if the following three conditions are satisfied:

• (i) $(\Delta , F)$ is stable, and

• (ii) each facet $F_i \subset F$ is stable, and

• (iii) if ${s}_{(\Delta ,F)}$ denotes the extremal affine function corresponding to $(\Delta , F)$ and, for each facet $F_i \subset F$, $s_{F_i}$ is the extremal affine function corresponding to the Delzant polytope $F_i$, then

  $$\begin{aligned} s_{F_i} - \big ({s}_{(\Delta ,F)} \big )_{|_{F_i}}= c_i >0, \end{aligned}$$

  for a constant $c_i$.

See Conjecture 4.14 in the body of the paper for the full statement when F is allowed to have intersecting facets. This is much stronger than the original conjecture made in [45], but we show that it is sharper too.

Theorem 1.3

(cf. Theorem 4.13) The conditions (i), (ii), (iii) of Conjecture 1.2 are necessary for $X\setminus Z$ to admit an extremal toric Kähler metric of Poincaré type in $[\omega _0]$.

The precise notions of stability for the pair $(\Delta , F)$ and F in the above statements are the ones corresponding to relative K-stability with respect to toric test configuration, introduced by Donaldson [22] (see Definitions 3.1, 4.10 and 4.4), but in the light of recent progress on the Yau–Tian–Donaldson conjecture in the compact toric case, we expect that a yet stronger notion of uniform stability with respect to the ${\varvec{L}}^1$-norm should be considered. Section 4.4 has a detailed discussion on these issues, as well as on some of the technical obstacles one would need to overcome in order to adapt the arguments in the compact case for proving the necessity of uniform stability to the Poincaré type setting.

In the last Sect. 5, we turn to explicit examples by using the methods of [5, 6]. These results together with [21] confirm a conjecture by Donaldson [22] (see Conjecture 4.11) concerning the existence of a complete extremal Kähler metric, in the special case when $(\Delta ,F)$ is a stable quadrilateral with some of its facets with measure 0, also allowing us to find the metric explicitly. Investigating the stability of such pairs is, on its own, a problem of formidable complexity but using the method from [40], we obtain a complete picture on the Hirzebruch complex surfaces.

Theorem 1.4

Let $X= {\mathbb {P}}({\mathcal {O}}\oplus {\mathcal {O}}(m)) \rightarrow {\mathbb {C}}P^1, m\ge 1$, be the mth Hirzebruch surface, considered as a toric complex surface under the action of a 2-dimensional torus ${\mathbb {T}}$, and $[\omega _0]$ be a Kähler class on X. Then,

• (a) If $Z \subset X$ is the divisor consisting of either the zero section or the infinity section, or the union of both, then the conditions (i)–(iii) of Conjecture 1.2 hold and $X\setminus Z$ admits a ${\mathbb {T}}$-invariant extremal Poincaré type Kähler metric in $[\omega _0],$ which is a Donaldson metric of $(X, Z, [\omega _0])$.

• (b) If $Z\subset X$ is the divisor consisting of a single fibre of X fixed by the ${\mathbb {T}}$-action, (or is the union of such a fibre with the zero or infinity section), then the conditions (i) and (ii) of Conjecture 1.2 (resp. Conjecture 4.14) hold but the condition (iii) fails, and $X\setminus Z$ admits a complete ${\mathbb {T}}$-invariant Donaldson extremal Kähler metric in $[\omega _0],$ which is not (and cannot be) of Poincaré type.

• (c) If Z consists of the union of the two fibres fixed by the torus action (or contains three curves fixed by the action), then the condition (i) of Conjecture 1.2 (resp. Conjecture 4.14) fails, and there are no Donaldson complete metrics on $X\setminus Z$.

In particular, Conjectures 1.2 and 4.14 hold true when X is a Hirzebruch surface.

Similar results are obtained for the toric surfaces $X= {\mathbb {C}}P^2$ and ${\mathbb {C}}P^1 \times {\mathbb {C}}P^1$, see Corollary 5.3 and Theorem 5.13.

We end the introduction by noticing that part (b) of Theorem 1.4 implies that while for the X, Z and $[\omega _0]$ considered here, the relative stability of $(X,Z,[\omega _0])$ does imply the existence of a complete extremal Kähler metric on $X \setminus Z$, this metric is not in general of Poincaré type, even though the Székelyhidi numerical constraint is satisfied. Thus, in general, one will need more conditions to guarantee that the extremal metric obtained for a relatively stable triple $(X, Z, [\omega _0])$ is of Poincaré type. Conjecture 1.2 is designed to incorporate this extra requirement in the toric setting.

Fig. 1

The rows illustrate cases (a), (b) and (c) of Theorem 1.4, where $S_0, S_{\infty }$ stand for the zero and infinity sections and $F_1, F_2$ for the torus invariant fibres

2 The Relative K-Stability of a Pair

2.1 Donaldson–Futaki Invariant of a Pair

We follow [45, §3.1.2].

Let (X, L) be a smooth polarized variety of complex dimension n and $Z\subset X$ a smooth divisor. We consider the embedding

$$\begin{aligned} H^0(X, L^k\otimes {\mathcal {O}}(-Z)) \subset H^0(X, L^k) \end{aligned}$$

via a section of ${\mathcal {O}}(Z)$ which vanishes along Z. Since L is ample,

$$\begin{aligned} H^1(X, L^k\otimes {\mathcal {O}}(-Z))=0 \ \ \mathrm{for} \ k\gg 0, \end{aligned}$$

and we have an exact sequence

$$\begin{aligned} 0 \longrightarrow H^0(X, L^k\otimes {\mathcal {O}}(-Z)) \longrightarrow H^0(X, L^k) \longrightarrow H^0(Z, L_{|_Z}^k) \longrightarrow 0. \end{aligned}$$

(1)

Let $d_k, d'_k, d^Z_k$ be the dimensions of $H^0(X,L^k), H^0(X, L^k\otimes {\mathcal {O}}(-Z)), H^0(Z, L_{|_Z}^k))$, respectively, and let ${\tilde{d}}_k$ be the average of $d_k$ and $d'_k$. By Riemann–Roch and (1),

$$\begin{aligned} \begin{aligned} d_k&= c_0k^n + c_1k^{n-1} + O(k^{n-2}); \ \ d^Z_k = \alpha _0k^{n-1} + \alpha _1 k^{n-2} + O(k^{n-3});\\ {\tilde{d}}_k&= \frac{d_k + d'_k}{2}= d_k - \frac{d^Z_k}{2}= c_0 k^{n} + \Big (c_1- \frac{\alpha _0}{2}\Big )k^{n-1} + O(k^{n-2}). \end{aligned} \end{aligned}$$

(2)

Suppose $\alpha$ is a ${\mathbb {C}}^{\times }$-action on (X, L) which preserves Z. We denote also by $\alpha$ the induced ${\mathbb {C}}^{\times }$ action on X and in what follows, we use the following convention for the infinitesimal generator $A^{\alpha }$ for the action of $\alpha$ on the space $\Gamma (L)$ of smooth sections of L:

$$\begin{aligned} (A^{\alpha } (s))(x) := \sqrt{-1} \frac{{\text {d}}}{{\text {d}}t}_{|_{t=0}} \Big (\alpha (e^{\sqrt{-1}t})\big (s(\alpha (e^{-\sqrt{-1}t})(x))\big ) \Big ), \ \ s \in \Gamma (L), x\in M. \end{aligned}$$

Letting $w_k, w'_k, w^Z_k$ be the respective weights of the induced actions of $\alpha$ on $H^0(X,L^k),$ $H^0(X, L^k\otimes {\mathcal {O}}(-Z)), H^0(Z, L_{|_Z}^k))$, respectively, and ${\tilde{w}}_k$ be the average of $w_k$ and $w'_k$, by the equivariant Riemann–Roch and (1) we have

$$\begin{aligned} \begin{aligned} w_k (\alpha )&= a_0k^{n+1} + a_1k^{n} + O(k^{n-1}); \ \ w^Z_k (\alpha ) = \beta _0k^{n} + \beta _1 k^{n-1} + O(k^{n-2});\\ {\tilde{w}}_k(\alpha )&= \frac{w_k (\alpha )+ w'_k(\alpha )}{2}= w_k(\alpha ) - \frac{w^Z_k(\alpha )}{2}= a_0 k^{n+1} + \Big (a_1- \frac{\beta _0}{2}\Big )k^{n} + O(k^{n-1}). \end{aligned} \end{aligned}$$

(3)

Definition 2.1

The Donaldson–Futaki invariant ${\widetilde{\mathcal {F}}}_{X,Z,L}(\alpha )$ of $\alpha$ with respect to (X, Z, L) is defined up to a sign as 4 times the residue at $k=0$ of the Laurent series of ${\tilde{w}}_k/(k {\tilde{d}}_k)$ with respect to k, i.e.

$$\begin{aligned} \begin{aligned} \frac{1}{4} {\widetilde{\mathcal {F}}}_{X,Z,L}(\alpha )&= \frac{c_0\big (a_1- \frac{\beta _0}{2}\big ) -a_0\big (c_1- \frac{\alpha _0}{2}\big )}{c_0^2}\\&= \frac{1}{4}{\mathcal {F}}_{X,L}(\alpha ) + \frac{1}{2}\Big (\frac{a_0\alpha _0 -c_0\beta _0}{c_0^2}\Big ), \end{aligned} \end{aligned}$$

(4)

where ${\mathcal {F}}_{X,L}(\alpha ) = 4(\frac{c_0a_1- a_0c_1}{c_0^2})$ is the convention for the Donaldson–Futaki invariant in [22], so that it coincides, up to a multiplicative factor of $\frac{1}{4(2\pi )^n}$, with the differential-geometric formula in [10] for the usual normalized Futaki invariant of $\alpha$, expressed in terms of the ${\varvec{L}}^2$-product of a normalized Killing potential for the ${\mathbb {C}}^{\times }$-action and the scalar curvature with respect to an $S^1$-invariant Kähler metric on X in $2\pi c_1(L)$, divided by the volume.

Following [45], one can also define a relative version of ${\widetilde{\mathcal {F}}}_{X,Z,L}(\alpha )$ with respect to another ${\mathbb {C}}^{\times }$-action $\beta$ in the group $\mathrm{Aut}(X, L, Z)$ of automorphisms of (X, L), preserving Z. Recall that the inner product $\langle \alpha , \beta \rangle$ is defined to be the coefficient of $k^{n+2}$ of the expansion of $\mathrm{Tr} (A_k B_k) - w_k(\alpha )w_k(\beta )/d_k$, where $A_k$ and $B_k$ are generators of the actions of $\alpha$ and $\beta$ on $H^0(X,L^k)$. This definition is consistent with the ${\varvec{L}}^2$ -norm of normalized Killing potentials (the so-called Futaki–Mabuchi bilinear form [26]).

Definition 2.2

The $\beta$-relative Donaldson–Futaki invariant (of $\alpha$, with respect to (X, Z, L)) is

$$\begin{aligned} {\widetilde{\mathcal {F}}}_{X,Z,L}^{\beta }(\alpha ) = {\widetilde{\mathcal {F}}}_{X,Z,L}(\alpha ) - \frac{\langle \alpha , \beta \rangle }{\langle \beta , \beta \rangle } {\widetilde{\mathcal {F}}}_{X,Z,L}(\beta ). \end{aligned}$$

(5)

The above definitions make sense for any rational multiples of $\alpha$ and $\beta$ (by linearity). We then consider a maximal complex torus ${\mathbb {T}}^c =({\mathbb {C}}^{\times })^{\ell }$ in $\mathrm{Aut}(X, L, Z)$ and define the extremal ${\mathbb {C}}^{\times }$-action $\chi$ of (X, L, Z) as the unique ${\mathbb {C}}^{\times }$ subgroup of ${\mathbb {T}}^c$ such that ${\widetilde{\mathcal {F}}}_{X,Z,L}^{\chi }(\alpha )=0$.

2.2 Test Configurations and K-Stability of a Pair

The ingredients of the previous section yield Székelyhidi’s extension [45] of K-stability to pairs.

Definition 2.3

The triple (X, Z, L) is called K-stable if for any test configuration $({\mathcal {X}}, {{\mathcal {L}}})$ of (X, L) with a flat ${\mathbb {C}}^{\times }$-invariant Cartier divisor ${\mathcal {Z}} \subset {\mathcal {X}}$ which restricts to Z on the non-zero fibres, the modified Donaldson–Futaki invariant of the central fibre satisfies

$$\begin{aligned} {\widetilde{\mathcal {F}}}_{X_0,Z_0,L_0}(\alpha ) \ge 0 \end{aligned}$$

(6)

with equality if and only if the test configuration is trivial in codimension 2 (see [44] for a precise definition of triviality). Similarly, one can define relative K-stability of (X, L, Z) by requiring

$$\begin{aligned} {\widetilde{\mathcal {F}}}^{\chi }_{X_0,Z_0,L_0}(\alpha ) \ge 0, \end{aligned}$$

(7)

with equality if and only if the test configuration is trivial in codimension 2. (Recall $\chi$ is the extremal ${\mathbb {C}}^{\times }$-action defined algebraically in the previous section.)

Investigating a ruled complex surface $X={\mathbb {P}}({\mathcal {O}}\oplus {\mathcal {L}}) \rightarrow \Sigma$ with Z being the infinity section, Székelyhidi [45] noticed that for some polarizations L, there are complete finite volume extremal Kähler metrics on $X\setminus Z$ in $c_1(L),$ which are not of Poincaré type, but have instead the asymptotics of

$$\begin{aligned} \frac{|{\text {d}}z|^2}{|z|^2 \big (- \log (|z|) \big )^{\frac{3}{2}}} + \ \mathrm{smooth}, \end{aligned}$$

where z is a (local) defining holomorphic function of Z. In order to exclude this behaviour, Székelyhidi furthermore proposes to use the notion of slope stability introduced by Ross–Thomas [39] for the triple (X, L, Z) as follows. Recall that for any (X, L, Z) as above, and any rational number $c\in (0, \epsilon (Z))$ (where $\epsilon (Z)$ is the Seshadri constant of Z with respect to (X, L)), one can associate a test configuration $({\mathcal {X}}, {\mathcal {L}}_c, {\mathcal {Z}})$, called the degeneration to the normal cone of Z: ${\mathcal {X}}$ is the blow-up of $X\times {\mathbb {C}}$ along $Z\times \{0\}$, ${\mathcal {L}}_c=\pi ^*(L)\otimes {\mathcal {O}}(-cP)$ where P is the exceptional divisor (naturally identified with the projective bundle $P=\mathbb {P}({\mathcal {O}}\oplus \nu _Z) \rightarrow Z$ where $\nu _Z$ is the normal bundle of $Z\subset X$), and $\pi : {\mathcal {X}} \rightarrow {\mathbb {C}}$ is the projection. Note that the central fibre $X_0$ of $\pi$ is isomorphic to X glued to P along the infinity section $\mathbb {P}(\nu _Z)\cong Z$. However, considering the zero section $Z_0 \subset P$, one gets the proper transform ${\mathcal {Z}}$ of $Z\times {\mathbb {C}}\subset X\times {\mathbb {C}}$ on the blow-up ${\mathcal {X}}$, so that $\pi : {\mathcal {Z}} \rightarrow {\mathbb {C}}$ is a trivial family. It follows from [39] that $({\mathcal {X}}, {\mathcal {L}}_c, {\mathcal {Z}})$ defines a test configuration for the triple (X, L, Z). This motivates:

Definition 2.4

In the notation above, we let

$$\begin{aligned} F(c):={\widetilde{\mathcal {F}}}_{X_0,{{\mathcal {L}}_c}_{|_{{\mathcal {X}}_0},Z_0}}(\alpha _c), \ \ \ F_{\chi }(c):={\widetilde{\mathcal {F}}}^{\chi }_{X_0, Z_0, {{\mathcal {L}}_c}_{|_{{\mathcal {X}}_0}}}(\alpha _c) \end{aligned}$$

be the corresponding modified Donaldson–Futaki invariant and relative modified Donaldson–Futaki invariant associated to the degeneration of the normal cone to $Z\subset (X, L)$.

Then, Székelyhidi conjectures:

Conjecture 2.5

(Székelyhidi [45]) The triple (X, Z, L) admits a constant scalar curvature (resp. an extremal) Kähler metric of Poincaré type if and only if (X, Z, L) is K-stable (resp. relative K-stable) and, additionally, $F''(0)>0$ (resp. $F_{\chi }''(0)>0{\text {)}}$.

Definition 2.6

We shall refer to the conditions $F''(0)>0$ (resp. $F_{\chi }''(0)>0$) as the Székelyhidi numerical constraint (resp. relative Székelyhidi numerical constraint).

The following observation is made in [45]:

Lemma 2.7

$F(c):={\widetilde{\mathcal {F}}}_{X_0,{{\mathcal {L}}_c}_{|_{{\mathcal {X}}_0},Z_0}}(\alpha _c)$ is a polynomial of degree $\le (n+1)$ in c satisfying $F(0)=F'(0)=0$. It is positive for $c\in (0, \epsilon (Z))$ if (X, L, Z) is K-stable. Furthermore, the Székelyhidi numerical constraint $F''(0)>0$ is equivalent to

$$\begin{aligned} \alpha _1c_0 > \alpha _0\left( c_1 - \frac{\alpha _0}{2}\right) , \end{aligned}$$

(8)

where $\alpha _i$, $c_i$ are defined by (2).

Proof

The (usual) Donaldson–Futaki invariant of $({\mathcal {X}}, {\mathcal {L}}_c)$ is computed in [39]:

$$\begin{aligned} \frac{1}{4}{\mathcal {F}}(\alpha _c) = \frac{1}{c_0^2}\left[ c_0\int _0^c \alpha _1(x)(x-c){\text {d}}x -c \frac{c_0\alpha _0}{2} - c_1\int _{0}^c \alpha _0(x)(x-c){\text {d}}x\right] , \end{aligned}$$

(9)

where $c_0,c_1$ are the coefficients of $k^n$ and $k^{n-1}$ of $d_k$ as defined in the previous section (with respect to (X, L, Z)), see (2), and

$$\begin{aligned} \begin{aligned} \alpha _0(x)&= \frac{1}{(n-1)!} \int _Z (c_1(L) + x c_1({\mathcal {O}}(Z))^{n-1}; \\ \alpha _1(x)&= \frac{1}{2(n-2)!}\int _Z c_1(TX)\wedge (c_1(L)+ x c_1({\mathcal {O}}(Z))^{n-2}. \end{aligned} \end{aligned}$$

(10)

By Riemann–Roch, $\alpha _i(0)$ is the constant $\alpha _i$ appearing in the previous section (first line of (2)).

The main ingredient in order to carry out the above calculation in the modified case is the weight space decomposition for the induced ${\mathbb {C}}^{\times }$-action $\alpha _c$ on the space $H^0(X_0, {{\mathcal {L}}_c}_{|_{{\mathcal {X}}_0}})$ (see [39, §4.2]):

$$\begin{aligned} H^0(X_0, {{\mathcal {L}}_c}_{|_{{\mathcal {X}}_0}})= H^0(X, L^{k}\otimes {\mathcal {O}}(kc Z)) \oplus \bigoplus _{i=0}^{ck-1} t^{ck-i}H^0\bigg (Z, L^k_{|_Z}\otimes (\nu ^*_Z)^i\bigg ), \end{aligned}$$

(11)

where the weight of $\alpha _c$ on the first factor is 0 and $-(ck-i)$ on the components of the second direct sum. Note that the factor $t^{ck}H^0(Z, L^k_{|_Z})$ in the above decomposition corresponds to $H^0(Z_0, {\mathcal {L}}_{|_{Z_0}})$ in (1). It follows that

$$\begin{aligned} d_k^{Z_0} = d_k^{Z} = \alpha _0 k^{n-1} + O(k^{n-2}); \ \ w_k^{Z_0} = ckd_k^Z= c\alpha _0 k^n + O(k^{n-1}) \end{aligned}$$

(12)

while the coefficients $a_0$ and $a_1$ of the weights induced on $H^0(X_0, {{\mathcal {L}}_c}_{|_{{\mathcal {X}}_0}})$ are given by (see [39, Eqn. (4.6)]):

$$\begin{aligned} a_0= \int _{0}^{c}(x-c)\alpha _0(x){\text {d}}x; \ \ a_1 = -c\frac{\alpha _0}{2c_0} + \int _0^c(x-c)\alpha _1(x) {\text {d}}x. \end{aligned}$$

(13)

We therefore compute the modified Donaldson–Futaki invariant given by (4)

$$\begin{aligned} \frac{1}{4}{\widetilde{\mathcal {F}}}_{X_0, Z_0, {{\mathcal {L}}_c}_{|_{{\mathcal {X}}_0}}}(\alpha _c) = \frac{1}{c_0^2}\left[ c_0\int _0^c(x-c)\alpha _1(x){\text {d}}x -\bigg (c_1-\frac{\alpha _0}{2}\bigg )\int _{0}^c(x-c)\alpha _0(x){\text {d}}x\right] . \end{aligned}$$

(14)

Form the above formula, the proof of Lemma 2.7 then follows easily. $\square$

Using Lemma 2.7, it is shown in [7]:

Theorem 2.8

[7] If there exists a CSCK metric of Poincaré type on $X\setminus Z$ in the class $c_1(L),$ then the Székelyhidi numerical constraint holds, i.e. (8) is satisfied.

Remark 2.9

It is plausible to expect a similar numerical expression for the relative Székelyhidi numerical constraint $F_{\chi }''(0)>0$ but we failed to see a neat way to compute $F_{\chi }(c)$ in a sufficient generality, especially if the extremal ${\mathbb {C}}^{\times }$-action is not trivial on Z.

We shall next turn to the toric case as a model example for the above theory, and where specific computations are manageable. We shall show (see in particular Corollary 5.9) that there are examples of relatively K-stable triples (X, Z, L) satisfying $F_{\chi }''(0) > 0,$ which cannot be of Poincaré type. We note, however, that these examples do admit a complete extremal metric on $X \setminus Z$ – it just cannot satisfy the Poincaré type condition. We shall thus propose a strengthened version of Conjecture 2.5 for when a relatively K-stable triple (X, Z, L) should admit an extremal Kähler metric of Poincaré type in $c_1(L)$ in the toric setting (see Conjecture 4.14).

3 Extremal Poincaré Type Kähler Metrics on Toric Varieties

In this section we consider the case when (X, L) is a (smooth) polarized toric variety. We denote by ${\mathbb {T}}$ the real n-dimensional torus and by ${\mathbb {T}}^c \cong ({\mathbb {C}}^{\times })^n$ its complexification. The material follows [22].

3.1 Stability of Pairs and Toric Test Configurations

Switching from complex to symplectic point of view, Delzant’s theorem [20] describes (X, L) in terms of a compact convex polytope $\Delta \subset \mathfrak {t}^*$ (where $\mathfrak {t}= \mathrm{Lie}({\mathbb {T}})$ is the Lie algebra of ${\mathbb {T}}$) such that $\Delta = \{\mu : L_j(\mu )= \langle e_j, \mu \rangle + \lambda _j \ge 0, j=1, \ldots , d\}$ with $e_j$ belonging to the lattice $\Lambda \subset \mathfrak {t}$ of circle subgroups of ${\mathbb {T}}$. The fact that X is smooth corresponds to requiring that at each vertex of $v\in \Delta$ the adjacent normals span the same lattice $\Lambda \subset {{\mathfrak {t}}}$ (see [20, 38]), while the polarization L forces $\Delta$ to have its vertices in the dual lattice $\Lambda ^* \subset \mathfrak {t}^*$. Taking any generators of $\Lambda$ as a basis of $\mathfrak {t}$, one identifies $\Lambda$ with ${\mathbb {Z}}^n$ and we consider the Lebesgue measure ${\text {d}}\mu$ on $\mathfrak {t}^*\cong {\mathbb {R}}^n$; furthermore, one defines a measure ${\text {d}}\nu$ on $\partial \Delta ,$ such that on each facet $F_j \subset \Delta$ (i.e. a face of co-dimension one), we let

$$\begin{aligned} -{\text {d}}L_j \wedge {\text {d}}\nu _{F_j} =-e_j \wedge {\text {d}}\nu _{F_j} ={\text {d}}\mu . \end{aligned}$$

(15)

A central fact in this theory (see e.g. [14, Sect. 6.6]) is the weight decomposition of $H^0(X, L^k)$ with respect to the (linearized) torus action of ${\mathbb {T}}$. It is isomorphic to $\{\mu \in k\Delta \cap {\mathbb {Z}}^n \}$ with the weights identified with corresponding elements of ${\mathbb {Z}}^n$. On the other hand, for any smooth function f on $\mathfrak {t}^*$, we have [32, 48]:

$$\begin{aligned} \sum _{\mu \in k\Delta \cap {\mathbb {Z}}^n} f(\mu ) = k^n \int _{\Delta } f {\text {d}}\mu + \frac{k^{n-1}}{2}\int _{\partial \Delta } f {\text {d}}\nu + O(k^{n-2}), \ \text {as }k\rightarrow \infty . \end{aligned}$$

If $\alpha$ is the ${\mathbb {C}}^{\times }$-action with Killing potential corresponding to an affine linear function $f_{\alpha }$ on ${{\mathfrak {t}}}^*$ normalized by $f_{\alpha }(0)=0$, the above formula allows us to compute the coefficients $c_0,c_1, a_0, a_1$ in (2) and (3) as follows:

$$\begin{aligned} c_0= \mathrm{Vol}(\Delta ); \ \ c_1= \frac{\mathrm{Vol}(\partial \Delta )}{2}; \ \ a_0 = \int _{\Delta } f_{\alpha } {\text {d}}\mu ; \ \ a_1= \frac{1}{2}\int _{\partial \Delta } f_{\alpha } {\text {d}}\nu , \end{aligned}$$

(16)

so that the Donaldson–Futaki invariant ${\mathcal {F}}(\alpha )$ of $\alpha$ is

$$\begin{aligned} \frac{(2\pi )^n\mathrm{Vol}(\Delta )}{2}{\mathcal {F}}(\alpha ) = \int _{\partial \Delta } f_{\alpha } {\text {d}}\nu - \frac{{\mathbf{s}}}{2}\int _{\Delta } f_{\alpha } {\text {d}}\mu , \end{aligned}$$

(17)

where $\mathbf{s}= 2\mathrm{Vol}(\partial \Delta )/\mathrm{Vol}(\Delta )=2n \Big (\int _X c_1(TX) \wedge c_1(L)^{n-1}/\int _X c_1(L)^n\Big )$ is the averaged scalar curvature of any compatible Kähler metric.

Similarly, if $Z\subset X$ is a divisor corresponding to the pre-image of the union $F=F_{i_1}\cup \cdots \cup F_{i_k}$ of some facets $\Delta$ by the momentum map, the coefficients $\alpha _0, \alpha _1, \beta _0, \beta _1$ in (2) and (3) are given by

$$\begin{aligned} \alpha _0 = \mathrm{Vol}(F); \ \ \alpha _1= \frac{1}{2} \mathrm{Vol} (\partial F); \ \ \beta _0 = \int _{F} f_{\alpha } {\text {d}}\nu ; \ \ \beta _1 = \int _{\partial F} f_{\alpha } {\text {d}}\sigma _F, \end{aligned}$$

(18)

where ${\text {d}}\sigma _{\partial F}$ is the induced measure on the boundary of each $F_i \in F$ (viewed itself as a Delzant polytope in ${{\mathbb {R}}}^{n-1}$). The modified Futaki invariant ${\widetilde{{\mathcal {F}}}_{X,L,Z}}(\alpha )$ is then

$$\begin{aligned} \frac{(2\pi )^n\mathrm{Vol}(\Delta )}{2} {\widetilde{{\mathcal {F}}}_{X,L,Z}}(\alpha ) = \int _{\partial \Delta \setminus F} f_{\alpha } {\text {d}}\nu - \frac{\mathbf{s}_{(\Delta , F)}}{2}\int _{\Delta } f_{\alpha } {\text {d}}\mu , \end{aligned}$$

(19)

with

$$\begin{aligned} \begin{aligned} \mathbf{s}_{(\Delta , F)}&=2\frac{\mathrm{Vol}(\partial \Delta \setminus F)}{\mathrm{Vol}(\Delta )}\\&=2n \left( \int _X (c_1(TX)+c_1({\mathcal {O}}(-Z)) \wedge c_1(L)^{n-1}\Big /\int _X c_1(L)^n\right) . \end{aligned} \end{aligned}$$

(20)

The extremal ${\mathbb {C}}^{\times }$-action $\chi$ has Killing potential which is an affine linear function $s_{\Delta }$ determined by requiring

$$\begin{aligned} \int _{\partial \Delta } f {\text {d}}\nu - \frac{1}{2}\int _{\Delta } f s_{\Delta } {\text {d}}\mu =0 \end{aligned}$$

for any affine function f (see e.g. [45]). Then, as shown in [22, 45], the relative Donaldson–Futaki invariant is given by

$$\begin{aligned} \frac{(2\pi )^n\mathrm{Vol}(\Delta )}{2}{\mathcal {F}}^{\chi }(\alpha ) = \int _{\partial \Delta } f_{\alpha } {\text {d}}\nu - \frac{1}{2}\int _{\Delta } f_{\alpha } s_{\Delta }{\text {d}}\mu \end{aligned}$$

(21)

while its modified version is

$$\begin{aligned} \frac{(2\pi )^n\mathrm{Vol}(\Delta )}{2} {\widetilde{{\mathcal {F}}}^{\chi }_{X,L,Z}}(\alpha ) = \int _{\partial \Delta \setminus F} f_{\alpha } {\text {d}}\nu - \frac{1}{2}\int _{\Delta } f_{\alpha } s_{(\Delta , F)} {\text {d}}\mu \end{aligned}$$

(22)

where $s_{(\Delta , F)}$ again is the unique affine function such that

$$\begin{aligned} \int _{\partial \Delta \setminus F} f {\text {d}}\nu - \frac{1}{2}\int _{\Delta } f s_{(\Delta , F)} {\text {d}}\mu =0 \end{aligned}$$

for any affine linear function f.

Donaldson generalizes the above expression for ${\mathcal {F}}(\alpha )$ by considering convex piecewise affine linear functions $f_{\alpha }$ with integer coefficients. He associates to such an $f_{\alpha }$ a test configuration $({\mathcal {X}}, {\mathcal {L}})$, called toric, and identifies the Donaldson–Futaki invariant of the central fibre $(X_0,L_0)$ with (17). Székelyhidi [45, § 4.1] shows that (21) computes the relative Donaldson–Futaki invariant for such test configurations. These computations generalize easily in the case of a pair (X, Z) where the divisor Z corresponds to the pre-image of a number of facets of $\Delta$ by the moment map. In this case, the toric test configurations come equipped with a divisor ${\mathcal {Z}}$ which defines a flat family for Z; furthermore, (19) and (22) compute the modified Donaldson–Futaki and relative Donaldson–Futaki invariant of toric test configurations, respectively. We are thus led to the following:

Definition 3.1

Let (X, L) be a toric polarized variety and $Z\subset X$ a divisor corresponding to the pre-image under the moment map of the union $F=F_{i_1}\cup \cdots \cup F_{i_k}$ of some facets of the momentum polytope $\Delta$. We say that (X, Z, L) is relative K-stable with respect to toric degenerations if

$$\begin{aligned} {\mathcal {L}}_{(\Delta ,F)} (f) := \int _{\partial \Delta \setminus F} f {\text {d}}\nu - \frac{1}{2}\int _{\Delta } f s_{(\Delta , F)} {\text {d}}\mu >0 \end{aligned}$$

(23)

for any convex, piecewise affine linear function f which is not affine linear on $\Delta$. Recall that $s_{(\Delta , F)}$ is by definition the unique affine linear function such that (23) vanishes for any affine linear function f, and is called the extremal affine linear function of $(\Delta , F)$. If (23) is satisfied, we shall refer to $(\Delta , F)$ as a stable pair.

Lemma 3.2

Let (X, L) be a toric polarized variety and $Z\subset X$ a divisor corresponding to the pre-image under the moment map of one facet F of the momentum polytope $\Delta$. Then, the relative Székelyhidi numerical constraint is equivalent to

$$\begin{aligned} F_{\chi }''(0) = \frac{1}{2} \int _{F} (s_F - {s}_{(\Delta ,F)})\mathrm{{d}}\nu _F >0, \end{aligned}$$

(24)

where $s_{(\Delta , F)}$ is the extremal affine linear function of $(\Delta , F)$ and $s_F$ is the extremal affine linear function of the facet F (seen as a Delzant polytope of an $(n-1)$ dimensional toric variety).

Proof

In the toric case, Ross–Thomas [39, § 4.3] link their construction of degenerations to the normal cone to toric test configurations: the degeneration to the normal cone of $Z \subset X$ corresponding to a facet $F \subset \Delta$ defined by the zero set of an affine linear function L (with $L \ge 0$ on $\Delta$) is given by Donaldson’s construction with $f_c= \mathrm{max}(0, c-L)$. Therefore, the corresponding relative modified Donaldson–Futaki invariant (22) is

$$\begin{aligned} \begin{aligned} F_{\chi }(c) =&\int _{\partial \Delta \setminus F} f_{c} {\text {d}}\nu - \frac{1}{2}\int _{\Delta } f_{c} s_{(\Delta , F)} {\text {d}}\mu \\ =&\int _{\partial \Delta \setminus F} f_{c} {\text {d}}\nu - \frac{1}{2}\int _{\Delta } f_c \mathbf{s}_{(\Delta , F)} {\text {d}}\mu \\&-\frac{1}{2} \int _{\Delta } f_c ({s}_{(\Delta ,F)}-\mathbf{s}_{(\Delta , F)}) {\text {d}}\mu . \end{aligned} \end{aligned}$$

(25)

Note that the sum on the second line is $c_0/4$ times the function F(c) introduced in Lemma 2.7 (and computed via (14)) and, for any affine function $\xi$,

$$\begin{aligned} \frac{\partial ^2}{\partial c^2}\left( \int _{\Delta } f_c \xi {\text {d}}\mu \right) \Big |_{c=0} = \int _{F} \xi {\text {d}}\nu _F, \end{aligned}$$

where, we recall, ${\text {d}}\nu _F$ is determined via the defining equation $L=0$ for F by letting $-{\text {d}}L\wedge {\text {d}}\nu _F = {\text {d}}\mu$. Using (16), (18), and (20) one then gets

$$\begin{aligned} \begin{aligned} F_{\chi }''(0)&= 2\left( \alpha _1- \bigg (c_1-\frac{\alpha _0}{2}\bigg )\frac{\alpha _0}{c_0}\right) -\frac{1}{2} \int _{F} ({s}_{(\Delta ,F)}-\mathbf{s}_{(\Delta ,F)}) {\text {d}}\nu _F\\&= \mathrm{Vol}(\partial F)- \frac{1}{2}\int _{F} {s}_{(\Delta ,F)}{\text {d}}\nu _F\\&\quad + \frac{\mathrm{Vol}(\partial \Delta \setminus F)}{\mathrm{Vol}(\Delta )} \mathrm{Vol}(F)-\frac{\mathrm{Vol}(\partial \Delta \setminus F)}{\mathrm{Vol}(\Delta )}\mathrm{Vol}(F)\\&= \int _{\partial F} {\text {d}}{\sigma }_F - \frac{1}{2}\int _{F} {s}_{(\Delta ,F)}{\text {d}}\nu _F\\&= \frac{1}{2} \int _{F} (s_F - {s}_{(\Delta ,F)}){\text {d}}\nu _F, \end{aligned} \end{aligned}$$

(26)

where $s_F$ denotes the extremal affine function corresponding to Z. $\square$

Lemma 3.3

Let (X, Z) be as in Lemma 3.2. If $(\Delta , F)$ is stable, then

$$\begin{aligned} \int _{F} (s_{F} - {s}_{(\Delta , F)})\mathrm{{d}}\nu _{F} \ge 0. \end{aligned}$$

Proof

Using the expression $F_{\chi }(c)= \frac{c_0}{4}F(c) -\frac{1}{2}\int _{\Delta }f_c(s_{(\Delta ,F)}- \mathbf{s}_{(\Delta ,F)}) {\text {d}}\mu$ in (25) and Lemma 2.7, one easily computes that $F_{\chi }(0)= F'_{\chi }(0)=0$. It thus follows from (26) that for c sufficiently small, the piecewise affine linear convex function $f_c= \mathrm{max}(0, c-L)$ will destabilize $(\Delta , F)$, should $\int _{F} (s_{F} - {s}_{(\Delta , F)}){\text {d}}\nu _{F} = 2 F_{\chi }''(0) < 0$. $\square$

4 Labelled Polytopes and the Abreu–Guillemin Theory for Kähler Metrics of Poincaré Type

4.1 Donaldson Metrics on a Labelled Polytope

Following [22], the discussion in Sect. 3 can be put in a broader framework which makes sense for any labelled convex compact simple polytope $(\Delta , \mathbf{L})$ in $({\mathbb {R}}^n)^*$.

Definition 4.1

Let $\Delta \subset ({\mathbb {R}}^n)^*={{\mathfrak {t}}}^*$ be a compact convex polytope defined by a system of d linear inequalities

$$\begin{aligned} \Delta =\{ x\in ({\mathbb {R}}^n)^* : L_j(x)=\langle e_j, x\rangle + \lambda _j \ge 0, \ \ j=1, \ldots , d\} \end{aligned}$$

where $\mathbf{L}=\{L_1(x), \ldots , L_d(x)\}$ are affine linear functions on $({\mathbb {R}}^n)^*$ and ${\text {d}}L_j :=e_j \in {\mathbb {R}}^n$ are inward normals to $\Delta$. We suppose that $\Delta$ is simple in the sense that for each vertex v, there are precisely n affine linear functions $L_{v,1}, \ldots , L_{v,n}$ in $\mathbf{L}$ which vanish at v and the corresponding inward normals $\{e_{v,1}, \ldots e_{v,n}\}$ form a basis of ${\mathbb {R}}^n$. We refer to such date $(\Delta , \mathbf{L})$ as a labelled (simple, compact, convex) polytope. Notice that, by Delzant’s theorem [20], $(\Delta , \mathbf L)$ is the momentum image of a compact smooth toric variety if the labelling $\mathbf{L}$ satisfies the integrality condition that at each vertex v, $\mathrm{span}_{{\mathbb {Z}}}\{u_{v,1}, \ldots , u_{v,n}\}$ is a fixed lattice $\Lambda \subset {\mathbb {R}}^n$. We shall refer to such labelled polytopes $(\Delta , \mathbf{L})$ as Delzant polytopes.

In the case when $(\Delta , \mathbf{L})$ is Delzant, the works [1, 31] give an effective way to parametrize ${\mathbb {T}}$-invariant, $\omega$-compatible Kähler metrics g on the toric symplectic manifold $(X, \omega )$ classified by $(\Delta , \mathbf{L})$ in terms of strictly convex smooth functions u(x) defined on the interior $\Delta ^0$ of $\Delta \subset ({\mathbb {R}}^n)^*$ and satisfying certain boundary conditions on $\partial \Delta$. Specifically, the Kähler metric g is written on $X^0=\mu ^{-1}(\Delta ^0)$ as

$$\begin{aligned} g =\sum _{i,j=1}^n \bigg ( u_{,ij} {\text {d}}x_i \otimes {\text {d}}x_j + u^{,ij} {\text {d}}t_i \otimes {\text {d}}t_j\bigg ) \end{aligned}$$

(27)

where $(x_1, \ldots , x_n)$ are the Euclidean coordinates on $({\mathbb {R}}^n)^*$, $(u_{,ij}) = \mathrm{Hess}(u)$ (and we tacitly identify smooth functions and tensors on $\Delta ^0$ with their pull-backs via $\mu$ on $X^0$) and $(t_1, \ldots , t_n)$ are angular ($2\pi$-periodic) coordinates obtained by fixing a point $p_0\in X^0$ and identifying $X^0 \cong ({\mathbb {C}}^{\times })^n$ with the principal orbit of $p_0$ under the complexified action ${\mathbb {T}}^c$ (with respect to the complex structure J determined by g and $\omega$). In this formalism, the symplectic form is

$$\begin{aligned} \omega = \sum _{i=1}^n {\text {d}}x_i \wedge {\text {d}}t_i. \end{aligned}$$

A central fact in this theory (see [2, 23]) is that (27) extends to a smooth Riemannian metric on X if and only if u satisfies the following Guillemin boundary conditions:

Definition 4.2

Let $(\Delta , \mathbf{L})$ be a labelled convex compact simple polytope in $({\mathbb {R}}^n)^*={{\mathfrak {t}}}^*$. We say that a strictly convex smooth function u on $\Delta ^0$ satisfies the Guillemin boundary conditions if

• $u - \frac{1}{2}\sum _{k=1}^{d} L_k \log L_k$ is smooth on $\Delta$, and

• the restriction of u to the relative interior $F^0$ of any face $F\subset \Delta$ is smooth and strictly convex.

We denote by ${\mathcal {S}}(\Delta , \mathbf{L})$ the space of such u.

An example of a function in ${\mathcal {S}}(\Delta , \mathbf{L})$ is (see [31])

$$\begin{aligned} u_0:= \frac{1}{2} \sum _{k=1}^d L_k \log L_k, \end{aligned}$$

(28)

which, in the Delzant case, characterizes the induced Kähler metric on X via the Kähler reduction of the flat metric on ${\mathbb {C}}^d$.

The space ${\mathcal {S}}(\Delta , \mathbf{L})$ can be equivalently characterized in terms of first-order boundary conditions:

Proposition 4.3

 [4] The space ${\mathcal {S}}(\Delta , \mathbf{L})$ consists of all smooth functions u on $\Delta ^0$ such that $\mathbf{H}^u:= (\mathrm{Hess}(u))^{-1}$ satisfies

• [smoothness] $\mathbf{H}^u$ extends smoothly on $\Delta$ as an $S^2({{\mathfrak {t}}}^*)$-valued function;

• [boundary conditions] For any facet $F_j \subset {\partial \Delta }$ with normal $e_j={\text {d}}L_j$, and $x \in F_j$

  $$\begin{aligned} \mathbf{H}^u_{x}(e_j, \cdot )=0; \ \ ({\text {d}}{} \mathbf{H}^u)_{x} (e_j,e_j) =2e_j; \end{aligned}$$

  (29)

• [positivity] $\mathbf{H}^u$ is positive definite on $\Delta ^0$, as well as on the relative interior $\Sigma ^0$ of any face $\Sigma \subset \Delta$, viewed there as a smooth function with values in $S^2(\mathfrak {t}/\mathfrak {t}_{\Sigma })^*$ where $\mathfrak {t}_{\Sigma }$ denotes the subspace spanned by normals to facets containing $\Sigma$.

The extremality of the Kähler metric (27) with $u\in {\mathcal {S}}(\Delta , \mathbf{L})$ reduces to solving the Abreu equation [1]

$$\begin{aligned} -\sum _{i,j=1}^n \frac{\partial ^2 \mathbf{H}^u_{ij}}{\partial x_i \partial x_j} = s_{(\Delta , \mathbf{L})}, \end{aligned}$$

(30)

for an affine linear function $s_{(\Delta , \mathbf{L})}$ determined from the labelled polytope $(\Delta , \mathbf{L})$ by the requirement that

$$\begin{aligned} {\mathcal {L}}_{(\Delta , \mathbf{L}) }(f) := 2\int _{\partial \Delta } f {\text {d}}\nu _\mathbf{L} - \int _{\Delta } s_{(\Delta , \mathbf{L})} f {\text {d}}\mu =0 \end{aligned}$$

for any affine linear function f, where ${\text {d}}\mu$ is a (fixed) Lebesgue measure on ${{\mathfrak {t}}}^* =({\mathbb {R}}^n)^*$ and ${\text {d}}\nu _\mathbf{L}$ is obtained from ${\text {d}}\mu$ and $\mathbf{L}$ via (15). In this setting, we recall the following:

Definition 4.4

A labelled compact convex simple polytope $(\Delta , \mathbf{L})$ in a vector space ${{\mathfrak {t}}}^*$ is called stable (or K-stable) if

$$\begin{aligned} {\mathcal {L}}_{(\Delta , \mathbf{L}) }(f) \ge 0, \end{aligned}$$

for any convex, piecewise affine linear function f, and the equality is achieved only when f is affine linear.

Using the integration by parts formula established in [22, Lemma 3.3.5], Zhou and Zhu have shown in [49] that the stability of $(\Delta , \mathbf{L})$ is a necessary condition for a solution $u\in {\mathcal {S}}(\Delta , \mathbf{L})$ of (30) to exist:

Proposition 4.5

[49] Suppose there exists a function $u\in {\mathcal {S}}(\Delta , \mathbf{L})$ which solves the Abreu equation (30). Then $(\Delta , \mathbf{L})$ is stable.

Furthermore, as observed in [22, 28], any solution $u\in {\mathcal {S}}(\Delta , \mathbf{L})$ of (30) must be, up to the addition of affine linear functions, the unique critical point ($=$ the minimum) of the convex relative Mabuchi functional

$$\begin{aligned} {\mathcal {M}}_{(\Delta , \mathbf{L})} (u) := {\mathcal {L}}_{(\Delta , \mathbf{L})}(u) - \int _{\Delta } \log \det (\mathbf{H}^u) {\text {d}}\mu , \end{aligned}$$

(31)

which is shown in [22] to take values in $(0, \infty ]$.

It is observed in [22, p. 344] that most of the above theory extends to the case when one takes $F=F_1\cup \cdots \cup F_k$ to be the union of facets of $(\Delta , \mathbf{L})$, and one modifies the induced measure ${\text {d}}\nu _\mathbf{L}$ to be zero on F. By (15), for each facet $F_i \subset F$, the modified measure can be thought of as the limit $\lim _{t\rightarrow \infty } {\text {d}}\nu _{tL_i}$, i.e. the measure obtained as in (15) when sending the corresponding label $L_i$ to infinity. There is a subtle point here, however. It is not immediately clear how to extend the Guillemin boundary conditions of Definition 4.2 over such limits. On the other hand, as observed in [5], the equivalent first-order boundary conditions given by Proposition 4.3 extend naturally:

Definition 4.6

Let $(\Delta , \mathbf{L})$ be a labelled convex compact simple polytope in ${{\mathfrak {t}}}^*$ and $F=F_1\cup \cdots \cup F_k$ the union of some of its facets. We denote by ${\mathcal {S}}(\Delta ,\mathbf{L},F)$ the functional space of $u\in {\mathcal {C}}^{\infty }(\Delta ^0)$ verifying the first-order boundary conditions

• [smoothness] $\mathbf{H}^u$ extends smoothly on $\Delta$;

• [boundary conditions] for any facet $F_i \subset F$ and any point $x \in F_i$,

  $$\begin{aligned} \mathbf{H}^u_{x}(e_i, e)=0; \ \ \big ({\text {d}}{} \mathbf{H}^u(e_i,e)\big )_x =0, \end{aligned}$$

  (32)

  where $e_i={\text {d}}L_i$ is the inward normal to $F_i$ defined by $\mathbf{L}$ and $e\in {{\mathfrak {t}}}$, and, for any facet $F_r$ which is not in F, and $x \in F_r$,

  $$\begin{aligned} \mathbf{H}^u_{x}(e_r, e)=0; \ \ \big ({\text {d}}{} \mathbf{H}^u(e_r,e_r)\big )_x =2e_r. \end{aligned}$$

  (33)

• [positivity] $\mathbf{H}^u$ is positive definite on $\Delta ^0$, as well as on the interior of any face $\Sigma \subset \Delta$, viewed there as a smooth function with values in $S^2(\mathfrak {t}/\mathfrak {t}_{\Sigma })^*$ where $\mathfrak {t}_{\Sigma }$ denotes the subspace spanned by normals to facets containing $\Sigma$.

Remark 4.7

Manifestly, the conditions (32) are independent of the choice of labels $L_i$ for the facets $F_i \subset F$, and are obtained from (33) by letting $e_i \rightarrow \infty$.

Remark 4.8

Symplectic potentials satisfying Definition 4.6 do not necessarily correspond to Poincaré type metrics. However, below we shall define subspaces ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ of ${\mathcal {S}}(\Delta , \mathbf{L}, F)$ depending on a positive parameter $\alpha$ and a real parameter $\beta$ which do induce metrics of Poincaré type on $X \setminus Z$.

We are thus interested to find solutions of (30) in ${\mathcal {S}}(\Delta , \mathbf{L}, F)$, where, by the integration by parts argument of [22, Lemma 3.3.5] (see also (36) for a precise statement), the right-hand side must be the unique affine linear function $s_{(\Delta , \mathbf{L}, F)}$, called extremal affine function, satisfying

$$\begin{aligned} {\mathcal {L}}_{(\Delta ,\mathbf{L}, F)}(f)=\int _{\partial \Delta \setminus F} f {\text {d}}\nu _\mathbf{L} - \frac{1}{2}\int _{\Delta } f s_{(\Delta , \mathbf{L}, F)} {\text {d}}\mu =0 \end{aligned}$$

(34)

for any affine linear function f. We also have the following straightforward extension of Proposition 4.5 to the case $(\Delta , \mathbf{L}, F)$:

Proposition 4.9

Suppose there exists a function $u\in {\mathcal {S}}(\Delta , \mathbf{L}, F)$ which solves the Abreu equation

$$\begin{aligned} -\sum _{i,j=1}^n \frac{\partial ^2 \mathbf{H}^u_{ij}}{\partial x_i \partial x_j} = s_{(\Delta , \mathbf{L}, F)}, \end{aligned}$$

(35)

where $s_{(\Delta , \mathbf{L}, F)}$ is the extremal affine linear function of $(\Delta , \mathbf{L}, F)$. Then ${\mathcal {L}}_{(\Delta , \mathbf{L},F)}(f) \ge 0$ for any convex, piecewise affine linear function f, with equality iff f is affine linear.

Definition 4.10

A labelled convex compact simple polytope $(\Delta , \mathbf{L})$ in $({\mathbb {R}}^n)^*$ with a fixed subset F of facets satisfying the conclusion of Proposition 4.9 will be referred to as stable triple $(\Delta , \mathbf{L}, F)$. A Kähler metric on $g_D$ on $\Delta ^0 \times {\mathbb {T}}$ defined by a solution $u\in {\mathcal {S}}(\Delta , \mathbf{L}, F)$ of (35) (if it exists) will be called a Donaldson metric on $(\Delta , \mathbf{L}, F)$.

The geometric interest of studying Donaldson metrics as above comes from the following:

Conjecture 4.11

(Donaldson [22]) Let $(\Delta , \mathbf{L})$ be the momentum polytope of a smooth compact toric Kähler manifold $(X, \omega _0)$ and Z the divisor in X corresponding to the momentum pre-image of the union F of facets of $\Delta$. If $(\Delta , \mathbf{L}, F)$ is stable, then there exists a complete extremal Kähler metric $g_D$ defined on $X\setminus Z$.

Remark 4.12

Notice that when $(\Delta , \mathbf{L})$ is a Delzant polytope corresponding to a smooth compact toric manifold $(X, \omega _0)$, the label $\mathbf{L}$ is uniquely determined from the Delzant condition. Thus, in order to simplify the notation in this case, and when there is no possible confusion, we shall skip the label $\mathbf{L}$. This is the convention we have taken in the previous Sect. 3.

We now use the results from [9, 10] in order to establish the following:

Theorem 4.13

Let $(X, \omega _0)$ be a smooth compact n-dimensional complex toric Kähler manifold, and $Z\subset X$ a smooth divisor corresponding to the pre-image under the moment map of a single facet F of the momentum polytope $\Delta$. If $X\setminus Z$ admits a ${\mathbb {T}}$-invariant extremal Kähler metric of Poincaré type in $[\omega _0]$, then $(\Delta , F)$ is stable and the relative Székelyhidi numerical constraint (24) holds. Furthermore, the Delzant polytope F is stable and

$$\begin{aligned} s_F - ({s}_{(\Delta ,F)})_{|_F} = {\text {const}} >0. \end{aligned}$$

Proof

The main point is to show that a ${\mathbb {T}}$-invariant extremal Kähler metric $(g, \omega )$ of Poincaré type on $X\setminus Z$ gives rise to a Donaldson metric in a slightly weaker sense, namely it corresponds to $\mathbf{H}^u \in C^{\infty }(\Delta ^0, S^2({{\mathfrak {t}}}^*))$ which extends smoothly on $\Delta \setminus F$ and $C^0$ on $\Delta$ and, moreover, the conditions (32) and (33) hold where the first-order condition at F is taken in the sense of limit, i.e.

$$\begin{aligned} \lim _{x\rightarrow F, x \in \Delta \setminus F} \big ({\text {d}}{} \mathbf{H}^u(e_F, \cdot )\big )_x =0, \end{aligned}$$

for $e_F\in {{\mathfrak {t}}}$ the inward normal to F. This will be enough in order to establish the integration by parts formula (compare with [22, Lemma 3.3.5]):

$$\begin{aligned} \int _{\Delta } \left( \sum _{i,j=1}^n H^u_{ij,ij}\right) \varphi {\text {d}}\mu = \int _{\Delta }\left( \sum _{i,j=1}^n H^u_{ij}\varphi _{,ij}\right) {\text {d}}\mu - 2 \int _{\partial \Delta \setminus F} \varphi {\text {d}}\nu _\mathbf{L} \end{aligned}$$

(36)

for any smooth function $\varphi$ on $\Delta$. The latter in turn implies that

• (a) $\mathrm{Scal}_g = s_{(\Delta , F)}$ and

• (b) $(\Delta , F)$ is stable (compare with Proposition 4.9).

With the conclusions (a) and (b) in place, the result follows easily from [9, 10]. Indeed, (a) and Lemma 3.2 together with [10, Thm. 4 and Prop. 2.1] show that (24) is a necessary condition for the existence of an extremal Kähler metric of Poincaré type on $X\setminus Z$. Furthermore, by [9, Thm. 4], Z must admit an extremal Kähler metric $\check{g}$ in the Kähler class $[\omega _0]_{|_Z}$, so that F must be a stable Delzant polytope by Proposition 4.5. It is also shown in [9, p.44] that the extremal vector fields $J\mathrm{grad}_{g} \mathrm{Scal}_g$ and $J\mathrm{grad}_{\check{g}} \mathrm{Scal}_{\check{g}}$ agree on Z, which in our case translates to say that $s_F-(s_{(\Delta , F)})_{|_F} =const$. The constant is positive because of (24).

We thus focus for the remainder of the proof to show that an extremal ${\mathbb {T}}$-invariant Poincaré type metric $(g, \omega )$ on $X\setminus Z$ is (weakly) Donaldson. To this end, we fix a ${\mathbb {T}}$-invariant Kähler metric $\omega _0 \in c_1(L)$ on X and denote by $(\Delta , \mathbf{L})$ the corresponding Delzant polytope. We shall write, for any basis $\{e_1, \ldots , e_n\}$ of ${{\mathfrak {t}}}$, $x^0=(x_1^0, \ldots , x^0_n)$ the corresponding momenta, viewed as functions from X to ${{\mathfrak {t}}}^*$ defined by $\imath _{K_j}\omega _0 = - {\text {d}}x^0_j$ where $K_j$ is the fundamental vector field of X corresponding to $e_j \in {{\mathfrak {t}}}$; thus $\Delta = \mathrm{Im}(x^0)$ and $Z= (x^0)^{-1}(F)$ for a facet $F\subset \Delta$. Let $v\in \Delta$ be a vertex of $\Delta$ and F, and $\{e_1, \ldots , e_n\}$ the basis of ${{\mathfrak {t}}}$ formed by the inward normals to the facets containing v, with $e_F=e_1$. By Delzant theory (see [20, 38]) there exists a $({\mathbb {C}}^{\times })^n$ equivariant chart ${\mathbb {C}}^n_v$ of X (with respect to the complexified $({\mathbb {C}}^{\times })^n$-action of ${\mathbb {T}}$ on X and the standard $({\mathbb {C}}^{\times })^n$-action on ${\mathbb {C}}_v^n ={\mathbb {C}}^n$) in which F is given by $z_1=0$. Furthermore, in this chart, $|z_j|^2= x_j^0 e^{\phi _j(z)}$ for smooth functions $\phi _j$ on X (see e.g. [22]) whereas the holomorphic vector fields $\frac{1}{2}(K_j - \sqrt{-1} JK_j)$ become $\sqrt{-1} z_j \frac{\partial }{\partial z_j}$.

According to Definition 1.1, we can write $\omega = \omega _0 + dd^c \varphi$ for a smooth ${\mathbb {T}}$-invariant function $\varphi$ on $X\setminus Z$, such that ${\text {d}}\varphi$ is bounded at any order with respect to the model metric

$$\begin{aligned} \omega _\mathrm{mod} =\sqrt{-1} \left( \frac{1}{|z_1|^2(\log |z_1|)^2} {\text {d}}z_1 \wedge {\text {d}}\bar{z}_1 + \sum _{j=2}^n {\text {d}}z_j \wedge {\text {d}}\bar{z}_j \right) \end{aligned}$$

defined on the chart ${\mathbb {C}}_v^n$: in particular

$$\begin{aligned} {\text {d}}\varphi (JK_j)= O(|z_j|), j=2, \ldots , n, \ \ \ {\text {d}}\varphi (JK_1) = O\left( \frac{1}{|\log (|z_1|^2)|}\right) . \end{aligned}$$

(37)

Writing

$$\begin{aligned} x_j = x_j^0 + {\text {d}}\varphi (JK_j), \ \ j=1, \ldots n, \end{aligned}$$

(38)

for the momenta of $(g, \omega )$, we see that the map $x^0 \rightarrow x$ sends $\Delta \setminus F$ to itself, preserving the faces. Furthermore, $x_1 : X\setminus Z \rightarrow \Delta \setminus F$ extends continuously as zero over Z.

We now let $\mathbf{H}_x(e_i,e_j)=(g_p(K_i, K_j))$ be the smooth $S^2({{\mathfrak {t}}}^*)$-valued function, defined on $\Delta ^0$ by using the extremal Kähler metric g and the momentum map x (with $x=x(p)$ for $p\in X\setminus Z$). Clearly, $\mathbf{H}$ extends smoothly over $\Delta \setminus F$. The proof of Proposition 4.3 (given in [4]) uses local arguments around a point on a facet $F_r \subset \overline{\Delta \setminus F}$, and thus shows that $\mathbf{H}$ satisfies the boundary conditions (33) on each $F_r$. We now focus on F. We use the chart ${\mathbb {C}}^n_v$ as above, and denote by $\pi _1 : {\mathbb {C}}_v^n \rightarrow {\mathbb {C}}^{n-1}$ the projection $\pi _1(z_1, z_2, \ldots , z_n) = (z_2, \ldots , z_n)$. Then, [9, Thm. 4] tells that as $z_1 \rightarrow 0$, $\omega$ is written as

$$\begin{aligned} \omega = a_1\bigg (\frac{\sqrt{-1}({\text {d}}z_1 \wedge {\text {d}}\bar{z}_1)}{|z_1|^2 \big (\log (|z_1|^2)^2}\bigg ) + \pi _1 ^* \omega _1 + O(|\log |z_1||^{-\delta }), \end{aligned}$$

(39)

where $a_1$ and $\delta$ are positive reals, $\omega _1$ is an extremal Kähler metric on $Z\cap {\mathbb {C}}_v^n=\{(0,z_2, \ldots , z_n)\}$, and $O(|\log |z_1||^{-\delta })$ is understood at any order with respect to the Kähler metric

$$\begin{aligned} a_1\bigg (\frac{\sqrt{-1}({\text {d}}z_1 \wedge {\text {d}}\bar{z}_1)}{|z_1|^2 \big (\log (|z_1|^2)^2}\bigg ) + \pi _1^* \omega _1. \end{aligned}$$

(40)

We compute from (39), with respect to the vector fields $\frac{1}{2}(K_j - \sqrt{-1} JK_j)=\sqrt{-1} z_j \frac{\partial }{\partial z_j}$,

$$\begin{aligned} \begin{aligned} \mathbf{H}(e_1,e_1)&= \omega (K_1, JK_1)=\frac{2a_1}{\big (\log (|z_1|^2)\big )^2} + O\big (|\log |z_1||^{-\delta -2}\big ), \\ \mathbf{H}(e_1, e_j)&= \omega (K_1, JK_j)= O\big (|z_j||\log |z_1||^{-\delta -1}\big ), \\ \mathbf{H}(e_i, e_j)&=\omega (K_i, JK_j)= \check{\mathbf{H}}^1(e_i,e_j) + O\big (|z_iz_j||\log |z_1||^{-\delta }\big ),\quad i,j \ge 2, \end{aligned} \end{aligned}$$

(41)

where $\check{\mathbf{H}}^1(e_i, e_j)= \pi ^*\omega _1(K_i, JK_j)$ is the $z_1$ independent smooth function computed from $\omega _1$ with respect to the induced vector fields on Z. It follows that $\mathbf{H}$ extends continuously on F, verifying $\mathbf{H}_x(e_1, \cdot )=0$ on F.

Taking interior product with $K_1$ and $K_i$, $i\ge 2$, in (39) we obtain

$$\begin{aligned} \begin{aligned} {\text {d}}x_1&= a_1 \left( \frac{{\text {d}}|z_1|^2}{|z_1|^2(\log |z_1|^2)^2}\right) + \sum _{j=1}^n f_j(z) {\text {d}}|z_j|^2, \\ {\text {d}}x_i&= 2\sum _{j=2}^n \check{\mathbf{H}}^1(e_i,e_j) {\text {d}}|z_j| + \sum _{j=1}^n f_{ij}(z) {\text {d}}|z_j|^2 = \pi _1^*{\text {d}}x_i^1 + \sum _{j=1}^n f_{ij}(z) {\text {d}}|z_j|^2, \end{aligned} \end{aligned}$$

where $f_1(z)= O\Big (\frac{1}{|z_1|^2(\log |z_1|^2)^{2+\delta }}\Big ), \ f_j(z)= O(|\log |z_1||^{-1-\delta }), \ j=2, \ldots , n$, and $f_{i1}(z)= O\Big (\frac{|z_i|^2}{|z_1|(\log |z_1|^2)^{1+\delta }}\Big ), \ f_{ij}(z)= O(|z_i|^2|\log |z_1||^{-\delta }), \ i,j=2, \ldots , n$; here ${\text {d}}x_i^1=-\iota _{K_i}\omega _1$ for $i=2, \ldots , n$, and ${\text {d}}x_i^1 = \sum _{j=2}^n A_{ij} {\text {d}}|z_j|^2$ for some invertible matrix $(A_{ij})$, locally uniformly bounded together with its inverse $(A^{ij})$ on $Z\cap {\mathbb {C}}^n_v$. Putting $\sigma _1 = \frac{{\text {d}}|z_1|^2}{|z_1|^2|\log (|z_1|^2)|}$ and $\sigma _i = d|z_i|^2$, the relations above can be recapped as

$$\begin{aligned} \begin{pmatrix} \big |\log |z_1|\big |{\text {d}}x_1 \\ {\text {d}}x_2 \\ \vdots \\ {\text {d}}x_n \end{pmatrix} = \Bigg [ \begin{pmatrix} a_1 &{}\quad 0 \\ 0 &{}\quad (A_{ij}) \end{pmatrix} + \varepsilon \Bigg ] \begin{pmatrix} \sigma _1 \\ \vdots \\ \sigma _n \end{pmatrix}, \end{aligned}$$

with $\varepsilon = O(|\log |z_1||^{-\delta })$; solving this system provides

$$\begin{aligned} \sigma _1 = \frac{|\log |z_1||}{a_1}{\text {d}}x_1 + \sum _{j=1}^n \eta _{1j} {\text {d}}x_j, \ \ \ \sigma _i = \eta _{i1} {\text {d}}x_1 + \sum _{j=2}^n (A^{ij}+\eta _{ij}) {\text {d}}x_j, \ i=2,\ldots ,n, \end{aligned}$$

with $\eta _{i1}= O(|\log |z_1||^{1-\delta })$, $i=1,\ldots ,n$, and $\eta _{ij} = O(|\log |z_1||^{-\delta })$, $i=1,\ldots ,n, \ j=2,\ldots ,n$. Differentiating the first two lines of (41) with respect to $z_1\log |z_1|\frac{\partial }{\partial z_1}$, the $\frac{\partial }{\partial z_i}$’s ($i\ge 2$), and their conjugates¹ implies

$$\begin{aligned} \begin{aligned} {\text {d}}{} \mathbf{H}(e_1, e_1)&= - \frac{4}{\log |z_1|}\Big ((1+\epsilon _{11}){\text {d}}x_1 + \sum _{j=2}^n \epsilon _{1j} {\text {d}}x_j\Big ), \\ {\text {d}}{} \mathbf{H}(e_1, e_i)&= \epsilon _{i1}{\text {d}}x_1 + \sum _{j=2}^n \epsilon _{ij} {\text {d}}x_j, \ \ i=2,\ldots ,n, \end{aligned} \end{aligned}$$

with $\epsilon _{i1}=O(|\log |z_1||^{-\delta })$, $i=1,\ldots ,n$, and $\epsilon _{ij}=O(|\log |z_1||^{-\delta -1})$, $i=1,\ldots ,n, \ j=2,\ldots ,n$. Hence in particular,

$$\begin{aligned} \lim _{x\rightarrow F} ({\text {d}}{} \mathbf{H}(e_1, e_i))_x =0, \ \ i=1, \ldots , n, \end{aligned}$$

as claimed. $\square$

4.2 Conjectural Picture for the Existence of Extremal Toric Metrics of Poincaré Type

Theorem 4.13 and Conjecture 4.11 motivate us to propose the following general conjectural picture in the toric case.

Conjecture 4.14

A smooth compact toric Kähler manifold $(X,\omega _0)$ with momentum polytope $\Delta$ and a divisor $Z \subset X$ corresponding to the pre-image of the union $F=\cup _{i}F_{i}$ of some facets $F_i$ of $\Delta$ admits an extremal toric Kähler metric of Poincaré type in $[\omega _0]$ if and only if the following three conditions are satisfied:

• (i) $(\Delta , F)$ is stable in a suitable sense, and

• (ii) for any facet $F_{i} \subset F$, the pair $(F_i, F_{i} \cap (\cup _{j\ne i \in I} F_j))$ is stable in a suitable sense, and

• (iii) if $s_{(F_i, F_{i} \cap (\cup _{j\ne i \in I} F_j)}$ is the extremal affine function corresponding to $(F_i, F_{i} \cap (\cup _{j\ne i \in I} F_j))$, then

  $$\begin{aligned} s_{(F_i, F_{i} \cap (\cup _{j\ne i} F_j))} - {s}_{(\Delta ,F)} = c_i>0, \end{aligned}$$

  (42)

  where $c_i$ are real constants.

Remark 4.15

Theorem 4.13 readily generalizes to the case when Z is a smooth toric submanifold of $(X, \omega _0)$, i.e. Z is the pre-image under the moment map of the union F of disjoint facets of $\Delta$. Thus, in this case, we have established the necessity of the conditions (i),(ii),(iii) of Conjecture 4.14 with respect to the notions of stability introduced in Definitions 4.4 and 4.10. The situation is not so clear in general, when Z has simple normal crossings. In this case we make the following remarks:

• (1) In order to establish (i) we would need to show that any toric extremal Kähler metric of Poincaré type on $X\setminus Z$ belongs to the class ${\mathcal {S}}(\Delta , \mathbf{L}, F)$, at least in the weaker sense as in the proof of Theorem 4.13.

• (2) (ii) would follow from (i), noting that the extremal Poincaré type metric on $Z_i\setminus Z'_i$ where $Z_i$ is the component of Z corresponding to $F_i$ and $Z'_i$ is the divisor of $Z_i$ induced by Z (which exists by virtue of [9, Thm. 4]) must be toric. Indeed, this can be derived from [9] as follows:

  • using toric-equivariant coordinates $(z_1,\ldots , z_n)\in {\mathbb {C}}^n_v$ centred at a point in $Z_i$ fixed by the torus action, and such that $Z_i \cap {\mathbb {C}}^n_v = \{z_1 = 0\}$ and $Z \cap {\mathbb {C}}^n_v = \{z_1 \cdots z_s = 0\}$, the induced metric is a ${\mathcal {C}}^{\infty }_\mathrm{loc}$-limit of $\omega _v^{\epsilon _j} := \omega _{|_{\{z_1=\epsilon _j\}\setminus Z}}$ (the pull-back of $\omega$ to $\{z_1=\epsilon _j\}\setminus Z$ by inclusion), with $\epsilon _j \rightarrow 0$;

  • the metric $\omega$ and the hypersurfaces $\{z_1=\epsilon _j\}\setminus Z$ are invariant by the action of ${\mathbb {T}}/{\mathbb {T}}_{F_i}$; therefore, the $\omega _v^{\epsilon _j}$ are invariant under this action, and their ${\mathcal {C}}^{\infty }_\mathrm{loc}$-limit is thus toric.

• (3) (iii) would follow by [10, Thm. 4 and Prop. 2.1], once we know that the scalar curvature of the extremal Kähler Poincaré type metric coincides with $s_{(\Delta , F)}$. This in turn would be the case if we establish point (i) above.

Another interesting question is how (i) and (ii) interrelated.

4.3 A Class of Poincaré Type toric Kähler Metrics

To link Conjectures 4.11 and 4.14, one needs a criterion ensuring that a Donaldson metric is of Poincaré type. We address this question in this section.

We start by introducing a class of toric metrics in the form (27) on $(\Delta ^0 \times {\mathbb {T}})$ via a certain type of Guillemin boundary conditions for the corresponding symplectic potential u, depending on the data $(\Delta , \mathbf{L}, F)$, and compare with Definition 4.6 in the case $F= \varnothing$. For simplicity, we shall assume that $(\Delta , \mathbf{L})$ is Delzant and $F=F_{1}$ is a single facet defined by the label $L_F(x):=L_{1}(x)=0$.

Definition 4.16

Let $\alpha >0$ and $\beta \in {\mathbb {R}}$ be fixed real numbers. The class ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ of symplectic potentials u is defined as the space of smooth and strictly convex functions on $\Delta ^0$, satisfying the following boundary conditions:

• $u + (\alpha +\beta L_{F}) \log L_F - \frac{1}{2}\sum _{j=2}^d L_j \log L_j$ is smooth on $\Delta$;

• if $\mathbf{f} \subset F$ is a sub-face of F, then $u_\mathbf{f}: = u + (\alpha +\beta L_{F})\log L_{F}$ restricts to the relative interior of $\mathbf{f}$ as a smooth strictly convex function;

• if $\Sigma \not \subset F$ is regular face, then u restricts to the relative interior of $\Sigma$ as a smooth strictly convex function.

Remark 4.17

Just as cusps can be seen as limits of cone singularities with angle tending to 0, one can observe that potentials of ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ appear as limits of symplectic potentials associated to cone singularities (as in [36, §6.3]). One elementary example is

$$\begin{aligned} u = \frac{1}{2}(1-x)\log (1-x) + \Big (\frac{1}{2} x - \frac{1}{2}\Big )\log x \in {\mathcal {S}}_{\frac{1}{2}, -\frac{1}{2}}([0,1], \mathbf{L}, \{0\}), \end{aligned}$$

(43)

where $\mathbf{L}$ is the standard labelling of [0, 1]. This potential is the limit of the family $(u_t)_{t\ge 1}$ as $t\rightarrow \infty$, where $u_t = \frac{1}{2}(1-x)\log (1-x) + \frac{t}{2}x\log x + v_t$, with $v_t$ smooth on [0, 1] given by²

$$\begin{aligned} v_t(x) = \left\{ \begin{aligned}&0,&\quad \text {if }\,t=1,\\&\frac{1}{2}(1-t)\Big (x+\frac{1}{t-1}\Big )\log \Big (x+\frac{1}{t-1}\Big ) +\Big [\frac{1}{2}\log \big (\frac{t}{t-1}\big )+\frac{t-1}{2t}\Big ],&\quad \text {if }\,t>1. \end{aligned} \right. \end{aligned}$$

With these notations, $u_1$ is associated to the Fubini–Study metric on ${\mathbb {C}}P^1$, and $u_t$, $t\in (1,\infty )$, to a “tear-drop” metric on ${\mathbb {C}}P^1$ with edge singularity at 0, of angle $\frac{2\pi }{t}$.

Our first observation concerning the potentials of ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ is the following result, whose proof is given in Appendix A.

Theorem 4.18

Let $(X, \omega )$ be a smooth, compact symplectic toric manifold with momentum Delzant polytope $(\Delta , \mathbf{L})$ and F be a single facet of $\Delta$. Then, for any $u\in {\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$, the metric (27) defines on X a ${\mathbb {T}}$-invariant complex structure J such that the momentum pre-image of F is a smooth divisor Z of (X, J),  and (27) is a Kähler metric of Poincaré type on $X\setminus Z$.

Using arguments similar to those in [4] (see Appendix A for more details), one can relate the spaces ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ and ${\mathcal {S}}(\Delta , \mathbf{L}, F)$ as follows:

Proposition 4.19

The space ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ is equivalently defined as the space of smooth functions on $\Delta ^0$ such that $\mathbf{H}^u = (\mathrm{Hess}(u))^{-1}$ satisfies

• [smoothness] $\mathbf{H}^u$ extends smoothly on $\Delta$;

• [boundary conditions on F] for any $x \in F$ we have

  $$\begin{aligned} \begin{aligned} \mathbf{H}^u_{x}(e_F, e)&=0; \ \ ({\text {d}}{} \mathbf{H}^u)_{x} (e_F, e) =0, \\ (d^2\mathbf{H}^u)_{x} (e_F,e_F)&=\frac{2}{\alpha } e_F \otimes e_F, \ \ (d^3\mathbf{H}^u)_{x} (e_F,e_F) = -\frac{6\beta }{\alpha ^2} e_F^{\otimes 3}, \end{aligned} \end{aligned}$$

  where $e_F={\text {d}}L_F$ is the inward normal to F defined by $\mathbf{L}$, e is any vector in ${{\mathfrak {t}}}$, and for a smooth function f on ${{\mathfrak {t}}}$, $d^k f$ denotes the kth covariant derivative of f with respect to the flat affine structure on ${{\mathfrak {t}}}^*$, so that $(d^k f)_x \in S^k({{\mathfrak {t}}})$;

• [regular boundary conditions] for any facet $F_r$ with inward normal $e_r$ which is not in F, and $x \in F_r$,

  $$\begin{aligned} \mathbf{H}^u_{x}(e_r, e)=0; \ \ ({\text {d}}{} \mathbf{H}^u)_{x} (e_r,e_r) =2e_r; \end{aligned}$$

• [positivity] $\mathbf{H}^u$ is positive definite on $\Delta ^0$, as well as on the relative interior of any face $\Sigma \subset \Delta$, viewed there as a smooth function with values in $S^2(\mathfrak {t}/\mathfrak {t}_{\Sigma })^*$ where $\mathfrak {t}_{\Sigma }$ denotes the subspace spanned by normals to facets containing $\Sigma$.

In particular, ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F) \subset {\mathcal {S}}(\Delta , \mathbf{L}, F)$.

Our next result shows that the extremality assumption in fact determines uniquely the space $\mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$.

Proposition 4.20

Suppose $u\in {\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ is a solution of (35). Then the real numbers $\alpha , \beta$ are determined from the data $(\Delta , \mathbf{L}, F)$. Furthermore, the solution u is unique modulo the addition of an affine linear function.

Proof

The uniqueness part is standard as each ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ is a linearly convex space and, choosing a reference point $u' \in {\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$, we can consider the following modification of relative Mabuchi functional (31):

$$\begin{aligned} {\mathcal {M}}_{(\Delta , \mathbf{L},F)} (u) := {\mathcal {L}}_{(\Delta , \mathbf{L},F)}(u-u') - \int _{\Delta } \log \bigg (\frac{\det (\mathbf{H}^u)}{\det (\mathbf{H}^{u'})}\bigg ) {\text {d}}\mu . \end{aligned}$$

The point is that ${\mathcal {M}}_{(\Delta , \mathbf{L},F)} (u)$ is well defined with values in $(-\infty , \infty )$, as $u-u'$ is a smooth function over $\Delta$ and $\det (\mathbf{H}^u)/\det (\mathbf{H}^{u'})$ is smooth and positive on $\Delta$ (the latter fact follows from the arguments in Appendix A.2). An argument from [22, 28] shows that ${\mathcal {M}}_{(\Delta , \mathbf{L},F)} (u)$ is convex and its minima, which are unique up to the addition of affine linear function, are precisely the solutions of (35).

The fact that $s_{(\Delta , \mathbf{L}, F)} - s_{(F, \mathbf{L}_F)}=const$ follows from [9, p. 44] when it is shown that the extremal vector of Z equals the vector field induced on Z by the extremal vector field of $X\setminus Z$. In the toric case, this condition reads as $d((s_{(\Delta , \mathbf{L}, F)})_{|_F} - s_{(F, \mathbf{L}_F)})=0$.

It remains to determine $(\alpha , \beta )$ from $(\Delta , \mathbf{L}, F)$, which will occupy the remainder of the proof.

Step 1. Determining $\alpha$. Let us choose a basis $\{e_1, e_2, \ldots , e_n\}$ of ${{\mathfrak {t}}}$ (and $\{e_1^*, \ldots , e_{n}^*\}$ denote the dual basis of ${{\mathfrak {t}}}^*$), by fixing a vertex $v\in F$ of $\Delta$ and taking $e_j$ be the inward normals to the facets meeting v with $e_1 = e_F$ (and therefore $e_i^*, i=2, \ldots , n$ are tangent to F). We assume furthermore that v is at the origin (so that $F\subset \{x_1=0\}$) and we write $\mathbf{H}^{u}= (H_{ij})$ in the chosen basis, where $H_{ij}(x)$ are smooth functions on $\Delta$, see Proposition 4.19. As u is a solution of (35), we have

$$\begin{aligned} s_{(\Delta , \mathbf{L}, F)} = -\sum _{i,j=1}^n H_{ij,ij}. \end{aligned}$$

(44)

We denote by $\check{\mathbf{H}}^u \in S^2(({{\mathfrak {t}}}/{{\mathfrak {t}}}_F)^*)$ the induced smooth positive definite bilinear form on F. It is easily seen (by continuity) that $\check{\mathbf{H}}^u$ satisfies the boundary conditions of Proposition 4.3 with respect to the labelling $\mathbf{L}_{F}$ of F, see [4, Rem. 1]. It thus defines an almost-Kähler metric $\check{g}_u$ on F (which can be shown to be Kähler). With respect to our choice of basis of ${{\mathfrak {t}}}$, we can identify ${{\mathfrak {t}}}/{{\mathfrak {t}}}_F \cong {\mathbb {R}}^{n-1}= \mathrm{span}_{{\mathbb {R}}}\{e_2, \ldots , e_n\}$, so that we have $\check{\mathbf{H}}^u=(H_{ij})_F, i,j = 2, \ldots , n$. It thus follows that on F we have

$$\begin{aligned} \begin{aligned} s_{(\Delta , \mathbf{L}, F)}&= -\sum _{i,j=1}^n H_{ij,ij} \\&= -H_{11,11} - 2\sum _{j=2}^n H_{1j,1j} - \sum _{i,j=2}^n H_{ij,ij} \\&= -\frac{2}{\alpha } - \sum _{i,j=2}^n\check{\mathbf{H}}^u_{ij,ij} = -\frac{2}{\alpha } + \mathrm{Scal}(\check{g}_u), \end{aligned} \end{aligned}$$

(45)

where we have used the boundary conditions of Proposition 4.19 (or equivalently the form (118) in our compatible coordinates) in order to see that $H_{1j,1j}=0$ on F for $j>1$. It thus follows that $\check{g}_u$ is an extremal almost-Kähler metric on F and

$$\begin{aligned} s_{(\Delta , \mathbf{L}, F)} = s_{(F, \mathbf{L}_F)} - \frac{2}{\alpha } \end{aligned}$$

(46)

Integrating over F, we thus have

$$\begin{aligned} -\frac{2}{\alpha } = \int _F(s_{(\Delta , \mathbf{L}, F)} - s_{(F, \mathbf{L}_F)}){\text {d}}\nu _F/\mathrm{Vol}(F), \end{aligned}$$

(47)

which determines $\alpha$.

Step 2: Determining $\beta$. In order to determine $\beta$, notice that (see (15)) at each point $p\in F$, we have $e_1 \wedge {\text {d}}\nu _F = - {\text {d}}\mu$ where we recall that we have set $e_1={\text {d}}L_{F}=e_F$. Furthermore, with our choice of basis we have $(e_j \wedge {\text {d}}\nu _F) =0$ for $j=2, \ldots , n$. We thus have, using (44),

$$\begin{aligned} (d s_{(\Delta , \mathbf{L}, F)} \wedge {\text {d}}\nu _F) = \left( \sum _{i,j=1}^n H_{ij,ij1}(p)\right) {\text {d}}\mu = c {\text {d}}\mu \end{aligned}$$

for a real constant $c=c(\Delta , \mathbf{L}, F)$ determined from the polytope $(\Delta , \mathbf{L}, F)$. In other words,

$$\begin{aligned} \begin{aligned} c&= \left( \sum _{i,j=1}^n H_{ij,ij1}\right) _F \\&= \big (H_{11,111}\big )_F + 2\left( \sum _{j=1}^nH_{1j, 11j}\right) _F +\left( \sum _{i,j=2}^n H_{ij,1ij}\right) _F \\&= -\frac{6 \beta }{\alpha ^2} + 2\left( \sum _{j=2}^nH_{1j, 11j}\right) _F +\left( \sum _{i,j=2}^n H_{ij,1ij}\right) _F, \end{aligned} \end{aligned}$$

(48)

where in the last line we have used $(H_{11,111})_F = -\frac{6\beta }{\alpha ^2}$, see Proposition 4.19. We are going to integrate (48) over F, and to this end we are going to use the integration by parts formula

$$\begin{aligned} \int _F \sum _{j=2}^n V_{j, j}{\text {d}}\nu _F = - \sum _{\Sigma \subset \partial F} \int _{\Sigma } \langle V, e_{\Sigma }\rangle {\text {d}}\sigma _{\Sigma }, \end{aligned}$$

(49)

where F belongs to the hyperplane $x_1=0$ of ${{\mathfrak {t}}}^*$, a smooth function V on F is seen as a smooth function of the variables $(x_2, \ldots , x_n)$, the sum is taken over the facets $\Sigma$ of F with inward normal $e_{\Sigma } \in {{\mathfrak {t}}}/{{\mathfrak {t}}}_F\cong {\mathbb {R}}^{n-1}= \mathrm{span}_{{\mathbb {R}}}\{e_2, \ldots , e_n\},$ and the induced measures ${\text {d}}\sigma _{\Sigma }$ are constructed from the label polytope $(F, \mathbf{L}_F)$ via (15). Thus, integrating (48) and using (49) give

$$\begin{aligned} \begin{aligned} \Big (c+ \frac{6 \beta }{\alpha ^2}\Big )\mathrm{Vol}(F) =&-2\sum _{\Sigma \in \partial F} \int _{\Sigma }\big (H(e_1, e_{\Sigma })\big )_{,11} {\text {d}}\sigma _{\Sigma } \\&- \sum _{\Sigma \in \partial F} \int _{\Sigma } \sum _{i=2}^n \big (H(e_i, e_{\Sigma }))_{,1i} {\text {d}}\sigma _{\Sigma }. \end{aligned} \end{aligned}$$

(50)

We recall that in (50), $\alpha$ and c have been already defined in terms of $(\Delta , \mathbf{L}, F)$, so in order to define $\beta$ it will be enough to show that each of the two sums at the right-hand side of (50) can also be defined by $(\Delta , \mathbf{L}, F)$.

We first deal with the term $\int _{\Sigma }\big (H(e_1, e_{\Sigma })\big )_{,11} {\text {d}}\sigma _{\Sigma }$. Notice that if $\Sigma$ is a facet of F which meets the chosen vertex ($=$ the origin), i.e. if $\Sigma$ belongs to $\{x_1=0, x_j=0\}, j>1$, then $\big ((H_{1, e_{\Sigma }})_{,11}\big )_{| \Sigma } = (H_{1j,11})_{\{x_1=0, x_j=0\}}=0$ by the expansion (118) of $H_{ij}$ near $\Sigma$. For a general facet $\Sigma$ of F, we let $P\subset \partial \Delta$ be the unique other facet of $\Delta ,$ such that $\Sigma = F\cap P$ and denote by $e_{F}, e_{P}$ the corresponding inward normals. Thus, ${{\mathfrak {t}}}_{\Sigma } = \mathrm{span}_{{\mathbb {R}}}\{e_F, e_P\}$ is the annihilator of $T_p\Sigma \subset {{\mathfrak {t}}}^*$ (where p in a interior point for $\Sigma$), equipped with a natural basis $\{e_F, e_{P}\}$. For any two vectors $e', e'' \in {{\mathfrak {t}}}$, the function $\mathbf{H}^u(e', e'')$ is smooth on $\Delta$ and we denote by $\mathrm{Hess}\Big (\mathbf{H}^u(e', e'')\Big )_p$ its Hessian at $p\in \Delta$, computed with respect to the affine structure of ${{\mathfrak {t}}}^*$. Thus, $\mathrm{Hess}\Big (\mathbf{H}^u(e', e'')\Big )_p \in S^2({{\mathfrak {t}}})$ and with respect to the chosen basis we have

$$\begin{aligned} H_{ij,kr} (p) = \Big \langle e_k^*\otimes e_r^*, \mathrm{Hess}\big (\mathbf{H}^u(e_i, e_j)\big )_p\Big \rangle . \end{aligned}$$

Using the boundary conditions of Proposition 4.19, we notice that for any $e\in {{\mathfrak {t}}}$, ${\text {d}}{} \mathbf{H}^u(e_F, e)=0$ along F, and hence along $\Sigma$. It thus follows that for each interior point $p\in \Sigma$, the symmetric bilinear form $\mathrm{Hess}\Big (\mathbf{H}^u(e_F, e)\Big )_p$ degenerates on $T_p\Sigma$, or in other words, for each $p\in \Sigma$, $\mathrm{Hess}\Big (\mathbf{H}^u(e_F, e)\Big )_p$ has values in ${{\mathfrak {t}}}_{\Sigma }\otimes {{\mathfrak {t}}}_{\Sigma }$. Using the basis $\{e_{F}, e_{P}\}$ of ${{\mathfrak {t}}}_{\Sigma }$, we have a natural decomposition at each point of $\Sigma$:

$$\begin{aligned} \begin{aligned} \mathrm{Hess}\Big (\mathbf{H}^u(e_F, e)\Big )&= \big (\mathbf{H}^u(e_F, e)\big )_{, e_Fe_F} e_F\otimes e_F\\&\quad + \big (\mathbf{H}^u(e_F, e)\big )_{, e_Fe_P} (e_F\otimes e_P + e_P\otimes e_F) \\&\quad + \big (\mathbf{H}^u(e_F, e)\big )_{, e_Pe_P} e_P\otimes e_P \end{aligned} \end{aligned}$$

By choosing a vertex of $\Delta$ which belongs to $\Sigma$ and a basis as above, and letting $e=\sum _{i=1}^n a_i e_i$ the coefficients above become

$$\begin{aligned} \begin{aligned} \big (\mathbf{H}^u(e_F, e)\big )_{, e_Fe_F}&= \sum _{i=1}^n a_i H_{1i, 11}, \\ \big (\mathbf{H}^u(e_F, e)\big )_{, e_Fe_P}&= \sum _{i=1}^n a_iH_{1i,1j}, \\ \big (\mathbf{H}^u(e_F, e)\big )_{, e_Pe_P}&= \sum _{i=1}^n a_iH_{1i,jj}, \end{aligned} \end{aligned}$$

where the index $j>1$ is determined by $\Sigma \subset \{x_1=0, x_j=0\}$. Using the boundary conditions of Proposition 4.19, which are equivalently expressed by the form (118) of $\mathbf{H}^u$ near $\Sigma$, we obtain that on $\Sigma$

$$\begin{aligned} \begin{aligned} \mathrm{Hess}\big (\mathbf{H}^u(e_F, e_F)\big )&= \frac{2}{\alpha } e_F \otimes e_F= \frac{2}{\alpha } e_1 \otimes e_1 \\ \mathrm{Hess}\big (\mathbf{H}^u(e_F, e_P)\big )&= 0. \end{aligned} \end{aligned}$$

(51)

Similarly, using the constancy of $d \mathbf{H}^u(e_P, e_P)$ along $\Sigma \subset P$ and (118) with respect to a suitable basis, we also conclude that

$$\begin{aligned} \mathrm{Hess}\big (\mathbf{H}^u(e_P, e_P)\big ) = \big (\mathbf{H}^u(e_P, e_P)\big )_{, e_Pe_P} e_P\otimes e_P. \end{aligned}$$

(52)

Turning back to the term $\int _{\Sigma }\big (H(e_1, e_{\Sigma })\big )_{,11} {\text {d}}\sigma _{\Sigma }$, we notice that the definition of the normal $e_{\Sigma }$ in the expression $H_{1e_{\Sigma }, 11}$ uses the initial basis $\{e_1, \ldots , e_n\}$. Indeed, decomposing

$$\begin{aligned} e_P= \sum _{i=1}^n c_i e_i, \end{aligned}$$

we have that $e_{\Sigma }= e_{P} - c_1 e_1=e_{P}-c_1e_{F}$ where $c_1= c_1(\Sigma )=-e_P\wedge {\text {d}}\nu _{F}/{\text {d}}\mu$ is a constant determined by the polytope $(\Delta , \mathbf{L}, F)$ and the facet $\Sigma$ of F. It thus follows from (51) that on $\Sigma$

$$\begin{aligned} \begin{aligned} H(e_1,e_{\Sigma })_{, 11}&= \Big \langle e_1^*\otimes e_1^*, \big (\mathbf{H}^u(e_F, e_P)- c_1 \mathbf{H}^u(e_F, e_F)\big )\Big \rangle \\&=- \frac{2c_1}{\alpha }, \end{aligned} \end{aligned}$$

and therefore

$$\begin{aligned} \int _{\Sigma }\big (H(e_1, e_{\Sigma })\big )_{,11} {\text {d}}\sigma _{\Sigma } =-\frac{2c_1(\Sigma )}{\alpha } \mathrm{Vol}(\Sigma ). \end{aligned}$$

(53)

We have thus shown that the first sum in (50) only depends on $(\Delta , F, \mathbf{L})$.

We now deal with the terms $\int _{\Sigma } \sum _{i=2}^n \big (H(e_i, e_{\Sigma }))_{,1i} {\text {d}}\sigma _{\Sigma }$ in the second sum of (50). First of all, notice that on F, the expression

$$\begin{aligned} \sum _{i=2}^n \big (H(e_i, e_{\Sigma }))_{,1i} = \sum _{i=2}^n \Big \langle e_i^*\otimes e_1^*, \mathrm{Hess} \big (\mathbf{H}^u(e_i, e_{\Sigma })\big ) \Big \rangle \end{aligned}$$

does not change if we replace the initial basis $\{e_1=e_F, e_2, \ldots , e_n\}$ of ${{\mathfrak {t}}}$ with a basis of the form $\{\bar{e}_1=e_F, \bar{e}_2, \ldots , \bar{e}_n\}$ with $\bar{e}_j \in \mathrm{span}_{{\mathbb {R}}}\{e_2, \ldots , e_n\} = {\mathbb {R}}^{n-1}$ for $j =2, \ldots , n$. We can thus assume that, on a given $\Sigma$, we have chosen the basis with $e_2=e_{\Sigma } \in {\mathbb {R}}^{n-1}$, and use then the integration by parts formula (49) to write

$$\begin{aligned} \int _{\Sigma } \sum _{i=2}^n \big (H(e_i, e_{\Sigma }))_{,1i} {\text {d}}\sigma _{\Sigma } = \int _{\Sigma } \big (H(e_{2}, e_{2}))_{,12} {\text {d}}\sigma _{\Sigma } - \sum _{ \mathbf{f} \in \partial \Sigma } \int _\mathbf{f} H(e_\mathbf{f}, e_{\Sigma })_{,1} {\text {d}}\sigma _\mathbf{f},\nonumber \\ \end{aligned}$$

(54)

where the sum is over the facets $\mathbf{f}$ of $\Sigma$ (taken to be zero if $n=2$), $e_\mathbf{f}$ is the induced inward normal of $\mathbf f$, seen as an affine hyperplane of the affine space supporting $\Sigma$, and ${\text {d}}\sigma _\mathbf{f}$ is the corresponding induced measure on $\mathbf{f} \subset \Sigma$. With these choices, we have

$$\begin{aligned} \begin{aligned} \big (H(e_2, e_2)\big )_{,12}&= \Big \langle e_1^*\otimes e_2^*, \mathrm{Hess}\big (\mathbf{H}^u(e_{\Sigma }, e_{\Sigma })\big )\Big \rangle \\&= \Big \langle e_1^*\otimes e_2^*, \mathrm{Hess}\big (\mathbf{H}^u(e_{P}- c_1e_F, e_{P}- c_1e_F)\big )\Big \rangle \\&= \Big \langle e_1^*\otimes e_2^*, \mathrm{Hess}\big (\mathbf{H}^u(e_{P}, e_{P})\big )\Big \rangle \\&= \mathbf{H}^u(e_P, e_P)_{, e_P,e_P} \big \langle e_1^*\otimes e_2^*, (c_1 e_1 + e_2)\otimes (c_1e_1 + e_2)\big \rangle \\&= c_1 \mathbf{H}^u(e_P, e_P)_{, e_Pe_P}, \end{aligned} \end{aligned}$$

for the same constant $c_1=c_1(\Sigma )$ as above. In order to compute the (base independent) quantity $\int _{\Sigma } \mathbf{H}^u(e_P, e_P)_{, e_Pe_P} {\text {d}}\sigma _{\Sigma }$, we are going to re-introduce a basis $\{e_1=e_F, e_2=e_P, e_3, \ldots , e_n \}$ with respect to a vertex of $v\in \Delta$. In this basis, $\mathbf{H}^u(e_P, e_P)_{, e_Pe_P} =H_{22,22}$. A computation along the lines of (45) yields

$$\begin{aligned} \begin{aligned} \int _{\Sigma } s_{(F, \mathbf{L}_F)} {\text {d}}\sigma _{\Sigma }&= \int _{\Sigma } -\bigg (\sum _{i,j=2}^n H_{ij,ij}\bigg ) {\text {d}}\sigma _{\Sigma }\\&= \int _{\Sigma } \bigg (\mathrm{Scal}\big ((g_u)_{|\Sigma }\big ) - H_{22,22} - \sum _{j=3}^n H_{2j,2j}\bigg ) {\text {d}}\sigma _\Sigma \\&= \int _{\Sigma } \bigg (s_{(\Sigma , \mathbf{L}_{\Sigma })} -H_{22,22}\bigg ){\text {d}}\sigma _{\Sigma } + \sum _{\mathbf{f} \subset \partial \Sigma } \int _\mathbf{f} H(e_2,e_\mathbf{f})_{, 2} {\text {d}}\sigma _\mathbf{f}, \end{aligned} \end{aligned}$$

where $\mathrm{Scal}\big ((g_u)_{|\Sigma }\big ) := - \sum _{i,j=2}^n H_{ij,ij}$ is the scalar curvature of the almost-Kähler metric $(g_u)_{|\Sigma }$ induced via $\mathbf{H}^u$ on the pre-image of $\Sigma$, and for passing from the second line to the third we have used that $\int _{\Sigma } \mathrm{Scal}\big ((g_u)_{|\Sigma }\big ) {\text {d}}\sigma _{\Sigma }= \int _{\Sigma } s_{(\Sigma , \mathbf{L}_{\Sigma })} {\text {d}}\sigma _{\Sigma }$ (see [22, Lem. 3.3.5]) and (49) applied to $(\Sigma , \mathbf{L}_{\Sigma })$. Note that in the last term (which is considered trivially 0 when $n=2$), the sum is over the facets $\mathbf{f}$ of $\Sigma$, and $e_\mathbf{f}$ denotes the corresponding inward normal of $\mathbf{f}$ (when considered as an affine hyperplane of the subspace $\{x_1=0, x_2=0\}$). We thus have

$$\begin{aligned} \int _{\Sigma } \mathbf{H}^u(e_P, e_P)_{, e_Pe_P} {\text {d}}\sigma _{\Sigma } = \int _{\Sigma } \big (s_{(\Sigma , \mathbf{L}_{\Sigma })} - s_{(F, \mathbf{L}_F)}\big ) {\text {d}}\sigma _{\Sigma } + \sum _{\mathbf{f} \subset \partial \Sigma }\int _\mathbf{f} H(e_\mathbf{f}, e_2)_{,2}{\text {d}}\sigma _\mathbf{f}, \end{aligned}$$

in an $(F,\Sigma )$-compatible basis with $e_1=e_F, e_2=e_P$. Notice that in any such a basis, we have $e_\mathbf{f}= e_Q - c_2 e_F - c_3 e_P= e_Q -c_2e_1-c_3e_2,$ where $e_Q$ is the normal of the unique facet $Q \subset \Delta$, such that $F\cap P \cap Q= \mathbf{f}$. Here, the constants $c_2=c_2(\Sigma , \mathbf{f})= -e_Q\wedge {\text {d}}\nu _{F}/({\text {d}}\mu )$ and $c_3=c_3(\Sigma , \mathbf{f})= -(e_Q \wedge {\text {d}}\nu _P)/({\text {d}}\mu )$ are determined in terms of $(\Delta , \mathbf{L})$. In particular, we have on $\mathbf{f}$

$$\begin{aligned} \begin{aligned} H(e_\mathbf{f}, e_2)_{,2}&= \Big \langle e_2^*, d \mathbf{H}^u(e_P, e_Q - c_2e_F-c_3 e_P) \Big \rangle \\&= -c_3 \Big \langle e_2^*, d \mathbf{H}^u(e_P, e_P) \Big \rangle \\&= - 2c_3 \langle e_2^*, e_P \rangle = - 2c_3\langle e_2^*, e-2\rangle = -2c_3, \end{aligned} \end{aligned}$$

where we have used the first-order boundary conditions along $\mathbf{f}$ (see Proposition 4.19):

$$\begin{aligned} \begin{aligned} \big ({\text {d}}{} \mathbf{H}^u(e_F, e)\big )_{|\mathbf f}&= 0 , \ \forall e\in {{\mathfrak {t}}}, \ \ \big ({\text {d}}{} \mathbf{H}^u(e_P, e_Q)\big )_{|\mathbf{f}} =0 \\ \big ({\text {d}}{} \mathbf{H}^u(e_P, e_P)\big )_{|\mathbf{f}}&= 2e_P, \ \ \ \ \ \ \ \ \big ({\text {d}}{} \mathbf{H}^u(e_Q, e_Q)\big )_{|\mathbf{f}} = 2 e_Q. \end{aligned} \end{aligned}$$

(55)

To summarize, we have shown that

$$\begin{aligned} \begin{aligned} \int _{\Sigma }\big (H(e_2,e_2)\big )_{,12} {\text {d}}\sigma _{\Sigma } =&c_1(\Sigma ) \left( \int _{\Sigma } \big ( s_{(\Sigma , \mathbf{L}_{\Sigma })} -s_{(F, \mathbf{L}_{F})}\big ) {\text {d}}\sigma _{\Sigma }\right. \\&\left. \quad - 2\sum _{\mathbf{f} \subset \partial \Sigma } c_3(\Sigma , \mathbf{f})\mathrm{Vol}(\mathbf{f}) \right) . \end{aligned} \end{aligned}$$

(56)

Finally, we deal with the terms $\int _\mathbf{f} H(e_\mathbf{f}, e_{\Sigma })_{,1} {\text {d}} \sigma _\mathbf{f}$ in (54) (where we recall $\mathbf{f} \subset \Sigma \subset F$ is sequence of co-dimension one sub-faces). Using (55) again, we have along $\mathbf{f}$

$$\begin{aligned} \begin{aligned} H(e_\mathbf{f}, e_{\Sigma })_{,1}&= \Big \langle e_1^*, {\text {d}}\mathbf{H}^u(e_Q-c_2e_F- c_3e_P, e_F - c_1 e_P) \Big \rangle \\&= 2c_1c_3 \langle e_1^*, e_P \rangle \\&= 2c_1c_3 \langle e_1^*, e_2 + c_1 e_1 \rangle = 2c_1^2c_3, \end{aligned} \end{aligned}$$

where for passing from the second line to the third we have used that we choose in (54) a base with $e_1=e_F, e_2=e_{\Sigma }=e_P - c_1 e_F=e_P - c_1e_1$. It follows that

$$\begin{aligned} \sum _{\mathbf{f} \subset \partial \Sigma } H(e_\mathbf{f}, e_{\Sigma })_{,1} {\text {d}}\sigma _\mathbf{f} = \big (c_1(\Sigma )\big )^2\left( \sum _{\mathbf{f} \subset \Sigma }c_3(\Sigma , \mathbf{f})\right) . \end{aligned}$$

(57)

Substituting (56) and (57) back in (54), and (53), (54) and (47) back in (50), we obtain an expression for $\beta$ in terms of $(\Delta , \mathbf{L}, F)$.$\square$

Remark 4.21

• (1) Theorem 4.18 and Proposition 4.20 extend without difficulty to the case when $F=F_{1} \cup \cdots \cup F_{k}$ is a union of non-intersecting facets, i.e. Z is a smooth toric divisor. In general, it is natural to extend Definition 4.16 by introducing a pair of real numbers $(\alpha _i, \beta _i)$ for each facet $F_i \subset F$ and, for each face $\mathbf{f}\subset F$, one should require the smoothness and convexity over the relative interior of $\mathbf{f}$ of the function

  $$\begin{aligned} u_\mathbf{f}:=u + \sum _{F_i \in F : \mathbf{f} \subset F_i} (\alpha _i + \beta _i L_i)\log L_i. \end{aligned}$$

  It will be interesting to see whether or not the above statements hold true for a general toric divisor as above, with respect to such spaces of symplectic potentials (compare with Conjecture 4.14).

• (2) The explicit examples in the next section suggest that the complete Donaldson metrics will have, more generally, symplectic potentials u such that

  $$\begin{aligned} u = \frac{1}{2} \left( \sum _{F_k\notin F} L_k(x)\log L_k(x) + \sum _{F_i \in F} f_{i}(x) \log L_{i}(x) \right) + \mathrm{smooth \ terms}, \end{aligned}$$

  where, for any facet $F_i\in F$, $f_{i}$ is some affine function.

• (3) As we noticed in the course of the proof of Proposition 4.20, the situation simplifies when $n=2$. In fact, one can then explicitly determine $(\alpha , \beta )$ as follows: suppose (without loss of generality) that $(\Delta , \mathbf{L}, F)$ is such that $\Delta \subset \{(x_1, x_2): x_1 \ge 0, x_2 \ge 0,\ \ell - x_2 - \lambda x_1 \ge 0\}$, F corresponds to the affine line $x_1=0$, whereas the two adjacent facets of $\Delta$ to F are defined by the affine lines $x_2=0$ and $\ell - x_2 - \lambda x_1 =0$. Suppose, furthermore, that the extremal affine function of $(\Delta , \mathbf{L}, F)$ is $s_{(\Delta , \mathbf{L}, F)} = a_0 + a_1 x_1 + a_2 x_2$. Then the real parameters $(\alpha , \beta )$ of Proposition 4.20 are given by

  $$\begin{aligned} \alpha = \frac{2\ell }{4-a_0\ell }, \ \ \ \beta = \frac{\alpha ^2}{6}\left( a_1 + \frac{2\lambda a_0}{\ell } - \frac{12\lambda }{\ell ^2}\right) . \end{aligned}$$

4.4 Uniform Stability

In the seminal work [22], Donaldson also introduces the notion of uniform stability of a compact convex simple labelled polytope $(\Delta , \mathbf{L})$. This has been later extended in [45] to the general (non-toric) context.

Following [22], let ${\mathcal {C}}(\Delta )$ denote the set of continuous convex functions on $\Delta$ (continuity follows from convexity on the interior of $\Delta$). The affine linear functions act on ${\mathcal {C}}(\Delta )$ by translation and let $\tilde{{\mathcal {C}}}(\Delta )$ be the slice for this action consisting of $f \in {\mathcal {C}}(\Delta )$ such that $f(x) \ge f(x_0)=0$ for a fixed reference point $x_0\in \Delta ^0$. Then any f in ${\mathcal {C}}(\Delta )$ can be written uniquely as

$$\begin{aligned} f=\pi (f)+g, \end{aligned}$$

where g is affine linear and $\pi (f)\in \tilde{{\mathcal {C}}}(\Delta )$ for a linear projection $\pi$. Functions in $\tilde{{\mathcal {C}}}(\Delta )$ are said to be normalized.

Let $||\cdot ||$ be a semi-norm on ${\mathcal {C}}(\Delta )$ which indices a norm on $\tilde{{\mathcal {C}}}(\Delta )$, which is tamed in the sense that there exists $C>0$ such that on $\tilde{{\mathcal {C}}}(\Delta )$

$$\begin{aligned} \frac{1}{C} || \cdot ||_1 \le || \cdot || \le C || \cdot ||_{\infty }, \end{aligned}$$

where $|| \cdot ||_1:=\int _\Delta | \cdot | \,dv$ is the ${\varvec{L}}^1$-norm and $|| \cdot ||_{\infty }$ is the $C^0$-norm on ${\mathcal {C}}(\Delta )$. One can take $|| \cdot ||$ to be the ${\varvec{L}}^p$-norm for some $p\ge 1$ as in [45], but Donaldson considers instead the boundary norm $||f||_b := \int _{\partial \Delta } f {\text {d}}\sigma$ (and shows it is tamed [22, Lemma 5.1.3]).

Definition 4.22

A labelled compact convex simple polytope $(\Delta , \mathbf{L})$ is called uniformly stable with respect to the a tamed norm $||\cdot ||$ if

$$\begin{aligned} \mathcal {L}_{(\Delta , \mathbf{L})} (f)\ge \lambda ||\pi (f)||, \end{aligned}$$

(58)

for all piecewise affine linear convex functions f, where ${\mathcal {L}}_{(\Delta , \mathbf{L})}$ is the linear functional introduced in Definition 4.4.

The uniform stability is, a priori, a stronger condition than the stability of $(\Delta , \mathbf{L})$ introduced in Definition 4.4. In [16], Chen, Li and Sheng have strengthened Proposition 4.5.

Proposition 4.23

 [16] If $(\Delta , \mathbf{L})$ is a labelled compact convex simple polytope such that the Abreu equation (30) admits a solution in ${\mathcal {S}}(\Delta , \mathbf{L})$, then $(\Delta , \mathbf{L})$ is uniformly K-stable with respect to $|| \cdot ||_b$.

We notice that the above result implies that $(\Delta , \mathbf{L})$ is also $||\cdot ||_1$-uniform stable as $|| \cdot ||_b$ is tamed. The 2-dimensional case appears to be special as Donaldson shows in [22, Prop. 5.2.1 and 5.3.1] that if the corresponding extremal affine linear function $s_{(\Delta , \mathbf{L})}$ is strictly positive on $\Delta$, then $(\Delta , \mathbf{L})$ is K-stable if and only if it is uniformly K-stable with respect to $||\cdot ||_b$.

As the space of convex piecewise affine linear convex functions is dense in ${\mathcal {C}}(\Delta )$ for any tamed norm, it follows that for a uniformly stable labelled polytope $(\Delta , \mathbf{L})$ the inequality (58) holds true for $f\in {\mathcal {C}}(\Delta )$ and, in particular, for any symplectic potential $u \in {\mathcal {S}}(\Delta , \mathbf{L})$. Using this and an argument from [22], Zhou and Zhu have established in [50] the following key result.

Proposition 4.24

 [50] If $(\Delta , \mathbf{L})$ is uniformly stable with respect to $||\cdot ||$, then there exists $\delta >0$ and C such that

$$\begin{aligned} {\mathcal {M}}_{(\Delta , \mathbf{L})}(u)\ge \delta ||\pi (u)||+C \ \text {for all} \ u\in {\mathcal {S}}(\Delta , \mathbf{L}), \end{aligned}$$

(59)

where $\mathcal {M}_{(\Delta , \mathbf{L})}$ is the relative Mabuchi functional introduced in (31).

Remark 4.25

The above result is established in [50] for the norm $||\cdot ||_b$ (the case $\delta =0$ is due to [22]). The extension to any tamed norm appears in [13].

The importance of the uniform stability discussed above has manifested recently in connection with the notion of $d_1$ -relative properness of the (relative) Mabuchi energy used by Chen–Cheng in their deep work [15], and in its extension to the extremal case found by He [33], who show that the latter is a sufficient condition for the existence of an extremal Kähler metric. More precisely, in the toric setting, it turns out that on a Delzant polytope $(\Delta , \mathbf{L})$ the condition (59) with respect to the (weakest) tamed norm $||\cdot ||_1$ yields that the relative $d_1$-properness of the relative Mabuchi energy on the corresponding smooth toric variety X (see e.g. [3, Sect. 7.1] or [37] for a detailed argument) which in turn leads to the following result, generalizing and extending the results in [17, 25] to arbitrary dimension:

Proposition 4.26

[15, 17, 25, 33] Suppose $(\Delta , \mathbf{L})$ is a uniformly stable Delzant polytope with respect to $||\cdot ||_1$. Then the (30) admits a solution in ${\mathcal {S}}(\Delta , \mathbf{L})$.

In the light of Proposition 4.26, one might consider in Conjecture 4.14 the uniform stability for the triple $(\Delta , \mathbf{L}, F)$ and the corresponding facets in F (with respect to the norm $||\cdot ||_1$) instead of the weaker notion of K-stability introduced in Definition 4.10. As a matter of fact, when Z is a smooth toric divisor, the arguments in the proof of Theorem 4.13 and Proposition 4.23 imply that the uniform stability of the Delzant facets in F is a necessary condition for the existence of an extremal Poincaré type metric on $X\setminus Z$. However, extending Proposition 4.23 to the triple $(\Delta , \mathbf{L}, F)$ is not obvious. Indeed, the arguments in [16] rely on the boundary norm $||\cdot ||_b$, which is not longer tamed if we integrate over $\partial \Delta \setminus F$. Another issue is the extension of Proposition 4.24 as the space $\mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L})$ no longer embeds in ${\mathcal {C}}(\Delta )$. Finally, it is not clear whether the properness of the relative Mabuchi energy can be used to obtain a solution in the non-compact context.

We end the discussion by noticing that the assumption that $(\Delta , \mathbf{L}, F)$ is $||\cdot ||_1$-uniformly stable is natural if one tries to approach the existence of an extremal Kähler metric of Poincaré type on the corresponding toric variety $X\setminus Z$, via the continuity method of [24]. Indeed, assuming that $(\Delta , \mathbf{L}, F)$ is $||\cdot ||_1$-uniform stable implies that so will be the labelled convex compact polytopes $(\Delta , \mathbf{L}_t)$ for $t>0$, where the labelling $\mathbf{L}_t$ is obtained from $\mathbf{L}$ by replacing $L_{i}$ with $tL_{i}$ for each defining function $L_i$ of a facet $F_i \subset F$. The resulting polytope $(\Delta , \mathbf{L}_t)$ describes a toric variety X with an edge singularity of angle $2\pi /t$ along the components of the divisor Z, see [36, Prop. 6.4]. If Proposition 4.26 can be extended to toric varieties with such edge singularities, then one could find a solution $u_t \in {\mathcal {S}}(\Delta , \mathbf{L}_t$) of (30) for each t big enough, and try to show that there exist affine linear functions $g_t$ such that $u_t + g_t$ converges as $t \rightarrow \infty$ to a solution of (35) in the space ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$. It can be checked explicitly that such a convergence does hold for some of the examples discussed in this section and the next one. For instance, the potentials $u_t$ of Remark 4.17 actually give rise to edge singular extremal metrics on ${\mathbb {C}}P^1$ and converge, as the angle of the singularity $2\pi /t$ goes to 0, to the symplectic potential u extremal of (43), itself associated to a Poincaré type metric on ${\mathbb {C}}P^1\backslash \{0\}$. Similarly, the Poincaré type extremal metric on ${\mathbb {C}}P^2\setminus {\mathbb {C}}P^1$ described in Theorem 5.13 was originally discovered in [2] as a smooth limit of Bochner-flat metrics on weighted projective spaces. Another motivating example for this approach, beyond the toric context consider in this paper, is the work of Guenancia [29] which produces Poincaré type negative Kähler–Einstein metrics as limits of Kähler–Einstein metrics with edge singularity.

5 Explicit Donaldson Metrics on Quadrilaterals

We will show in this section, by using the explicit constructions of [5, 6], that Conjecture 4.11 is true for $X={\mathbb {C}}P^2, {\mathbb {C}}P^1\times {\mathbb {C}}P^1$ or the mth Hirzebruch complex surface ${\mathbb {F}}_{m} = \mathbb {P}({\mathcal {O}}\oplus {\mathcal {O}}(m))\rightarrow {\mathbb {C}}P^1, m\ge 1$.

By [5, Thm. 1 and Rem. 7], any stable compact convex quadrilateral $(\Delta , \mathbf{L}, F)$ in ${\mathbb {R}}^2$ admits a Donaldson metric, which is explicit and ambitoric. Dixon [21] showed that when $(\Delta , \mathbf{L}, F)$ corresponds to a compact toric complex orbi-surface X with a divisor Z, the metric is complete on $X\setminus Z$. In other words, Conjecture 4.11 holds true for compact toric surfaces whose momentum polytope is a quadrilateral. On the other hand, a detailed study of the stability of the triples $(\Delta , \mathbf{L}, F)$ was carried out by the third named author in [40]. In the next subsections we shall combine these results in order to obtain a complete picture in the case when $X={\mathbb {C}}P^2, {\mathbb {C}}P^1 \times {\mathbb {C}}P^1$ or $\mathbb {P}({\mathcal {O}}\oplus {\mathcal {O}}(k))\rightarrow {\mathbb {C}}P^1$, i.e. $(\Delta , \mathbf{L})$ is a Delzant triangle, parallelogram or a trapezoid.

5.1 Ambitoric Structures

Here we briefly review the explicit construction of extremal toric metrics for $n=2$ via the ambitoric ansatz of [6].

Definition 5.1

An ambikähler structure on a real 4-manifold or orbifold M consists of a pair of Kähler metrics $(g_+, J_+, \omega _+)$ and $(g_-, J_-, \omega _-)$ such that

• $g_+$ and $g_-$ induce the same conformal structure (i.e. $g_- = f^2g_+$ for a positive function f on M);

• $J_+$ and $J_-$ have opposite orientations (equivalently the volume elements $\frac{1}{2}\omega _+\wedge \omega _+$ and $\frac{1}{2}\omega _-\wedge \omega _-$ on M have opposite signs).

The structure is said to be ambitoric if in addition

• there is a 2-dimensional subspace ${{\mathfrak {t}}}$ of vector fields on M, linearly independent on a dense open set, whose elements are hamiltonian and Poisson-commuting Killing vector fields with respect to both $(g_+, \omega _+)$ and $(g_-, \omega _-)$.

Thus M has a pair of conformally equivalent but oppositely oriented Kähler metrics, invariant under a local 2-torus action, and both locally toric with respect to that action. There are three classes of examples of ambitoric structures.

5.2 Toric Products

Let $(\Sigma _1,g_1,J_1,\omega _1)$ and $(\Sigma _2,g_2,J_2,\omega _2)$ be (locally) toric Kähler manifolds or orbifolds of real dimension 2, with hamiltonian Killing vector fields $K_1$ and $K_2$. Then $M=\Sigma _1\times \Sigma _2$ is ambitoric, with $g_\pm =g_1\oplus g_2$, $J_\pm =J_1\oplus (\pm J_2)$, $\omega _\pm =\omega _1\oplus (\pm \omega _2)$ and ${{\mathfrak {t}}}$ spanned by $K_1$ and $K_2$. The metric $g_+$ is extremal (resp. CSCK) iff $g_{-}$ is extremal (resp. CSCK) iff both $g_1$ and $g_2$ are extremal (resp. CSCK). Writing $(\Sigma _1,g_1)$ and $(\Sigma _2, g_2)$ as toric Riemann surfaces

$$\begin{aligned} g_1 = \frac{{\text {d}}x^2}{A(x)} + A(x){\text {d}}t_1^2; \ \ g_2=\frac{{\text {d}}y^2}{B(y)} + B(y) {\text {d}}t_2^2, \end{aligned}$$

for positive functions A, B of one variable, and momentum/angular coordinates

$$\begin{aligned} x_1=x, \ x_2=y, \ t_1, \ t_2, \end{aligned}$$

the extremal metrics are given by taking A and B to be polynomials of degrees $\le 3$. In this case, we obtain solutions to Abreu’s equation on labelled parallelograms $(\Delta , \mathbf{L}, F)$ (which are affine equivalent to a square) by taking

$$\begin{aligned} \mathbf{H}^{A,B} =\mathrm{diag} (A(x), B(y)) \end{aligned}$$

(60)

for A and B polynomials of degree $\le 3$ and noting that the positivity and boundary conditions of Definition 4.6 reduce to $A>0$ on $(\alpha _0, \alpha _\infty )$, $B>0$ on $(\beta _0, \beta _\infty )$ and

$$\begin{aligned} A(\alpha _k)=0=B(\beta _k)=0, \quad A'(\alpha _k)=-2r_{\alpha ,k}, \quad B'(\beta _k) =2r_{\beta ,k} \quad (k=0,\infty ), \end{aligned}$$

(61)

where $r_{\alpha ,0} \le 0 \le r_{\alpha ,\infty }, \ r_{\beta ,0} \ge 0 \ge r_{\beta ,\infty }$ are determined by the choice of inward normals $e_{\alpha , k}=\frac{1}{r_{\alpha , k}}(-1,0)$ (resp. $e_{\beta , k}= \frac{1}{r_{\beta , k}}(0,1)$) if the facet $F_{\alpha ,k}$ (resp. $F_{\beta ,k}$) defined by $x=\alpha _k$ (resp. $y=\beta _k$) does not belong to F, and $r_{\alpha ,k}=0$ (resp. $r_{\beta ,k}=0$) otherwise. The above boundary conditions can be solved for polynomials of degree 3, A(x) and B(y), if and only if $|r_{\alpha ,0}| + |r_{\alpha , \infty }|>0$ and $|r_{\beta ,0}| + |r_{\beta ,\infty }|>0$, i.e. iff no two opposite sides of $\Delta$ belong to F: in this case, the positivity of A and B automatically follows from the boundary conditions. On the other hand, when two opposite sides of $\Delta$ belong to F there is no solution to (35) verifying the positivity condition. Indeed, if $r_{\alpha ,0}=r_{\alpha ,\infty }=0$ say, then $\mathbf{H}^{A,B} =\mathrm{diag} (A(x), B(y))$ with $A\equiv 0$ and B(y) a polynomial of degree $\le 3$ determined from (61). This provides a formal solution of (35). The latter can be used (by using integration by parts, as in [5, 35]) to compute that ${\mathcal {L}}_{\Delta ,u,F}(f_{\alpha })=0$ for any simple crease function $f_{\alpha }$ with crease at $x=\alpha$ ($\alpha \in (\alpha _0,\alpha _\infty )$), showing that $(\Delta , \mathbf{L}, F)$ is not stable in this case.

Whenever it exists, the solution $u_{A,B}$ is determined from (60) by the formula

$$\begin{aligned} u_{A,B} = \int ^x\left( \int ^s \frac{{\text {d}}t}{A(t)}\right) {\text {d}}s + \int ^y \left( \int ^s \frac{{\text {d}}t}{B(t)}\right) {\text {d}}s, \end{aligned}$$

which leads to the intrinsic expression

$$\begin{aligned} u_{A,B} = \frac{1}{2}\left( \sum _{F_j \in \partial \Delta } L_j \log L_j - \sum _{F_k \in F} a_k \log L^c_k \right) . \end{aligned}$$

(62)

where, for each facet $F_j \in \partial \Delta \setminus F$, $L_j(x)= \langle e_j, x\rangle + \lambda _j$ is the corresponding label from $\mathbf{L}$ and, for each $F_k \in F$, we define the label $L^c_k(x) := \langle e_k,x \rangle + \lambda _k$ by requiring that $e_k:=-e_{\tilde{k}}= -{\text {d}}L_{\tilde{k}}$ where $L_{\tilde{k}}$ is the label form $\mathbf{L}$ of the opposite side $F_{\tilde{k}}$ to $F_k$ (by the discussion above, $F_{\tilde{k}} \in \partial \Delta \setminus F$) and let $a_k := L_k + L_{\tilde{k}}>0$ be a real constant.

We notice that when the solution exists, the degree $\le 3$ polynomials A(x) and B(y) must satisfy $A''(x)=A''(\alpha _k) >0$ (resp. $B''(y)=B''(\beta _k)>0$) on facets in F. The formula for the scalar curvature

$$\begin{aligned} s_+ = - (A''(x) + B''(y)) \end{aligned}$$

then confirms that $s_{(\Delta , \mathbf{L}, F)} - s_{(F, \mathbf{L}_F, \check{F})}$ restricts to F as a negative constant.

We conclude that

Proposition 5.2

Let $(\Delta ,\mathbf{L},F)$ be a labelled parallelogram in ${\mathbb {R}}^2$. Then the Abreu equation (35) admits a solution in ${\mathcal {S}}(\Delta , \mathbf{L}, F)$ iff F does not contain opposite sides. In this case, there exists a solution explicitly given by (62).

Turning to the compact smooth case, there exists only one compact complex toric surface whose Delzant polytopes are parallelograms, namely $X= {\mathbb {C}}P^1 \times {\mathbb {C}}P^1$. The result above trivially produces products of a cusp metric on ${\mathbb {C}}P^1\setminus \{\mathrm{pt} \}$ with a Fubini–Study metric on another copy of ${\mathbb {C}}P^1$, or the product of two cusp metrics on $({\mathbb {C}}P^1\setminus \{\mathrm{pt} \}) \times ({\mathbb {C}}P^1\setminus \{\mathrm{pt} \})$, according to whether $Z \subset {\mathbb {C}}P^1\times {\mathbb {C}}P^1$ is a copy of ${\mathbb {C}}P^1$ (i.e. F is a one facet) or is the union of two copies of ${\mathbb {C}}P^1$ (i.e. F consists of two adjacent facets). We thus can conclude that

Corollary 5.3

Let $X= {\mathbb {C}}P^1 \times {\mathbb {C}}P^1$ endowed with the product of circle actions on each factor, and Z be either ${\mathbb {C}}P^1 \times \{p_2\}$ or $({\mathbb {C}}P^1 \times \{p_2\}) \cup (\{p_1\} \times {\mathbb {C}}P^1)$ where $p_1$ and $p_2$ are fixed points for the $S^1$ actions on each factor. Then, in each Kähler class of ${\mathbb {C}}P^1\times {\mathbb {C}}P^1$, there exists a complete extremal Donaldson metric on ${\mathbb {C}}P^1 \times {\mathbb {C}}P^1 \setminus Z$ which is of Poincaré type.

If Z contains $({\mathbb {C}}P^1 \times \{p_2'\}) \cup ({\mathbb {C}}P^1 \times \{p_2''\})$ where $p_2'$ and $p_2''$ are the two distinct fixed points for the $S^1$ action, then (X, Z) is K-unstable, and admits no extremal Donaldson metric at all.

5.3 Toric Calabi Type Metrics

The construction in this section is not new, see e.g. [34]. For the sake of completeness, and to make the link with toric geometry more explicit, we follow the formalism from [6].

Let $(\Sigma ,g,J,\omega )$ be a toric real 2-dimensional Kähler manifold with hamiltonian Killing vector field V (with momentum y). Let $\pi :P\rightarrow \Sigma$ be a circle bundle with connection $\theta$ and curvature $d\theta =\pi ^*\omega _{\Sigma }$, and A(x) be a positive function defined on an open interval $I \subset {\mathbb {R}}^+$. Then $M=P\times I$ is ambitoric, with

$$\begin{aligned} g_\pm&= x^{\pm 1} \Bigl (g_{\Sigma }+\frac{{\text {d}}x^2}{A(x)}+ \frac{A(x)}{x^2}\theta ^2\Bigr ), \ d\theta = \omega _{\Sigma },\\ \omega _\pm&= x^{\pm 1}\bigl ( \omega _{\Sigma } \pm x^{-1} {\text {d}}x\wedge \theta \bigr ),\qquad J_\pm (x {\text {d}}x) = \pm A(x) \theta , \end{aligned}$$

and the local torus action spanned by the generator K of the circle action on P and the lift $\tilde{V} = V^H + y K$ of the hamiltonian Killing field of $(\Sigma , g_{\Sigma }, \omega _{\Sigma })$ to M. Here, $x:M\rightarrow {\mathbb {R}}^+$ is the projection onto $I\subset {\mathbb {R}}^+$. It is easily seen that $g_+$ is extremal (resp. CSCK) iff $g_-$ is extremal iff $(\Sigma ,g)$ has constant Gauss curvature $\kappa$ and A(x) is a polynomial of degree $\le 4$ with coefficient of $x^2$ equal to $\kappa$. Because of this equivalence, we shall focus on $(g_+,\omega _+)$, say.

Writing the toric metric $(g_{\Sigma }, \omega _{\Sigma })$ in momentum/angle coordinates as

$$\begin{aligned} g_{\Sigma }= \frac{{{\text {d}}y}^2}{B(y)} + B(y) {\text {d}}t_2^2, \ \omega _{\Sigma } = {\text {d}}y \wedge {\text {d}}t_2, \end{aligned}$$

(63)

for a positive function B(y), the Kähler metric $(g_+, \omega _+)$ becomes (see [35])

$$\begin{aligned} \begin{aligned} g_+&= x \frac{{\text {d}}x^2}{A(x)} + x \frac{{\text {d}}y^2}{B(y)} + \frac{A(x)}{x}({\text {d}}t_1 + y {\text {d}}t_2)^2 + x B(y) {\text {d}}t_2^2 , \\ \omega _+&= {\text {d}}x \wedge {\text {d}}t_1 + d(xy) \wedge {\text {d}}t_2= {\text {d}}x_1\wedge {\text {d}}t_1 + {\text {d}}x_2 \wedge {\text {d}}t_2, \end{aligned} \end{aligned}$$

(64)

with

$$\begin{aligned} (x_1, x_2)=(x, xy) \end{aligned}$$

(65)

being the momentum coordinates and $(t_1,t_2)$ the angular coordinates. The corresponding symplectic potential is then

$$\begin{aligned} u_{A,B}(x,y)= x\int ^y \left( \int ^s \frac{{\text {d}}t}{B(t)}\right) {\text {d}}s + \int ^x \left( \int ^s \frac{t{\text {d}}t}{A(t)}\right) {\text {d}}s. \end{aligned}$$

(66)

In order to obtain functions in ${\mathcal {S}}(\Delta ,\mathbf{L},F)$ for some compact convex labelled polytope $(\Delta , \mathbf{L}, F)$, we fix the data of real numbers

$$\begin{aligned} 0 \le \beta _0<\beta _\infty , \ 0<\alpha _0< \alpha _\infty , \ r_{\alpha ,0} \le 0 \le r_{\alpha ,\infty }, \ r_{\beta ,0} \ge 0 \ge r_{\beta ,\infty } \end{aligned}$$

(67)

and impose the following positivity and boundary conditions on the smooth functions of one variable A(x) and B(y)

$$\begin{aligned} A(x)>0 \ \mathrm{on} \ (\alpha _0, \alpha _\infty ) \ \mathrm{and} \ B(y)>0 \ \mathrm{on} \ (\beta _0, \beta _\infty ), \end{aligned}$$

(68)

$$\begin{aligned} A(\alpha _k)=0, A'(\alpha _k)=-2r_{\alpha ,k}, \ B(\beta _k)=0, B'(\beta _k)= -2r_{\beta ,k}, \quad (k=0,\infty ). \end{aligned}$$

(69)

Note that the line $\{x=\alpha \}$ transforms in the $(x_1,x_2)$-coordinates (65) to the affine line $\ell _{\alpha } = \{(\alpha , x_2)\}$ with normal $p_{\alpha }=(\alpha ,0)$ and $y=\beta$ ($\beta >0$) to the affine line $\ell _{\beta }=\{(x_1,\beta x_1) \}$ with normal $p_{\beta }=(-\beta , 1)$. Thus, the image of $D=[\alpha _0, \alpha _\infty ]\times [\beta _0, \beta _\infty ]$ under (65) is a trapezoid $\Delta$ with facets $F_{\alpha ,k}, F_{\beta ,k}$ determined by the lines $\ell _{\alpha _k}, \ell _{\beta _k}$, and the inverse Hessian $\mathbf{H}^{A,B}$ of $u_{A, B}$ is given by

$$\begin{aligned} \mathbf{H}^{A,B} = \frac{1}{x}\left( \begin{array}{ll} A(x) &{}\quad yA(x) \\ yA(x) &{}\quad x^2B(y) + y^2A(x) \end{array} \right) . \end{aligned}$$

(70)

We write for the normals $e_{\alpha , k}= p_{\alpha _k}/ r_{\alpha ,k}, e_{\beta ,k}= p_{\beta _k}/r_{\beta ,k}$. Then, $\mathbf{H}^{A,B}$ satisfies the smoothness, positivity and boundary conditions (32)–(33) iff $r_{\alpha ,k}=0$ (resp. $r_{\beta ,k}=0$) on a facet $F_{\alpha ,k}\in F$ (resp. $F_{\beta ,k} \in F$).

Conversely, by [35, Lem. 4.7], if $(\Delta ,\mathbf{L}, F)$ is a labelled trapezoid, there exist real numbers $\alpha _k, r_{\alpha ,k}, \kappa$ $(k=0,\infty )$, subject to the inequalities (67), such that $\Delta$ the image of $D=[\alpha _0, \alpha _\infty ]\times [\beta _0, \beta _\infty ]$ under $(\mu _1, \mu _2)$, and $r_{\alpha ,k}$ are determined from the normals $\mathbf{L}$ and F as explained above. It is easily seen [35, Prop. 4.12] that (70) satisfies (35) if and only if A(x) is a polynomial of degree $\le 4$, and B(y) is a polynomial of degree $\le 2$, which satisfy

$$\begin{aligned} A''(0) + B''(0)=0. \end{aligned}$$

(71)

For such polynomials to satisfy (69), one must have $r_{\beta ,0}= -r_{\beta ,\infty }= r \ge 0$. This is also a sufficient condition to determine the polynomials A(x) and B(y) from (69), subject to the relation (71). In particular, $B(y)= \frac{r}{(\beta _2-\beta _1)}(y-\beta _1)(\beta _\infty -y)$, showing that the positivity conditions (68) imply $r>0$, i.e. $r_{\beta ,k} \ne 0$. This also implies positivity for A(x) on $(\alpha _0, \alpha _\infty )$: otherwise A(x) will have all of its roots between $[\alpha _0, \alpha _\infty ]$ and, by the boundary conditions (69), it must satisfy $\lim _{x\rightarrow \infty } A(x)= -\infty$. The latter contradicts $A''(0)= -B''(0)>0$. The corresponding Kähler metric has scalar curvature

$$\begin{aligned} s_+ = -\frac{A''(x) + B''(y)}{x}, \end{aligned}$$

(72)

showing that the extremal affine function $s_{(\Delta , \mathbf{L}, F)}$ determines an affine line parallel to $F_{\alpha , 0}$ and $F_{\alpha , \infty }$. Conversely, the proofs of [35, Lem. 4.2, Thm. 1.4] show that if $(\Delta , \mathbf{L}, F)$ is a labelled trapezoid $(\Delta ,\mathbf{L}, F)$ which is not a parallelogram, and the extremal affine function $s_{(\Delta , \mathbf{L}, F)}$ is constant on each of the pair of parallel facets of $\Delta$, then one can associate to $(\Delta , \mathbf{L}, F)$ data (67) satisfying the relation $r_{\beta ,0}=-r_{\beta ,\infty }=r \ge 0$. The case $r=0$ (i.e. when two opposite non-parallel facets of $\Delta$ belong to F) implies $B(y)\equiv 0$. As observed in [45], and similarly to the case of a parallelogram, this contradicts the stability of $(\Delta , \mathbf{L}, F)$. Indeed, substituting in (70), we still obtain a smooth matrix $\mathbf{H}^{A,B}$ on $\Delta$ verifying (35) and the boundary conditions of Definition 4.6. This can be used to compute ${\mathcal {L}}_{(\Delta ,u,F)} (f_{\alpha })$ for a simple crease function $f_{\alpha }$ with crease on the line $\ell _{\alpha }=\{x=\alpha \}$: integration by parts reduces to an integral over the crease of the quantity $\mathbf{H}^{A,0}(p_{\alpha }, p_{\alpha })=0$, showing that ${\mathcal {L}}_{(\Delta ,\mathbf{L}, F)} (f_{\alpha })=0$, i.e. $(\Delta , \mathbf{L}, F)$ is not stable. We summarize the discussion in the following:

Proposition 5.4

[35, 45] Let $(\Delta ,\mathbf{L}, F)$ be a labelled trapezoid in ${\mathbb {R}}^2$ which is not a parallelogram. Suppose that the corresponding extremal affine function $s_{(\Delta , \mathbf{L} ,F)}$ is constant on each of the pair of parallel facets of $\Delta$. Then $(\Delta , \mathbf{L}, F)$ admits a solution to (35) in ${\mathcal {S}}(\Delta , \mathbf{L}, F)$ if and only if $(\Delta , F)$ is stable, if and only if F is one or the union of 2 of the parallel facets of $\Delta$. In these cases, the solution is of Calabi type, i.e. given by (66) for polynomials A(x) and B(y) as described above.

In order to derive further geometric applications, we use [35, Cor. 1.6] which identifies the choice of labels $\mathbf{L}$ of a given trapezoid $\Delta$ for which $s_{(\Delta , \mathbf{L},F)}$ is constant on each of the pair of parallel facets with one single linear constraint on the pair of inward normals to non-parallel facets. Up to an overall positive rescaling of $\mathbf{L}$, this fixes the choice of these normals, but leaves no constraint on the pair of normals corresponding to the parallel opposite facets. In our notation, this corresponds to fixing the boundary condition for B(y) and allowing $r_{\alpha ,0} \le 0 \le r_{\alpha ,\infty }$ to be arbitrary real numbers. It thus follows that if $(\Delta ,\mathbf{L})$ is a labelled trapezoid for which the corresponding extremal affine function $s_{(\Delta ,\mathbf{L})}$ is parallel to the pair of parallel facets, then, by taking F to be either one or two of the parallel facets of $\Delta$, the extremal affine function $s_{(\Delta ,\mathbf{L},F)}$ must also be parallel to the pair of parallel facets. We now apply this observation to Delzant trapezoids $(\Delta , \mathbf{L})$.

The compact toric complex surfaces X for which the Delzant polytopes are trapezoids (but not parallelograms) are the Hirzebruch surfaces ${\mathbb {F}}_m= \mathbb {P}({\mathcal {O}}\oplus {\mathcal {O}}(m)) \rightarrow {\mathbb {C}}P^1, \ m \ge 1$. Calabi [12] has shown that these surfaces admit extremal Kähler metric of Calabi type in each Kähler class. In particular, the extremal affine functions are always constant on the pair of parallel facets of the corresponding Delzant polytopes. Thus, Proposition 5.4 yields the following natural extension of Calabi’s result.

Corollary 5.5

Let $X\cong {\mathbb {F}}_m=\mathbb {P}({\mathcal {O}}\oplus {\mathcal {O}}(m))\rightarrow {\mathbb {C}}P^1$ be the mth Hirzebruch surface and $Z\subset X$ the divisor consisting of either the zero section $S_0$, the infinity section $S_{\infty }$ or the union of both. Then $X\setminus Z$ admits a complete extremal Donaldson–Kähler metric in each Kähler class of ${\mathbb {F}}_m$. Furthermore, this metric is of Poincaré type.

Proof

The only additional clarification we need to supply is whether the explicit extremal Calabi type metrics are of Poincaré type. This follows from the expression (70), noting that the only the facets $F_{\alpha , k}$ are in F, and on such a facet (having normal vector ($\alpha _k, 0)$), the boundary conditions of Proposition 4.19 reduce to $A(\alpha _k)= A'(\alpha _k)=0$ and $A'''(\alpha _k)\ne 0$. If these hold, the extremal Kähler metric of Calabi type is manifestly in some class ${\mathcal {S}}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$, and thus is of Poincaré type according to Theorem 4.18. The vanishing conditions are always satisfied. The only condition we need to verify is $A'''(\alpha _k)\ne 0$. To this end, we describe the solutions explicitly.

Letting

$$\begin{aligned} \begin{aligned} \alpha _0&=1, \alpha _\infty =a>1, r_{\alpha ,k}=0, \\ \beta _0&=0, \beta _\infty = m, r_{\beta ,0}= - r_{\beta ,\infty } = 1, \end{aligned} \end{aligned}$$

(73)

we obtain in the case $Z= S_0 \cup S_{\infty }$ an extremal Kähler metric on $X\setminus Z$ given by (64) with

$$\begin{aligned} B(y) = -\frac{2}{m}y(y - m), \ A(x)=-\frac{2}{m(a^2 + 4a +1)}(x-1)^2(x-a)^2 \end{aligned}$$

(74)

where $a>1$ parametrizes (up to a scale) the Kähler cone of ${\mathbb {F}}_m$. This is a complete extremal Kähler metric defined on the total space of the principal ${\mathbb {C}}^{\times }$-bundle over ${\mathbb {C}}P^1$ classified by $c_1({\mathcal {O}}(m)) \in H^2({\mathbb {C}}P^1, {\mathbb {Z}})$, with cusp singularities at 0 and $\infty$. The conditions $A'''(1) \ne 0 \ne A'''(a)$ obviously hold.

Similarly, when $Z=S_0$ say, for the same choice of $\alpha _k, \beta _k$ the extremal solution is given by (64) with

$$\begin{aligned} B(y) = -\frac{2}{m}y(y - m), \ A(x) =-(px+q)(x-1)^2(x-a), \end{aligned}$$

(75)

where the constants p, q are given by

$$\begin{aligned} \begin{aligned} p&= \frac{2\left( \frac{r_{\alpha ,\infty }(a+2)}{(a-1)^2} -\frac{1}{m}\right) }{(a^2 + 4a +1)}, \\ q&= \frac{2\left( \frac{r_{\alpha ,\infty }(2a+1)}{(a-1)^2} +\frac{a}{m}\right) }{(a^2+4a+1)}. \end{aligned} \end{aligned}$$

(76)

Such a metric compactifies smoothly at $S_{\infty }$ precisely when the real parameter $r_{\alpha ,\infty }=1$, which gives the complete extremal Kähler metrics in Corollary 5.5; for other values of $r_{\alpha ,\infty }>0$, one gets a complete metric on $X\setminus S_0$ with a cone singularity of angle $2\pi r_{\alpha ,\infty }$ along $S_{\infty }$. Now, the condition $A'''(1)=0$ is equivalent to $p=-q$. With $r_{\alpha , \infty }=1$, this reads as

$$\begin{aligned} 3m(a+1) + (a-1)^3 =0, \end{aligned}$$

which is impossible for $a>1$.

The case of $Z=S_{\infty }$ can be treated similarly. $\square$

Remark 5.6

As a special case of the ansatz (75), one can construct CSCK metrics by putting $r_{\alpha ,\infty }= \frac{(a-1)^2}{m(a+2)}$ (i.e. setting the coefficient p in (76) to be zero). For each $m\ge 1$, this defines a CSCK metric on ${\mathbb {F}}_m\setminus S_0$, in each Kähler class of ${\mathbb {F}}_m$ (parametrized by $a>1$) with a cusp singularity along $S_0$ and a cone singularity of angle $2\pi \Big (\frac{(a-1)^2}{ma(a+2)}\Big )<2\pi$ along $S_{\infty }$.

5.4 Regular Ambitoric Structures

Let $q(z)=q_0z^2+2q_1 z+ q_2$ be a quadratic polynomial, M a 4-dimensional manifold with real-valued functions $(x,y,\tau _0,\tau _1,\tau _2),$ such that $x>y$, $2q_1\tau _1=q_0\tau _2+q_2\tau _0$, and at each point of M, the 1-forms ${\text {d}}x, {\text {d}}y, {\text {d}}\tau _0, {\text {d}}\tau _1, {\text {d}}\tau _2$ span the cotangent space. Let ${{\mathfrak {t}}}$ be the 2-dimensional space of vector fields K on M satisfying ${\text {d}}x(K)=0={\text {d}}y(K)$ and ${\text {d}}\tau _j(K)$ constant. Then, for any smooth and positive functions of one variable, A(x) and B(y), defined on the images of x and y in ${\mathbb {R}}$, respectively, M is ambitoric with respect to ${{\mathfrak {t}}}$ and the Kähler structures

$$\begin{aligned} g_\pm = \biggl (\frac{x-y}{q(x,y)}\biggr )^{\pm 1} \biggl (&\frac{{\text {d}}x^2}{A(x)} + \frac{{\text {d}}y^2}{B(y)} + A(x) \Bigl (\frac{y^2 {\text {d}}\tau _0 + 2y {\text {d}}\tau _1 + {\text {d}}\tau _2}{(x-y)q(x,y)}\Bigr )^2 \nonumber \\&+ B(y) \Bigl (\frac{x^2 {\text {d}}\tau _0 + 2x {\text {d}}\tau _1 + {\text {d}}\tau _2}{(x-y)q(x,y)}\Bigr )^2 \biggr ), \end{aligned}$$

(77)

$$\begin{aligned} \omega _\pm = \biggl (\frac{x-y}{q(x,y)}\biggr )^{\pm 1}&\frac{{\text {d}}x\wedge (y^2 {\text {d}}\tau _0 + 2y {\text {d}}\tau _1 + {\text {d}}\tau _2) \pm {\text {d}}y \wedge (x^2 {\text {d}}\tau _0 + 2x {\text {d}}\tau _1 + {\text {d}}\tau _2)}{(x-y)q(x,y)},\nonumber \\ J_\pm {\text {d}}x= A(x)&\frac{y^2 {\text {d}}\tau _0 + 2y {\text {d}}\tau _1 + {\text {d}}\tau _2}{(x-y)q(x,y)}, \quad J_\pm {\text {d}}y= \pm B(y)\frac{x^2 {\text {d}}\tau _0 + 2x {\text {d}}\tau _1 + {\text {d}}\tau _2}{(x-y)q(x,y)}, \end{aligned}$$

(78)

where $q(x,y)=q_0xy+q_1(x+y)+q_2$. The metric $g_+$ is extremal iff $g_-$ is extremal iff

$$\begin{aligned} \begin{aligned} A(z)&=q(z)\pi (z)+P(z),\\ B(z)&=q(z)\pi (z)-P(z),\\ \end{aligned} \end{aligned}$$

(79)

where $\pi (z)=\pi _0z^2 + 2\pi _1z + \pi _2$ is a polynomial of degree at most two satisfying $2\pi _1 q_1 -(q_2\pi _0 + q_0\pi _2)=0$, and P(z) is polynomial of degree at most four.

The space of Killing fields of $g_{\pm }$ for the torus is naturally isomorphic to the space of $S^2_{0,q}$ of polynomials p(z) of degree $\le 2$ which are orthogonal to q with respect to the inner product $\langle \cdot , \cdot \rangle$ defined by the discriminant, i.e. $p(z)= p_0z^2 + 2p_1z + p_2$ satisfying

$$\begin{aligned} \langle p, q \rangle =2p_1 q_1 -(q_2p_0 + q_0p_2)=0. \end{aligned}$$

The space $S^2_{0,q}$ is in turn isomorphic to the quotient space $S^2 /\langle q \rangle$ of all polynomials of degree $\le 2$ by the subspace generated by q, by using $\frac{1}{2} \mathrm{ad}_q$ with respect to the Poisson bracket

$$\begin{aligned} \mathrm{ad}_q(w)=\{q,w\}= q'w - w'q \end{aligned}$$

on $S^2$. Thus, if $\{p_1,p_2\}$ is a basis of $S^2_{q,0}$ and $\{w_1,w_2\}$ the corresponding basis of $S^2 /\langle q \rangle$ (with $p_i = \frac{1}{2}\{q, w_i\}$) momentum/angular coordinates for $g_{\pm }$ are given by

$$\begin{aligned} \begin{aligned} x_i^+&= w_i(x,y)/q(x,y), \ t_i, \ (i=1,2) \\ x_i^-&= p_i(x,y)/(x-y), \ t_i, \ (i=1,2). \end{aligned} \end{aligned}$$

(80)

It follows that the lines $x=\alpha$ (resp. $y=\beta$) transform to lines $\ell ^+_{\alpha }= \frac{(x-\alpha )(y-\alpha )}{q(x,y)}=0$ (resp. $\ell ^+_{\beta }= \frac{(x-\beta )(y-\beta )}{q(x,y)}$) in the $(x_1^+, x_2^+)$ plane, which are tangent to the non-degenerate conic $C^*_+ \subset {{\mathfrak {t}}}^*$ corresponding to $\Big (\frac{x-y}{q(x,y)}\Big )^2=0$; similarly, $\ell ^-_{\alpha }= \frac{(x-\alpha )q(y, \alpha )}{(x-y)}$ (resp. $\ell _{\beta }^-=\frac{(y-\beta )q(x,\beta )}{x-y}$) are lines in the $(x_1^-,x_2^-)$-plane (corresponding to $x=\alpha$ and $y=\beta$ in the (x, y)-plane) which are tangent to the (possibly degenerate) conic $C^*_-\subset {{\mathfrak {t}}}^*$ defined by $\Big (\frac{q(x,y)}{x-y}\Big )^2=0$. In both cases, the corresponding normals are

$$\begin{aligned} p_{\alpha }(z) = q(\alpha , z)(z-\alpha ) ; \ \ p_{\beta } (z) = q(z, \beta )(z-\beta ), \end{aligned}$$

(81)

viewed as elements of $S^2_{0,q}$.

It is straightforward to compute the matrix ${{\mathbf {H}}}^{A,B}_{\pm }$ of $g_\pm$:

$$\begin{aligned} \begin{aligned} {{\mathbf {H}}}^{A,B}_{-}(p_i,p_j)&=\frac{A(x) p_i(y) p_j(y) + B(y) p_i(x) p_j(x)}{(x-y)^3\, q(x, y) },\\ {{\mathbf {H}}}^{A,B}_{+}(p_i,p_j)&=\frac{A(x) p_i(y) p_j(y) + B(y) p_i(x) p_j(x)}{(x-y)\, q(x, y)^3 }, \end{aligned} \end{aligned}$$

(82)

whose inverses are the Hessians in momenta of the symplectic potentials

$$\begin{aligned} \begin{aligned} u^+_{A,B}(x,y)&= - \int ^x \frac{(t-x)(t-y){\text {d}}t}{q(x,y)A(t)} + \int ^y \frac{2(t-x)(t-y){\text {d}}t}{q(x,y)B(t)},\\ u^-_{A,B}(x,y)&= \int ^x \frac{2(x-t)q(y,t){\text {d}}t}{(x-y)A(t)} + \int ^y \frac{2(y-t)q(x,t){\text {d}}t}{(x-y)B(t)}. \end{aligned} \end{aligned}$$

(83)

In order for $u^{\pm }_{A,B}$ be in ${\mathcal {S}}(\Delta , \mathbf{L}, F)$ for some labelled compact convex quadrilateral $(\Delta ,\mathbf{L},F)$, one has to choose real numbers $\alpha _k, \beta _k, r_{\alpha ,k}, r_{\beta ,k}$ $(k=0,\infty )$ satisfying the inequalities

$$\begin{aligned} \beta _0< \beta _\infty< \alpha _0< \alpha _\infty , \qquad r_{\alpha ,0} \le 0 \le r_{\alpha ,\infty }, \quad r_{\beta ,0} \ge 0 \ge r_{\beta ,\infty }, \end{aligned}$$

and such that $q(x,y)>0$ on $D=[\alpha _0,\alpha _\infty ]\times [\beta _0,\beta _\infty ]$, and then impose on the smooth functions A(x), B(y) the positivity conditions

$$\begin{aligned} A(x)>0 \ \mathrm{on} \ (\alpha _0,\alpha _\infty ) \ \mathrm{and} \ B(y)>0 \ \mathrm{on} \ (\beta _0, \beta _\infty ), \end{aligned}$$

(84)

and the boundary conditions

$$\begin{aligned} A(\alpha _k)=0=B(\beta _k),\ A'(\alpha _k)=-2r_{\alpha ,k}, \ B'(\beta _k) =2r_{\beta ,k} \ (k=0,\infty ). \end{aligned}$$

(85)

Considering $\mathbf{H}^{A,B}_+$ for simplicity (and dropping the $+$ script), the data as above give rise to a convex compact quadrilateral $\Delta$ (determined by the affine lines $\ell _{\alpha _k}$ and $\ell _{\beta _k}$ introduced above via (80)) which is endowed with the canonical set $\{p_{\alpha _k}, p_{\beta _k}, k=0,\infty \}$ of normals (81). We take F be the union of all facets $\ell _{\alpha _k}=0$ and $\ell _{\beta ,k}=0$ for which $r_{\alpha ,k}=0$ and $r_{\beta ,k}=0$, and normalize the remaining normals by

$$\begin{aligned} e_{\alpha ,k}: = p_{\alpha _k}/r_{\alpha ,k}, \ e_{\beta ,k}:= p_{\beta _k}/r_{\beta ,k}. \end{aligned}$$

One can easily check that these become inward normals to $\Delta$ and that $\mathbf{H}^{A,B}$ verifies the boundary conditions (32)–(33) on $(\Delta , \mathbf{L}, F)$ if and only if (85) holds. Furthermore, as it is shown in [6], $\mathbf{H}^{A,B}$ gives rise to a solution of the Abreu equation (35) on $(\Delta ,\mathbf{L},F)$ iff A, B are polynomials of degree $\le 4$ which satisfy (79) and the positivity and boundary conditions (84)–(85).

Conversely, the following is established in [5].

Proposition 5.7

[5] Let $(\Delta , \mathbf{L})$ be a compact convex labelled quadrilateral in ${\mathbb {R}}^2$, and F the union of some of its facets. Suppose that $\Delta$ is neither a parallelogram nor a trapezoid whose extremal affine function $s_{(\Delta , \mathbf{L}, F)}$ is constant on the parallel facets $\Delta$. Then there exist real numbers $\alpha _k, \beta _k, r_{\alpha ,k}, r_{\beta ,k}$ $(k=0,\infty )$, subject to the inequalities

$$\begin{aligned} \beta _0< \beta _\infty< \alpha _0< \alpha _\infty , \qquad r_{\alpha ,0} \le 0 \le r_{\alpha ,\infty }, \quad r_{\beta ,0} \ge 0 \ge r_{\beta ,\infty }, \end{aligned}$$

and a quadratic q(z) satisfying $q(x,y)>0$ on $D=[\alpha _0,\alpha _\infty ]\times [\beta _0,\beta _\infty ]$, such that

• $\Delta$ is the image of D either under $(x_1^+, x_2^+)$ or $(x_1^-, x_2^-)$ in (80);

• For each facet $F_{\alpha ,k}$ of $\Delta$ obtained as the image of $x=\alpha _k$ under (80) (resp. $F_{\beta ,k}$ obtained as the image of $y=\beta _k$), which does not belong to F, $r_{\alpha ,k}\ne 0$ (resp. $r_{\beta ,k} \ne 0$) and the corresponding inward normal is $e_{\alpha ,k} = \frac{p_{\alpha _k}}{r_{\alpha ,k}}$ (resp. $e_{\beta ,k} = \frac{p_{\beta _k}}{r_{\alpha ,k}}$), where $p_{\alpha _k}$ and $p_{\beta _k}$ are the the normals defined by (81);

• For each facet $F_{\alpha ,k}$ of $\Delta$ (resp. $F_{\beta ,k}$) which belongs to F, the corresponding $r_{\alpha ,k}= 0$ (resp. $r_{\beta ,k} =0$);

• There exist polynomials P(z) of degree $\le 4$ and $\pi (z)$ of degree $\le 2$, satisfying $\langle q, \pi \rangle =0$, such that the A(z) and B(z) defined by (79) satisfy the boundary conditions (85) (but not necessarily the positivity condition (84)).

Furthermore, the corresponding $\mathbf{H}_{+}^{A,B}$ or $\mathbf{H}_-^{A,B}$ defined by (82) satisfies (35) and defines a solution $u_{A,B}\in {\mathcal {S}}(\Delta , \mathbf{L}, F)$ if and only if $(\Delta , \mathbf{L}, F)$ is stable. The latter condition is equivalent to (84).

As an illustration of the theory, let us again take $(\Delta , \mathbf{L})$ to be a trapezoid but not a parallelogram, and F to be either a facet which is not parallel to another facet or the union of two adjacent facets. We have shown in Sect.  5.3 that in this case the extremal affine linear function $s_{(\Delta ,\mathbf{L},F)}$ is not constant on the parallel facets of $\Delta$ and, therefore, the solution of (35) (if it exists) must be given by Proposition 5.7. On the other hand, we have the following:

Proposition 5.8

Let $(\Delta , \mathbf{L})$ be a labelled trapezoid corresponding to a Hirzebruch surface, and F be one facet, or the union of 2 adjacent facets. Then $(\Delta , \mathbf{L}, F)$ is stable.

Putting Propositions 5.7 and 5.8 together, we obtain

Corollary 5.9

Let $X= {\mathbb {F}}_m$ be the mth Hirzebruch surface and Z be the divisor consisting of a single fibre fixed by the ${\mathbb {T}}$ action, or the union of such a fibre with either the zero section or the infinity section. Then, $X\setminus Z$ admits a complete extremal Donaldson metric in each Kähler class of X,  which is not of Poincaré type.

The proofs of Proposition 5.8 and Corollary 5.9 are presented in Appendix B.

Example 5.10

In the light of Corollary  5.9, we use the explicit description of the extremal Donaldson metrics in order to determine their asymptotic behaviour in normal direction to Z.

The parametrization of a regular ambitoric metric by the data

$$\begin{aligned} \alpha _k, \beta _k, r_{\alpha ,k}, r_{\beta ,k}, q(z), A(z), B(z) \end{aligned}$$

as above is not effective: there is a natural $\mathrm{SL}(2, {\mathbb {R}})$ action on the space of degree 2 polynomials q(z), as well as a homothety freedom for the metric. This can be normalized by taking q(z) to be either 1, 2z or $z^2+1$ (see [6, Sect. 5.4]), thus referring to the corresponding ambitoric metric as being of parabolic, hyperbolic or elliptic type, respectively. Moreover, it is observed in [5, Sect. 5.4] that the solution corresponding to a trapezoid is given by a (positive) hyperbolic ambitoric metric, i.e.

$$\begin{aligned} \begin{aligned} g =&\frac{(x-y)}{(x+y)}\bigg (\frac{{\text {d}}x^2}{A(x)} + \frac{{\text {d}}y^2}{B(y)}\bigg ) \\&+ \frac{1}{(x-y)(x+y)^3}\bigg (A(x)({\text {d}}t_1 + y^2 {\text {d}}t_2)^2 + B(y)({\text {d}}t_1 + x^2 {\text {d}}t_2)^2\bigg ),\\ \omega =&\frac{{\text {d}}x\wedge ({\text {d}}t_1 + y^2 {\text {d}}t_2)}{(x+y)^2} + \frac{{\text {d}}y\wedge ({\text {d}}t_1 + x^2 {\text {d}}t_2)}{(x+y)^2}, \end{aligned} \end{aligned}$$

(86)

for $(x, y) \in [\alpha _0, \alpha _\infty ]\times [\beta _0, \beta _\infty ]$ with

$$\begin{aligned} \beta _{0}<\beta _{\infty }< \alpha _{0} < \alpha _{\infty }, \ \ \ \beta _{0} + \alpha _{0} >0 \end{aligned}$$

and polynomials $A(z)= \sum _{i=0}^4 a_i z^{4-i}$ and $B(z)=\sum _{i=0}^4 b_i z^{4-i}$ satisfying

$$\begin{aligned} a_0+b_0=a_2+b_2 = a_4+b_4=0, \end{aligned}$$

(87)

and the positivity and boundary conditions (84)–(85). The momentum coordinates of (86) then become

$$\begin{aligned} x_1= -\frac{1}{x+y}, \ x_2 = \frac{xy}{x+y} \end{aligned}$$

(88)

so that the image of the interval $[\alpha _0, \alpha _\infty ]\times [\beta _0, \beta _\infty ]$ under (88) is a quadrilateral $\Delta$ determined by the affine lines

$$\begin{aligned} \ell _{\alpha ,k}= -\alpha _k^2x_1 +x_2 - \alpha _k=0, \ \ \ell _{\beta ,k}= -\beta _k^2x_1 + x_2 - \beta _k =0, \ k=0, \infty , \end{aligned}$$

whose normals are $p_{\alpha ,k}=(-\alpha _k^2, 1)$ and $p_{\beta ,k}=(-\beta _k^2, 1),$ respectively. It follows that $\Delta$ is a trapezoid iff $\beta _\infty =-\beta _0=b>0$, see Figure 2.

Fig. 2

The Delzant polytope of a Hirzebruch surfaces (in blue) obtained by the hyperbolic ambitoric construction (Color figure online)

As observed in [21], each Hirzebruch surface ${\mathbb {F}}_m$ can be obtained from a labelled trapezoid $(\Delta , \mathbf{L})$ as above, by taking inward normals $e_{\alpha ,k}= p_{\alpha _k}/r_{\alpha ,k}$ and $e_{\beta ,k}=p_{\beta _k}/r_{\beta ,k}$ satisfying

$$\begin{aligned} e_{\beta ,0}= -e_{\beta ,\infty }, \ e_{\alpha ,\infty } + m e_{\beta , 0}= -e_{\alpha , 0}. \end{aligned}$$

Equivalently, the labelling $\mathbf{L} =\{L_{\alpha ,k}=\frac{1}{r_{\alpha ,k}} \ell _{\alpha ,k}, L_{\beta , k} = \frac{1}{r_{\beta , k}} \ell _{\beta , k}, k=0, \infty \}$ satisfies

$$\begin{aligned} r_{\beta ,0}=-r_{\beta , \infty }=r>0, \ r_{\alpha ,0}= \frac{r}{m} \bigg (\frac{\alpha _0^2-\alpha _\infty ^2}{\alpha _\infty ^2-b^2}\bigg ), \ r_{\alpha , \infty }= \frac{r}{m} \bigg ( \frac{\alpha _\infty ^2 -\alpha _0^2}{\alpha _0^2-b^2}\bigg ). \end{aligned}$$

(89)

The positive constant r is just a scale factor for the Kähler class and can be taken $r=1$. Thus, by considering the lattice generated by $e_{\alpha ,k}, e_{\beta ,k}$ as above, the corresponding labelled trapezoid corresponds to a toric Hirzebruch surface ${\mathbb {F}}_m$.

We now take $F_{\alpha , \infty }$ (defined by $\ell _{\alpha ,{\infty }}=0$) be the facet of $\Delta$ corresponding to a fibre of ${\mathbb {F}}_m$ fixed by the torus action. We are thus looking for extremal metrics given by (86) for polynomials

$$\begin{aligned} A(x)= -c(x-\alpha _0)(x-\alpha _\infty )^2(x-\alpha _3), \ B(y)= c(y-b)(y+b)(y^2 + py + q) \end{aligned}$$

(90)

where $0<b<\alpha _0< \alpha _\infty$ and $\alpha _3, c, p, q$ are real parameters (which we are going to express as functions of $(b, \alpha _{0}, \alpha _{\infty })$).

The extremality conditions (87) then read as

$$\begin{aligned} \begin{aligned}&\alpha _3(2\alpha _\infty + \alpha _0) + \alpha _\infty ^2 + 2\alpha _0\alpha _\infty = q- b^2\\&\alpha _3\alpha _\infty ^2\alpha _0= -q b^2 \end{aligned} \end{aligned}$$

(91)

from which we get

$$\begin{aligned} \begin{aligned} \alpha _3&=- \left( \frac{b^2 + \alpha _\infty ^2 + 2\alpha _0\alpha _\infty }{2\alpha _\infty + \alpha _0+ \frac{\alpha _\infty ^2\alpha _0}{b^2}}\right) , \\ q&= \frac{\alpha _\infty ^2\alpha _0}{b^2}\left( \frac{b^2 + \alpha _\infty ^2 + 2\alpha _0\alpha _\infty }{2\alpha _\infty + \alpha _0+ \frac{\alpha _\infty ^2\alpha _0}{b^2}}\right) . \end{aligned} \end{aligned}$$

(92)

From the boundary condition (85) at $\alpha _{0}$ we obtain

$$\begin{aligned} A'(\alpha _0)=-c(\alpha _0-\alpha _3)(\alpha _0-\alpha _\infty )^2 =-2r_{\alpha , 0}= \frac{2}{m} \left( \frac{\alpha _\infty ^2 -\alpha _0^2}{\alpha _\infty ^2-b^2}\right) \end{aligned}$$

so that we determine

$$\begin{aligned} \begin{aligned} c&= -\frac{2}{m} \left( \frac{(\alpha _{\infty } + \alpha _{0})}{(\alpha _{0}-\alpha _3)(\alpha _{\infty }-\alpha _{0})(\alpha _{\infty }^2 - b^2)}\right) \\&= -\frac{2}{m} \left( \frac{(\alpha _{\infty } + \alpha _{0})(2\alpha _{\infty } + \alpha _{0} + \frac{\alpha _{\infty }^2\alpha _{0}}{b^2})}{\big (\alpha _{0}^2 + \alpha _{\infty }^2 + 4\alpha _{0}\alpha _{\infty } + b^2 + \frac{\alpha _{0}^2\alpha _{\infty }^2}{b^2}\big )(\alpha _{\infty }-\alpha _{0})(\alpha _{\infty }^2 - b^2)}\right) . \end{aligned} \end{aligned}$$

(93)

Consider first the case when $F=F_{\alpha , {\infty }}$ consists of only one facet. The boundary conditions (85) at $\pm b$ read as

$$\begin{aligned} B'(b) =2cb(b^2 + pb + q) =2r_{\beta , \infty }=-2, \ B'(-b)=-2cb(b^2 - pb + q) = 2r_{\beta , 0}= 2. \end{aligned}$$

We then have $p=0$ and the additional relation $-1 = cb(b^2 + q)$ which can be used in order to express $\alpha _{0}$ as a function of $(\alpha _{\infty }, b)$. This last step, however, is not obvious (and is implicit) as $\alpha _{0}$ appears to be a real root of a polynomial of degree 4, which also needs to satisfy $0<b< \alpha _{0} < \alpha _{\infty }$. The existence of such a root is thus guaranteed by Propositions 5.7 and 5.8, so we shall not develop this step any further. We also notice that homotheties in (x, y) preserve the form (86) (but change A and B by scale) so we can assume $b=1$. Thus, on a fixed Hirzebruch surface ${\mathbb {F}}_m$ we obtain a one-dimensional family of complete extremal Kähler metrics (defined on ${\mathbb {F}}_m \setminus Z$) parametrized by $a=\alpha _{\infty }>1$, which is precisely the dimension of the Kähler cone of ${\mathbb {F}}_m$ modulo scales.

Notice that, by (92), the third root $\alpha _3$ of A is negative, and thus $\alpha _{\infty }$ has always multiplicity 2. Using (83) with $q(x,y)=x+y$ and A, B given by (90), we observe that up to smooth terms on $\Delta$, the symplectic potential is of the form

$$\begin{aligned} \begin{aligned} u&= \bigg (A \frac{(2\alpha _{\infty } -(x+y))}{(x+y)} + B L_{\alpha , {\infty }}\bigg ) \log L_{\alpha , {\infty }} + \frac{1}{2} \sum _{k=0, \infty }L_{\alpha ,k} \log L_{\alpha ,k} + L_{\beta ,k} \log L_{\beta , k} \\&= \big (A (-2\alpha _{\infty }x_1 -1) + B L_{\alpha , {\infty }}\big ) \log L_{\alpha , {\infty }} + \frac{1}{2} \sum _{k=0, \infty }L_{\alpha , k} \log L_{\alpha , k} + L_{\beta ,k} \log L_{\beta , k} \end{aligned} \end{aligned}$$

for some real constants $A \ne 0, B$. As $\alpha _{\infty }>0$, the affine function $\big (A (-2\alpha _{\infty } x_1 -1) + B L_{\alpha , {\infty }}\big )$ is not constant when restricted to the facet $F_{\alpha , \infty }$ (on this facet $x = \alpha _{\infty }$ and $y \in [\beta _{0}, \beta _{\infty }]$).

Similarly, when $F= F_{\alpha , \infty }\cup F_{\beta , \infty }$ say, we must have $y^2+py + q= (y-b)(y-\beta _3)$, so that $q= b\beta _3$ and $p= -(b + \beta _3)$, and from (92) we determine

$$\begin{aligned} \beta _3 = \frac{\alpha _\infty ^2\alpha _0}{b^3}\left( \frac{b^2 + \alpha _\infty ^2 + 2\alpha _0\alpha _\infty }{2\alpha _\infty + \alpha _0+ \frac{\alpha _\infty ^2\alpha _0}{b^2}}\right) . \end{aligned}$$

The above formula together with the inequalities $0<b<\alpha _{0} < \alpha _{\infty }$ show that $\beta _3 >b$, and thus b is double root of B(y). Similarly to the previous case, the symplectic potential of the extremal metric then takes the form

$$\begin{aligned} u = f_{\alpha , \infty } \log L_{\alpha , \infty } + f_{\beta , \infty }\log L_{\beta , \infty } + \frac{1}{2}\Big (L_{\alpha , 0} \log L_{\alpha , 0} + L_{\beta , 0} \log L_{\beta , 0}\Big ) + {\text {smooth}} \end{aligned}$$

where $f_{\alpha , \infty }$ and $f_{\beta , \infty }$ are affine functions in momenta which are not constant on the corresponding facets in F. One can also check that in this case too the condition (42) fails.

We notice also

Proposition 5.11

Let X be a Hirzebruch surface ${\mathbb {F}}_m$ or ${\mathbb {C}}P^1 \times {\mathbb {C}}P^1$, viewed as a toric variety endowed with a Kähler class $[\omega ]$ corresponding to a Delzant polytope $(\Delta , \mathbf{L})$. Let $Z\subset X$ be the toric divisor corresponding to the union F of 3 facets of $\Delta$. Then $(\Delta , F)$ is unstable and $X\setminus Z$ admits neither a Donaldson extremal Kähler metric nor an extremal Kähler metric of Poincaré type in $[\omega ]$.

Proof

The proof of instability of $(\Delta , F)$ follows from the arguments in Appendix B, see in particular Remark A.6. Thus $(\Delta , F)$ cannot admit a Donaldson metric by Proposition 4.9.

To rule out the existence of a (non-toric) complete extremal metric of Poincaré type, we can use [9, Thm. 5] which asserts that each rational curve corresponding to the pre-image of a facet in F must admit a complete Poincaré type extremal Kähler metric. Taking the ${\mathbb {C}}P^1$ corresponding to the facet in F which intersects the other two facets in F, we conclude that ${\mathbb {C}}P^1 \setminus (\{p\} \cup \{q\})$ admits a complete extremal metric of Poincaré type. But if it did, it would have to be scalar-flat, as the Poincaré–Futaki invariant vanishes and the average scalar curvature is 0. This would then violate the numerical constraint in [7, Thm. 1.2] for Poincaré type metrics of constant scalar curvature. So no such metric can exist. $\square$

Corollary 5.12

Conjecture 4.14, with respect to the stability of pairs introduced in Definition 4.10, holds true on the Hirzebruch surface ${\mathbb {F}}_m$ and on ${\mathbb {C}}P^1 \times {\mathbb {C}}P^1$.

Proof

Using Corollaries 5.3, 5.5 and 5.9 together with Proposition 4.9 at one hand, and Propositions 5.4 and 5.11 at the other hand, we conclude that the conditions (i) and (ii) of Conjecture 4.14 limit the possibilities as follows:

• (a) $X= {\mathbb {C}}P^1 \times {\mathbb {C}}P^1$ and Z is the union of the pre-image of one or two adjacent facets;

• (b) $X = {\mathbb {F}}_m$ and Z consist of either the zero section, the infinity section, or the the union of both;

• (c) $X= {\mathbb {F}}_m$ and Z consist of a single fibre or the union of such a fibre and either the zero section or the infinity section.

In the cases (a) and (b), there exists an explicit extremal Poincaré type metric by the Corollaries 5.3 and 5.5. In the case (c), there exists a Donaldson complete extremal metric which is not of Poincaré type, but in this case the condition (iii) of Conjecture 4.14 fails, as shown in Example 5.10. $\square$

5.5 Triangles as a Limiting Case

This case is already treated in [11] (see also [2]), but it can also be viewed as a limiting case of the ambitoric ansatz with $\pi =0$ (i.e. $A=-B$). The corresponding extremal metrics $(g_+, J_+, \omega _+)$ provide solutions of (35) on labelled triangles, and compactify on weighted projective planes as extremal Bochner–flat (i.e. self-dual) orbifold metrics, see [2, 11].

Indeed, putting $\pi =0$ and $P(z)= - \prod _{j=0}^{3} (z-\beta _j)$ with $\beta _0\le \beta _1<\beta _2 < \beta _3$ in (77) and (78), the degree 4 polynomial $B(y)=-P(y)$ is positive on $(\beta _1, \beta _2)$ while $A(x)=P(x)$ on $(\beta _2, \beta _3)$. When $\beta _0< \beta _1$, the Kähler metric $(g_+,\omega _+)$ defines an extremal Bochner-flat Kähler metric on a labelled simplex $(\Delta ,\mathbf{L})$, while taking $\beta _0=\beta _1$ gives rise to a solution to the Abreu equation (35) on a labelled simplex minus one facet (corresponding to the image of $y=\beta _1$ under the momentum map (80)). One can always take the two normals to form a basis of a lattice, so that the metric extends smoothly over the corresponding faces and has a complete end towards the third, see [21]. We get, in fact, one of the complete Bochner-flat metrics described in [11, Thm. 4.2.7] (see also [19]).

To see this explicitly, let us identify (by an affine map) the simplex $\Delta$ with the standard simplex of ${\mathbb {R}}^2$ (with vertices at (0, 0), (1, 0) and (0, 1)) and assume (without loss) that F corresponds to the facet defined by the equation $L_3(x)= a_3(1-x_1-x_2)=0$ whereas the other labels are $L_1(x)=a_1x_1$ and $L_2(x)=a_2x_2$ with $a_i>0$. The Bryant complete extremal Bochner-flat metric has symplectic potential in ${\mathcal {S}}(\Delta , \mathbf{L}, F)$, given by

$$\begin{aligned} u_{B} = \frac{1}{2} \Big (a_1x_1\log (x_1) + a_2 x_2\log (x_2) - (a_1x_1 + a_2x_2)\log (1-x_1-x_2)\Big ). \end{aligned}$$

(94)

If we take $(\Delta , \mathbf{L})$ be a labelled simplex and $F= F_1 \cup F_2$ the union of two facets, then by identifying $\Delta$ with the standard simplex of ${\mathbb {R}}^2$ and $F_i$ with the affine line $x_i=0, i=1,2$, respectively, one sees that the reflection along the line $x_1=x_2$ is a symmetry of $(\Delta , \mathbf{L}, F)$. By uniqueness, $s_{(\Delta ,\mathbf{L},F)}$ must be invariant under this reflection, i.e. $s_{(\Delta , \mathbf{L}, F)}= r(x_1+x_2) + c$ for some real numbers r, c. Using the definition (3.1) with $f= 1-(x_1 +x_2)$ (which vanishes on $F_3$), one gets $r=-2c$ for a real number c (which must be inverse proportional to the normal $e_3$). It follows that $s_{(\Delta , \mathbf{L}, F)}$ vanishes at the affine line parallel to $F_3$ and passing though the midpoint $m=(1/4,1/4)$ of its median d. Let $f_d$ be a simple crease function with crease along d and non-zero on the sub-triangle $\Delta ' \subset \Delta$ (cut from $\Delta$ by d). Two of the facets of $\Delta '$ inherit the measures of the facets of $\Delta$ and we put measure zero to the facet along d. Thus, $(\Delta ', {\text {d}}\nu _{\partial \Delta '})$ and $(\Delta , {\text {d}}\nu _{\partial \Delta })$ are equivalent under an affine transformation of ${\mathbb {R}}^2$. From the affine characterization of $s_{(\Delta , \mathbf{L}, F)}$, it follows that the extremal affine linear function of $\Delta '$ is a multiple of $s_{(\Delta , \mathbf{L}, F)}$; it is not hard to see (e.g. by using the definition (3.1) with $f= 1-(x_1 +x_2)$ and $f\equiv 1$) that the extremal affine linear function of $\Delta '$ equals to $s_{(\Delta , \mathbf{L}, F)}$. Thus, ${\mathcal {L}}_{(\Delta , \mathbf{L}, F)}(f_d)$ also computes the Donaldson–Futaki invariant of the affine linear function $f_d$ over $\Delta '$, and hence is zero. It follows that $(\Delta , \mathbf{L}, F)$ is unstable. We thus conclude

Theorem 5.13

Let $(\Delta ,\mathbf{L},F)$ be a labelled simplex in ${\mathbb {R}}^2$. Then $(\Delta , \mathbf{L}, F)$ is stable if and only if F consist of a single facet. In this case (35) admits an explicit solution $u_B$ in ${\mathcal {S}}(\Delta , \mathbf{L}, F)$ given by

$$\begin{aligned} u_{B} = \frac{1}{2}\Big ( L_1 \log L_1 + L_2\log L_2 - (L_1 + L_2)\log L_3\Big ), \end{aligned}$$

(95)

where $L_3$ vanishes on F. The corresponding metric (27) extends to the complete Bochner-flat metric on ${\mathbb {C}}^2$ found in [11]. In particular, ${\mathbb {C}}P^2\setminus {\mathbb {C}}P^1$ admits a complete extremal Donaldson metric, which is of Poincaré type (and conformal to the Taub-NUT metric).

Appendices

Appendix A: Proof of Theorem 4.18 and Proposition 4.19

1.1 Appendix A.1: Proof of Theorem 4.18

We follow the notation of Sects. 3 and 4. Thus, X is a smooth compact toric variety classified by the labelled Delzant polytope $(\Delta , \mathbf{L})$. We fix once and for all a ${\mathbb {T}}$-invariant Kähler metric $\omega _0$ on X with momentum map $\mu _0 : X \rightarrow \Delta \subset {{\mathfrak {t}}}^*$. To simplify the discussion, we can take $\omega _0$ to be the Kähler quotient metric on X obtained from the flat Kähler structure on ${\mathbb {C}}^d$ via the Delzant construction, see Eq. (28). We denote by $X^\circ =\mu _0^{-1}(\Delta ^0)$ the pre-image of the interior of $\Delta$, which is also the subspace of regular points for the action of ${\mathbb {T}}$. Complexifying the ${\mathbb {T}}$-action, we obtain a holomorphic action of the complex n-torus ${\mathbb {T}}^c =({\mathbb {C}}^{\times })^n$ with $X^{\circ }$ being the principal orbit for the ${\mathbb {T}}^c$-action. Choosing (once for all) a reference point $z^{\circ }\in X^{\circ }$, for each fixed point for the ${\mathbb {T}}$-action, corresponding to a vertex $v\in \Delta$, we introduce a $({\mathbb {C}}^{\times })^n$-equivariant chart ${\mathbb {C}}_v^n \cong {\mathbb {C}}^n$ as follows. Using a basis of ${{\mathfrak {t}}}$ obtained by the inward normals of the facets of $\Delta$ meeting at v, we identify ${\mathbb {T}}^c$ with $({\mathbb {C}}^{\times })^n$ and consider the equivariant map $\Phi _v : ({\mathbb {C}}^{\times })^n \rightarrow X^{\circ }$

$$\begin{aligned} \Phi _{v}\left( r_1e^{\sqrt{-1}t_1}, \ldots , r_n e^{\sqrt{-1}t_n}\right) : =\left( r_1e^{\sqrt{-1}t_1}, \ldots , r_n e^{\sqrt{-1}t_n}\right) \cdot z^{\circ }, \end{aligned}$$

where $r_ie^{\sqrt{-1}t_i} \in {\mathbb {C}}^{\times }$ stand for the polar coordinates on each factor. It follows by the holomorphic slice theorem that $\Phi _v : X^{\circ } \rightarrow ({\mathbb {C}}^{\times })^n$ extends equivariantly to a holomorphic embedding of ${\mathbb {C}}^n_v$ to X, thus defining an equivariant atlas of affine charts ${\mathbb {C}}^n_v$ of X (where v runs among the vertices of $\Delta$).

The theory of toric varieties (see e.g. [30]) yields that in such a chart, the (smooth) divisor Z corresponding to the pre-image under $\mu _0$ of a facet F meeting v has the equation $z_j=0$ where $(z_1, \ldots , z_n)=(r_1 e^{\sqrt{-1}t_1}, \ldots , r_n e^{\sqrt{-1}t_n})$ are the affine coordinates on ${\mathbb {C}}_v^n$. In what follows, we will suppose without loss that

$$\begin{aligned} Z\cap {\mathbb {C}}_v^n =\{(z_1, \ldots , z_n) \in {\mathbb {C}}^n : z_1 =0\}. \end{aligned}$$

(96)

To connect with the description (27) of the Kähler metric $\omega _0$, one needs to apply the Legendre transform to the strictly convex smooth function $u_0$ on $\Delta ^0$, given by (28). More precisely, if $x^0 = \mu _0(z^{\circ })\in \Delta ^0$ and $u\in \mathcal {S}(\Delta , \mathbf{L})$ is any symplectic potential, we let

$$\begin{aligned} y(x):={\text {d}}u(x) - {\text {d}}u(x^0) = (u_{,1}(x), \ldots , u_{,n}(x)) - (u_{,1}(x^0), \ldots , u_{,n}(x^0)) \end{aligned}$$

and define a smooth function $\varphi _u(y)$ by

$$\begin{aligned} \varphi _{u}(y) + u(x) = \sum _{i=1}^n y_ix_i. \end{aligned}$$

(97)

We notice the following elementary:

Lemma A.1

Let $\Omega$ be a non-empty bounded convex open subset of a finite-dimensional affine space ${{\mathfrak {t}}}^*$, and $u:\Omega \rightarrow {\mathbb {R}}$ a smooth strictly convex function such that $|{\text {d}}u|$ tends to $\infty$ near $\partial \Omega$, where $|\cdot |$ is any Euclidean norm on ${{\mathfrak {t}}}$. Then ${\text {d}}u$ is a diffeomorphism from $\Omega$ onto ${{\mathfrak {t}}}$.

Proof

Clearly, ${\text {d}}u(x)=\big (u_{,1}(x), \ldots , u_{,n}(x)\big )$ is a local diffeomorphism as its differential at x (represented by the matrix $(u_{,ij}(x))$ is invertible for all $x\in \Omega$ by the strict convexity assumption.

Moreover ${\text {d}}u$ is injective by using the convexity of $\Omega$ and the fact that ${\text {d}}u(p(t))$ is strictly monotone in t on each linear segment $p(t) \in \Omega$ (which again follows from the strict convexity of u).

Finally, ${\text {d}}u$ is surjective: This can be proven by checking that ${\text {d}}u(\Omega )$ is closed, and hence coincides with ${\mathbb {R}}^n$ as it is also non-empty and open. Indeed, let $(q_k)$ be a sequence in ${\mathbb {R}}^n\cong {{\mathfrak {t}}}$, with limit q, such that $q_k = {\text {d}}u(p_k)$ for $p_k\in \Omega$ for all $k\ge 0$. There is a subsequence, still denoted by $(p_k)$, converging to some $p\in \overline{\Omega }$. Now if $p\in \partial \Omega$, then by assumption, $|q_k|=|{\text {d}}u(p_k)|\rightarrow \infty$, a contradiction, and thus $p\in \Omega$, and $q={\text {d}}u(p)$. $\square$

It follows that for each $u\in \mathcal {S}(\Delta , \mathbf{L})$, we have a ${\mathbb {T}}^c$-equivariant biholomorphism $\Phi _{u} : (\Delta ^0 \times {\mathbb {T}}, J_u) \rightarrow ({\mathbb {C}}^{\times })_v^n \cong X^{\circ },$ given by

$$\begin{aligned} \Phi _{u}(x_1, \ldots , x_n, t_1, \ldots , t_n) := (e^{y_1 + \sqrt{-1} t_1}, \ldots , e^{y_n + \sqrt{-1} t_n}), \end{aligned}$$

(98)

where, we recall, $y(x) = {\text {d}}u(x) - {\text {d}}u(x^0)$ is a diffeomorphism from $\Delta ^0$ to ${\mathbb {R}}^n$ by virtue of Lemma A.1, and $J_u$ is the ${\mathbb {T}}$-invariant $\omega _0$-compatible complex structure corresponding to $u\in \mathcal {S}(\Delta , \mathbf{L})$. The central fact in this theory is the following identity on $({\mathbb {C}}^{\times })^n_v$ (see [30]):

$$\begin{aligned} \omega _u:=(\Phi _{u}^{-1})^*(\omega _0) = dd^c \Big (\big (\varphi _u(\log |z_1|, \ldots , \log |z_n|)\big )\Big ), \end{aligned}$$

(99)

where $(z_1, \ldots , z_n)$ are the complex coordinates associated to the chart ${\mathbb {C}}_v^n,$ and $d^c$ is taken with respect to the standard complex structure. The fact that $u\in \mathcal {S}(\Delta , \mathbf{L})$ guarantees the smooth extension of the right-hand side to a positive definite (1, 1)-form on ${\mathbb {C}}^n_v$.

Let us now suppose $u\in \mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ (instead of being in $\mathcal {S}(\Delta , \mathbf{L})$). It is easily checked that such a u still verifies the condition that $|{\text {d}}u|$ tends to $\infty$ near $\partial \Delta$, so that, by using Lemma A.1, (98) and (99), we obtain a Kähler metric $\omega _u$ on $({\mathbb {C}}^{\times })^n_v,$ which can be written as

$$\begin{aligned} \omega _u = \omega _{0} + dd^c \varphi , \end{aligned}$$

(100)

where $\omega _0$ is the (globally defined on ${\mathbb {C}}^n_v$) Kähler metric corresponding to (28) and

$$\begin{aligned} \varphi (z_1, \ldots , z_n) :=(\varphi _u-\varphi _{u_0})(\log |z_1|, \ldots , \log |z_n|) \end{aligned}$$

(101)

is a smooth function on $({\mathbb {C}}^{\times })^n_v \subset {\mathbb {C}}^n_v$. We notice that through this identification (which depends upon u!), $Z\cap {\mathbb {C}}^n_v$ still corresponds to a hyperplane in the ${\mathbb {C}}^n_v$ affine chart, as it follows from the following:

Lemma A.2

Let $u_0$ be the symplectic potential in $\mathcal {S}(\Delta , \mathbf{L})$ given by (28) and let $u \in \mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$. Then,

$$\begin{aligned} ({\text {d}}u_0)^{-1} \circ {\text {d}}u :\Delta ^0 \longrightarrow \Delta ^0 \end{aligned}$$

extends continuously to a homeomorphism of $\Delta$, inducing a diffeomorphism on every open face of the polytope $\Delta$ and preserving its vertices.

Proof

The proof is elementary. Let $p\in \partial \Delta$, and consider some sequence $(p_k)$ in $\Delta ^0$ converging to p. We want to see that

• (a) $\lim _{k\rightarrow \infty } (({\text {d}}{u_0})^{-1} \circ {\text {d}}u) (p_k)$ lies in the same open facet as p (or equals p if p is a vertex of $\Delta$);

• (b) this limit does not depend on the sequence $(p_k)$.

The claimed regularity of $({\text {d}}{u_0})^{-1} \circ {\text {d}}u$ will follow from the proof (a) and (b) above.

Let us assume that $p \in \mathbf {f} := (F_{i_1}\cap \cdots \cap F_{i_{\ell }})\backslash (F_{i_{\ell +1}}\cup \cdots \cap F_{i_{d}}$), $\ell \le n$, where $i_1<\cdots <i_{\ell }$ and $\{i_1, \ldots , i_{d} \}= \{1, \ldots ,d\}$, and also that $F_{i_1}\cap \cdots \cap F_{i_n}$ is not empty.

As already noticed, $\big ({\text {d}}u(p_k)\big )$ tends to $\infty$ in $\mathfrak {t}$; thus, as ${\text {d}}u_0$ is a diffeomorphism $\Delta \rightarrow \mathfrak {t}$ whose norm tends to $\infty$ near $\partial \Delta$,

$$\begin{aligned} (\underline{p}{}_k) := \big (({\text {d}}{u_0})^{-1}\circ {\text {d}}u (p_k) \big ) \end{aligned}$$

tends to $\partial \Delta$. We now prove that any limit point of $(\underline{p}{}_k)$ must belong to $\mathbf {f}$. For this, observe that if $(\overline{p}_k)$ tends to $\partial \Delta$, then $(\overline{p}_k)$ tends to $\mathbf {f}$ if and only if $u_{0,1}(\overline{p}_k),\ldots , u_{0,\ell }(\overline{p}_k)$ tend to $-\infty$ while $u_{0,\ell +1}(\overline{p}_k), \ldots , u_{0,n}(\overline{p}_k)$ remain bounded (we have set $u_{0,i}(x) = \frac{\partial u_0}{\partial x_i}(x)$). We thus want to prove that, as $k\rightarrow \infty$,

$$\begin{aligned} \left\{ \begin{aligned}&{u}_{0,1}(\underline{p}{}_k), \ldots , {u}_{0,\ell }(\underline{p}{}_k) \longrightarrow -\infty , \\&{u}_{0,\ell +1}(\underline{p}{}_k), \ldots , {u}_{0,n}(\underline{p}{}_k) \, = {O}(1), \end{aligned} \right. \end{aligned}$$

or equivalently

$$\begin{aligned} \left\{ \begin{aligned}&u_{,1}(p_k), \ldots , u_{,\ell }(p_k) \longrightarrow -\infty , \\&u_{,\ell +1}(p_k), \ldots , u_{, n}(p_k) \, = {O}(1). \end{aligned} \right. \end{aligned}$$

(102)

Since $u\in \mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$, up to a smooth $\mathfrak {t}$-valued function near p we have

$$\begin{aligned} u_{, j} =\left\{ \begin{aligned}&\Big (-\frac{\alpha }{L_1}+\beta \log L_1 \Big )\quad \text { if }\, j=1, \\&\frac{1}{2}\log L_{j}\quad \qquad \qquad \qquad \text { if }\, j\ge 2, \end{aligned} \right. \end{aligned}$$

which yield the estimates (102) and concludes the point (a) of our proof.

We now address (b). Let $p_k \rightarrow p$ and ${\underline{p}}{}_k\rightarrow {\underline{p}}$. Our task is to prove that $\underline{p}$ does not depend on $(p_k)$; if p is a vertex, this already follows from (a), so we assume that $\mathbf {f}$ is an open face of $\Delta$. Letting $u_{0, \mathbf{f}}: = (u_0)_{|_\mathbf{f}},$ $u_{\mathbf {f}}:=( u + \alpha \log L_1)_{|_{\mathbf {f}}}$ if $\mathbf {f}\subset F=F_1$ and $u_{\mathbf {f}}=u_{|_{\mathbf {f}}}$ if $\mathbf {f}\not \subset F$, the definitions of the spaces $\mathcal {S}(\Delta , \mathbf{L})$ and $\mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ ensure that ${u}_{0,\mathbf {f}}$ and $u_{\mathbf {f}}$ are strictly convex on $\mathbf {f}$. With the notations above, observe that the ${u}_{0,j}$, $j=\ell +1, \ldots , n$, are smooth in a neighbourhood of $\mathbf {f}$, and ${u}_{0, j}= ({u}_{0,\mathbf {f}})_{,j}$ along $\mathbf {f}$; similarly, the $u_{,j}$ are smooth around $\mathbf {f}$ and $u_{,j} = (u_{\mathbf {f}})_j$ along $\mathbf {f}$. In this way, letting $k \rightarrow \infty$ in the equality ${\text {d}}{u_0}(\underline{p}{}_k) = {\text {d}}u(p_k),$ we obtain

$$\begin{aligned} ({\text {d}}{u}_{0,\mathbf {f}})(\underline{p}) = ({\text {d}}u_{\mathbf {f}})(p). \end{aligned}$$

Using the strict convexity of ${u}_{0,\mathbf {f}}$ and $u_{\mathbf {f}}$ on $\mathbf {f}$, and that the norms of their differentials tend to $\infty$ near $\partial \mathbf {f}$, we conclude that ${\text {d}}{u}_{0,\mathbf {f}}$ and ${\text {d}}u_{\mathbf {f}}$ are diffeomorphisms $\mathbf {f}\rightarrow (\mathfrak {t}/\mathfrak {t}_{\mathbf {f}})^*$ (see Lemma A.1 above), and thus,

$$\begin{aligned} \underline{p}{} = ({\text {d}}{u}_{0,\mathbf {f}})^{-1} \circ {\text {d}}u_{\mathbf {f}} (p) \end{aligned}$$

does not depend on $(p_k)$.

This completes the proof of Lemma A.2. $\square$

The general theory [30] (which uses local arguments around the pre-image of each face) ensures that $\varphi (z)$ extends smoothly over ${\mathbb {C}}^n_v \setminus ({\mathbb {C}}^n_v \cap Z)$, and that $\omega _u$ defines a Kähler metric on $X \setminus Z$. We shall thus focus our analysis on ${\mathbb {C}}^n_v$ in order to understand the behaviour of $\omega _u$ near $Z\cap {\mathbb {C}}^n_v$, see (96).

Let $p\in Z \cap {\mathbb {C}}^n_v$. We shall consider the following limiting cases:

• (a) $p=(0,\ldots , 0)$ corresponds to the vertex v of $\Delta$;

• (b) $\mu _0(p)$ belongs to the relative interior of F, i.e. in the chart ${\mathbb {C}}^n_v$, p has coordinates $(0, z_2, \ldots , z_n)$ with $z_j \ne 0, j=2, \ldots , n$.

The case when $\mu _0(p)$ belongs to an $\ell$-codimensional face of F with $1\le \ell \le n-1$ can be dealt with by combining the arguments for the cases (a) and (b). We shall also assume at first that $\beta =0$.

Case (a): $p=(0, \ldots , 0)$. We can assume without loss that the vertex $\mu _0(p)=v$ of $\Delta$ is at the origin of ${{\mathfrak {t}}}^*\cong {\mathbb {R}}^n$, i.e. $L_j (x)= x_j, j=1, \ldots , n$. We thus have, near $0 \in \Delta$,

$$\begin{aligned} u(x)=-\alpha \log (x_1)+\frac{1}{2}\left( \sum _{j=2}^m x_j\log (x_j) - x_j\right) + w(x) \end{aligned}$$

(103)

with w(x) smooth on ${\mathbb {R}}^n$. We can further modify u by adding an affine linear function (which does not change neither the induced Kähler metric nor the belonging of u to $\mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$) so that ${\text {d}}u(x^0)=0$. It then follows that in the chart ${\mathbb {C}}^n_v$ the functions $y_j = u_{,j } = \log |z_j|$ are given by

$$\begin{aligned} \begin{aligned} y_1&= \log |z_1|=-\frac{\alpha }{x_1} + w_{,1}(x),\\ y_j&= \log |z_j| = \frac{1}{2}\log x_j + w_{,j}(x), \ j=2, \ldots , n, \end{aligned} \end{aligned}$$

or equivalently,

$$\begin{aligned} \left\{ \begin{aligned}&-\frac{1}{\log |z_1|} = f_1(x) \quad , \\&|z_j|^2= f_j (x) \quad \text {for }j=2,\ldots , n, \end{aligned} \right. \end{aligned}$$

(104)

with $f_1(x) =\frac{x_1}{\alpha - x_1w_{,1}(x)}$ and $f_j(x)=x_je^{2w_{,j}(x)}, j=2, \ldots , n.$ We want to use (104) in order to express the momentum coordinates $x=(x_1, \ldots , x_n)$ of $\omega _u$ in terms of $|z_j|'s$. To this end, we notice that $f(x)=(f_1(x), \ldots , f_n(x))$ extends smoothly near the origin $v=0\in \Delta$. (Similarly, the LHS of (104) extends continuously on ${\mathbb {C}}^n_v$ by letting $\frac{-1}{\log |z_1|}$ be 0 along $z_1=0$.) Computing the Jacobian of f(x) at 0, we conclude that the local inverse $h(\zeta )$ of f(x) is defined on a small neighbourhood of $0\in {\mathbb {R}}^n$ and has the form

$$\begin{aligned} h(\zeta )=\big (\alpha \zeta _1\tilde{h}_1(\zeta ), \zeta _2\tilde{h}_2(\zeta ), \ldots , \zeta _n\tilde{h}_n(\zeta )\big ), \end{aligned}$$

(105)

with $\tilde{h}_1(\zeta ),\ldots , \tilde{h}_n(\zeta )$ smooth and non-vanishing near 0. It thus follows from (104) that

$$\begin{aligned} \left\{ \begin{aligned} x_1&=\Big (\frac{-\alpha }{\log |z_1|}\Big )\tilde{h}_1\Big (\frac{-1}{\log |z_1|}, |z_2|^2,\ldots ,|z_n|^2\Big ),\\ x_j&= |z_j|^2\tilde{h}_j\Big (\frac{-1}{\log |z_1|},|z_2|^2,\ldots ,|z_m|^2\Big ) \qquad \text {for }j=2,\ldots , n. \end{aligned} \right. \end{aligned}$$

(106)

By (97), and using that $y_j = \log |z_j|$ together with (106), we find that

$$\begin{aligned} \begin{aligned} \varphi _u(z) =&\frac{1}{2}\sum _{j=1}^n \log (|z_j|^2)x_j - u(x) \\ =&- \alpha \log \big (-\log |z_1|\big ) + W\Big (\frac{-1}{\log |z_1|},|z_2|^2,\ldots ,|z_n|^2\Big ) \end{aligned} \end{aligned}$$

(107)

where

$$\begin{aligned} W(\zeta ):= \alpha \log \big [\alpha \tilde{h}_1(\zeta )\big ]-\alpha \tilde{h}_1(\zeta ) -\frac{1}{2}\left( \sum _{j=2}^m \zeta _j\tilde{h}_j(\zeta )\big (\log \tilde{h}_j(\zeta ) -1\big )\right) -w\big [h(\zeta )\big ] \end{aligned}$$

(108)

for w(x) defined in (103). Thus, $W(\zeta )$ is a smooth function near 0.

A similar (and well-established) argument using $u_0$ instead of u shows that $\varphi _{u_0}$ can be viewed as a smooth function $\psi _0(z)$ on ${\mathbb {C}}^n_v$, so that the relative potential $\varphi (z)$ in (101) is written as

$$\begin{aligned} \varphi (z) = - \alpha \log \big (-\log |z_1|\big ) +\psi _0(z) + W\left( \frac{-1}{\log |z_1|},|z_2|^2,\ldots ,|z_n|^2\right) \end{aligned}$$

and thus has the required behaviour of a Poincaré type potential near $Z=\{z_1=0\}$, see Definition 1.1.

We now examine the asymptotic behaviour of the Kähler form $\omega _u=dd^c \varphi _u$ near the origin $0 \in Z\cap {\mathbb {C}}^n_v$. Using (107), we find that

$$\begin{aligned} \omega _u = dd^c\varphi _u = 2\alpha \frac{\sqrt{-1}{\text {d}}z_1\wedge {\text {d}}\overline{z_1}}{|z_1|^2\log ^2(|z_1|^2)} +\Omega +\eta , \end{aligned}$$

(109)

where $\Omega +\eta = dd^c\big [W\big (\frac{-1}{\log (|z_1|)},|z_2|^2,\ldots ,|z_n|^2\big )\big ]$ are given by

$$\begin{aligned} \left\{ \begin{aligned} \Omega =&\sum _{j=2}^n W_{j}\Big (\frac{-1}{\log (|z_1|},|z_2|^2,\ldots ,|z_n|^2\Big ) \, \sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_j} \\&+\sum _{j,k=2}^n W_{jk}\Big (\frac{-1}{\log (|z_1|)},|z_2|^2,\ldots ,|z_n|^2\Big ) \, \overline{z_j}z_k \sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_k}, \\ \eta =&W_1\Big (\frac{-1}{\log (|z_1|)},|z_2|^2,\ldots ,|z_n|^2\Big )\frac{ \sqrt{-1}{\text {d}}z_1\wedge {\text {d}}\overline{z_1}}{|z_1|^2\log ^2(|z_1|)}\\&+\sum _{j=2}^n W_{1j}\Big (\frac{-1}{\log (|z_1|)},|z_2|^2,\ldots ,|z_n|^2\Big ) \Big (\frac{z_j\sqrt{-1}{\text {d}}z_1\wedge {\text {d}}\overline{z_j}}{z_1\log (|z_1|)} +\frac{\overline{z_j} \sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_1}}{\overline{z_1}\log (|z_1|)}\Big ), \end{aligned} \right. \end{aligned}$$

(110)

and we have put

$$\begin{aligned} \begin{aligned} W_1(\zeta )&=\zeta _1^2\Big (\frac{\partial ^2W}{\partial \zeta _1 \partial \zeta _1}\Big )(\zeta ) +\frac{1}{2}\zeta _1\Big (\frac{\partial W}{\partial \zeta _1}\Big )(\zeta ), \ \ W_j(\zeta ) = \Big (\frac{\partial W}{\partial \zeta _j}\Big )(\zeta ), \ j\ge 2;\\ W_{1j}(\zeta )&=-\zeta _1\Big (\frac{\partial ^2 W}{\partial \zeta _1 \partial \zeta _j}\Big )(\zeta ), \ j\ge 2; \ \ W_{jk}(\zeta ) = \Big (\frac{\partial ^2 W}{\partial \zeta _j \partial \zeta _k}\Big )(\zeta ), \ j,k\ge 2. \end{aligned} \end{aligned}$$

Since $W_k(\zeta ), W_{pq}(\zeta )$ are smooth, $||\nabla ^{s}\eta ||={O}\big (\frac{1}{\log (|z_1|)}\big )$ near Z for all $s\ge 0$, where $\nabla$ is the Levi-Civita connection of the model Poincaré type metric

$$\begin{aligned} \omega _\mathrm{mod}=\frac{\sqrt{-1}{\text {d}}z_1\wedge {\text {d}}\overline{z_1}}{|z_1|^2\log ^2(|z_1|^2)}+\sum _{j=2}^n \sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_j} \end{aligned}$$

on ${\mathbb {C}}^n_v\backslash Z,$ and the norms are computed with the help of $\omega _\mathrm{mod}$.

It follows from (109)–(110) that $\omega _u$ has the Poincaré type behaviour in the normal $z_1$-direction, as well as in the $(z_1,z_j)$-directions for $j\ge 2$.

We are thus left to examine the metric over the hyperplane $z_1=0$. Letting $\tilde{W}_j(\zeta )$ and $\tilde{W}_{jk}(\zeta )$ be the smooth functions determined near 0 by

$$\begin{aligned} W_j(\zeta )=W_j(0,\zeta _2,\ldots ,\zeta _n)+\zeta _1\tilde{W}_j(\zeta ); \, \, W_{jk}(\zeta )=W_{jk}(0,\zeta _2,\ldots ,\zeta _n)+\zeta _1\tilde{W}_{jk}(\zeta ), \end{aligned}$$

one has the decomposition $\Omega =\Omega _0+\varepsilon$, where

$$\begin{aligned} \left\{ \begin{aligned} \Omega _0=&\sum _{j=2}^n W_{j}(0,|z_2|^2,\ldots ,|z_n|^2) \,\sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_j}\\&+\sum _{j,k=2}^n W_{jk}(0,|z_2|^2,\ldots ,|z_n|^2) \,\overline{z_j}z_k \sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_k},\\ \varepsilon =&\frac{-1}{\log (|z_1|)}\sum _{j=2}^n \tilde{W}_{j}\Big (\frac{-1}{\log (|z_1|)},|z_2|^2,\ldots ,|z_n|^2\Big ) \,\sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_j} \\&-\frac{1}{\log (|z_1|)}\sum _{j,k=2}^n \tilde{W}_{jk}\Big (\frac{-1}{\log (|z_1|)},|z_2|^2,\ldots ,|z_n|^2\Big ) \,\overline{z_j}z_k \sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_k}. \end{aligned} \right. \end{aligned}$$

(111)

We notice that $\Omega _0$ is smooth around the origin, whereas $\varepsilon$ satisfies, for all $s\ge 0$, $||\nabla ^{s}\varepsilon ||={O}\big (\frac{1}{\log (|z_1|)}\big )$ near Z (with covariant derivatives and norms taken with respect to the model Poincaré metric $\omega _\mathrm{mod}$). Computing the value of $\Omega _0$ at $z=0$, we get

$$\begin{aligned} (\Omega _0)_{|_{z=0}} = \sum _{j=2}^m W_j(0) \, \sqrt{-1}{\text {d}}z_j\wedge {\text {d}}\overline{z_j}. \end{aligned}$$

Using (108), we have

$$\begin{aligned} \begin{aligned} W_j(0)&= \Big (\frac{\partial W}{\partial \zeta _j} \Big )(0) \\&= -\frac{1}{2}\tilde{h_j}(0)\log \big (\tilde{h_j}(0)\big )- \Big (\frac{\partial h_j}{\partial \zeta _j}\Big ) (0)\Big (w_{,j}(0) -\frac{1}{2}\Big ) \\&= -\frac{1}{2}\tilde{h}_j(0)\log \big (\tilde{h}_j(0)\big )- \tilde{h}_j(0)\Big (w_{,j}(0)-\frac{1}{2}\Big ). \end{aligned} \end{aligned}$$

(112)

By the definition (105) (and inverting the diagonal Jacobian of f at 0) we have that for any $j=2, \ldots , n$, ${\tilde{h}}_j (0)= e^{-c_j}$ where $c_j=2w_{,j}(0)$. Substituting back to (112), we conclude

$$\begin{aligned} W_j(0)= e^{-c_j}\Big (-\frac{1}{2}(-c_j) - \frac{c_j-1}{2} \Big ) = \frac{e^{-c_j}}{2} >0, \end{aligned}$$

thus showing the positivity of $\omega _u$ in the directions parallel to Z.

Case 2: $p=(0, z_2, \ldots , z_n)$ with $z_j \ne 0$ for $j=2, \ldots , n$. Now

$$\begin{aligned} u(x)=-\alpha \log x_1 +w(x) \end{aligned}$$

with w(x) smooth in a neighbourhood of $x_p=\mu (p) \in F^0$. By assumption, u is strictly convex on $\Delta ^0$, and $u_{F}= w_{|_F}$ is strictly convex on $F^0$. The relations (104) now become

$$\begin{aligned} \left\{ \begin{aligned}&\frac{-1}{\log (|z_1|)} = f_1(x), \\&|z_j|^2 = f_j (x)\quad \text {for }j=2,\ldots , n, \end{aligned} \right. \end{aligned}$$

(113)

with $f_1(x)= \frac{x_1}{\alpha -x_1w_{,1}(x)}$, and $f_j(x)=e^{2w_{,j}(x)}$, $j=2,\ldots ,n$. Using the strict convexity of w along $F^0=\{x_1=0\}\cap \Delta$, we see that $f(x)=(f_1(x), \ldots , f_n(x))$ is smooth and locally invertible around $x_p=(0, b_2, \ldots , b_n)$. We denote by $h(\zeta )=(h_1(\zeta ), \ldots , h_n(\zeta ))$ the local inverse of f around $x_p$, which must be of the form

$$\begin{aligned} h(\zeta )= \big (\alpha \zeta _1\tilde{h}_1(\zeta ), \tilde{h}_2(\zeta ), \ldots , \tilde{h}_{n}(\zeta )\big ) \end{aligned}$$

with ${\tilde{h}}_1(0) >0$. Thus,

$$\begin{aligned} x = h\left( \frac{-1}{\log |z_1|},|z_2|^2,\ldots ,|z_n|^2 \right) , \end{aligned}$$

(which extends along $z_1=0$ near p). We obtain again

$$\begin{aligned} \varphi _u(z)= -\alpha \log \big (-\log |z_1|\big ) +W\left( \frac{-1}{\log |z_1|},|z_2|^2,\ldots ,|z_m|^2\right) , \end{aligned}$$

(114)

with W smooth and given by

$$\begin{aligned} W(\zeta ) =&\alpha \log \big (\alpha \tilde{h_1}(\zeta )\big )-\alpha \tilde{h_1}(\zeta ) +\frac{1}{2}\sum _{j=2}^m \tilde{h_j}(\zeta )\log \zeta _j -w\big [h(\zeta )\big ], \end{aligned}$$

(notice that $\zeta _j(p)=|z_j(p)|^2>0$ for $j=2,\ldots , n$). Thus,

$$\begin{aligned} \varphi (z)= \varphi _u(z) - \varphi _{u_0}(z)= -\alpha \log (-\log |z_1|)- \psi (z)+ W\left( \frac{-1}{\log |z_1|},|z_2|^2,\ldots ,|z_n|^2\right) , \end{aligned}$$

with $\psi$ and W are smooth near p. Consequently, $\varphi$ has the right asymptotics near p.

We now address the positivity of $\omega _u$ near p. Writing

$$\begin{aligned} \omega _u = dd^c\varphi _u =\frac{2\alpha \sqrt{-1}{\text {d}}z_1\wedge {\text {d}}\overline{z_1}}{|z_1|^2\log ^2(|z_1|^2)}+ \Omega +\eta , \end{aligned}$$

(115)

with $\Omega +\eta =dd^c\big (W\big (\frac{-1}{\log |z_1|},|z_2|^2,\ldots ,|z_n|^2\big )\big )$ given by (110), we have that $\eta ={O}\big (\frac{1}{\log |z_1|}\big )$ at any order; it is thus enough to show the positivity of $\Omega$ in (115). Decomposing $\Omega =\Omega _0+\varepsilon$ with $\varepsilon =O\big (\frac{1}{\log |z_1|}\big )$ as in (111), we need to establish the positivity of $(\Omega _0)_p$. By its very definition, $\Omega _0|_{Z}=dd^c\Big (W(0,|z_2|^2,\ldots ,|z_n|^2)\Big )$. A careful examination of the definition of W reveals that, up to additive pluriharmonic terms $\log |z_j|^2$, $W(0,|z_2|^2,\ldots ,|z_n|^2)$, seen as a function on the hypersurface $\{z_1=0\}$, coincides with the Kähler potential corresponding to the Legendre transform (97) of the strictly convex function $u_F(x) = \alpha \log x_1 + u(x)= w(x)$ restricted to the relative interior of F. Thus, $(\Omega _0)_p>0$.

We finally comment on the case when $u\in \mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ with $\beta \ne 0.$ The main difficulty is that the equations (104) hold with

$$\begin{aligned} f_1(x)= \frac{x_1}{\alpha -\beta x_1\log x_1 - x_1(\beta +w_{,1}(x))} \end{aligned}$$

(116)

which is no longer smooth (nor even $C^2$) around $x_1=0$.

One way to bypass this difficulty is to use a suitable change of variables. We detail below the case $n=1$, for the general case is treated similarly by considering the change of variables with respect to $|z_1|^2$ and leaving the variables $|z_j|^2, j =2, \ldots , n$ unchanged.

We set

$$\begin{aligned} s:=\frac{-1}{\log x_1}, \ \ t_1:=\log (-\log |z_1|), \end{aligned}$$

so that the first equation in (104) becomes

$$\begin{aligned} \frac{1}{t_1} = \frac{s}{1+s\log \big (\alpha +\beta e^{-1/s}/s + e^{-1/s}(\beta +w_{,1}(e^{-1/s}))\big )} =: f(s). \end{aligned}$$

(117)

As $w_{,1}$ is smooth in a neighbourhood of 0 and the functions

$$\begin{aligned} s\longmapsto \left\{ \begin{aligned}&0&\text { if }s\le 0,\\&e^{-1/s}&\text { if }s> 0, \end{aligned} \right. \qquad \text {and} \qquad s\longmapsto \left\{ \begin{aligned}&0&\text { if }s\le 0,\\&\frac{1}{s} e^{-1/s}&\text { if }s > 0, \end{aligned} \right. \end{aligned}$$

are smooth on ${\mathbb {R}}$, we can extend f(s) as a smooth function in a neighbourhood of 0 by letting $f(s)=\frac{s}{1-s\log \alpha }$ for $s\le 0$.

As $\partial _sf(0)= 1$, we get that $s=h(\frac{1}{t_1})$ for some h smooth around 0, satisfying $h(0)=0$, $h'(0)=1$. Thus, $s=\frac{1}{t_1}\big (1+\frac{\gamma }{t_1}+{\tilde{h}}(\frac{1}{t_1})\big )$ for some constant $\gamma$, and $\tilde{h}$ a smooth function vanishing at order 2 at 0. In fact, one must have $\gamma =\log \alpha$, and thus

$$\begin{aligned} \frac{1}{s}=t_1-\log \alpha +H\Big (\frac{1}{t_1}\Big ), \end{aligned}$$

with H a smooth function vanishing at order (at least) 1 at 0. We can be more precise, and rewrite (117) as

$$\begin{aligned} \frac{1}{t_1}=\frac{1}{1/s+\log \alpha +P(s)}, \quad \text {i.e.} \quad \frac{1}{s}=t_1-\log \alpha -P(s) \end{aligned}$$

with $P(s)=\log \big (1+(\beta /\alpha )e^{-1/s}/s+e^{-1/s}(\beta +w_{,1}(e^{-1/s}))/\alpha \big )$. Replacing 1 / s with $t_1-\log \alpha +H\big (\frac{1}{t_1}\big )$ in the explicit expression of P(s), we see by induction that $H(\frac{1}{t_1})$, as well as its derivatives with respect to $t_1$ at any order, are ${O}(t_1e^{-t_1})$ when $t_1\rightarrow +\infty$. Therefore,

$$\begin{aligned} x= e^{-1/s} = \exp \big (-t_1 +\log \alpha + P(s)\big ) = \frac{-\alpha }{\log |z_1|} \big (1+O(t_1e^{-t_1})\big ), \end{aligned}$$

at any order with respect to differentiation in $t_1$. In particular, $x=\frac{-\alpha }{\log |z_1|} + {O}(t_1e^{-2t_1})$ (instead of ${O}(e^{-2t_1})$ in the case $\beta =0$) at any order with respect to $\frac{|{\text {d}}z_1|^2}{|z_1|^2\log ^2(|z_1|^2)}$. We thus get for the relative potential (101)

$$\begin{aligned} \varphi = -\log \big (-\log |z_1|\big ) + \log \alpha + O(t_1e^{-t_1}), \end{aligned}$$

and recover the asymptotic behaviour $\omega _u$ near Z with arguments identical with the ones in the case $\beta =0$. This ends the proof of Theorem 4.18.

1.2 Appendix A.2: Proof of Proposition 4.19

The arguments are local, near a point $p\in \partial \Delta$, and not materially different from ones in [4, §1.3]. In fact, we need to only consider the case when $p\in F=F_1$ (otherwise the result follows from [4]). To this end, we fix a vertex $v\in F$ of $\Delta$, which without loss can be taken to be at the origin of ${{\mathfrak {t}}}^*$, and consider a basis of ${{\mathfrak {t}}}$ corresponding to the inward normals of the n facets $F_1, \ldots , F_n$ meeting at v. We can also assume that $F=F_1$ is defined by the equation $x_1=0$, i.e. $p=(0, x_2, \ldots , x_n)$ with $x_j \ge 0,$ and $L_j(x)=x_j$.

In one direction, we want to show that if $u \in \mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F)$ then u satisfies the four conditions of Proposition 4.19.

Let us define the mutually inverse matrices

$$\begin{aligned} \mathbf {G}^0_{\alpha , \beta }= \begin{pmatrix} \frac{\alpha + \beta x_1}{x_1^2} &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad \frac{1}{2x_2} &{}\quad \ddots &{}\quad \vdots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad 0 \\ 0 &{}\quad \cdots &{}\quad 0 &{}\quad \frac{1}{2x_n} \end{pmatrix} \quad \text {and}\quad \mathbf {H}^0_{\alpha , \beta } = \begin{pmatrix} \frac{(x_1)^2}{\alpha +\beta x_1} &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 2x_2 &{}\quad \ddots &{}\quad \vdots \\ \vdots &{} \quad \ddots &{}\quad \ddots &{}\quad 0 \\ 0 &{} \quad \cdots &{}\quad 0 &{}\quad 2x_n \end{pmatrix}. \end{aligned}$$

Given $u\in \mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F_1)$, we first prove that $\mathbf {G}^u =\mathrm{Hess}(u)$ and its inverse $\mathbf {H}^u$ satisfy the property that $\mathbf {G}^u-\mathbf {G}^0_{\alpha , \beta }$ and $\mathbf {G}^0_{\alpha , \beta }\mathbf {H}^u\mathbf {G}^0_{\alpha , \beta }-\mathbf {G}^0_{\alpha , \beta }$ extend smoothly through $(F_1\cup \cdots \cup F_n)\backslash (F_{n+1}\cup \cdots \cup F_d)$ in the region $\{\alpha +\beta x_1>0\}$, see Figure 3. Then we will show that if $\mathbf {G}^u-\mathbf {G}^0_{\alpha , \beta }$ and $\mathbf {G}^0_{\alpha , \beta }\mathbf {H}_u\mathbf {G}^0_{\alpha , \beta }-\mathbf {G}^0_{\alpha , \beta }$ extend smoothly, then the conditions of Proposition 4.19 must be satisfied.

Fig. 3

The polytope ${\Delta }$ and the domain near the face $F_1$ (grey)

We notice that $\mathbf {G}^0_{\alpha ,\beta }=\mathrm{Hess}(u^0_{\alpha ,\beta })$ with

$$\begin{aligned} u^0_{\alpha ,\beta }=-(\alpha -\beta x_1)\log x_1+\frac{1}{2}\sum _{j=2}^n x_j\log x_j. \end{aligned}$$

Thus, $\mathbf {G}^u-\mathbf {G}^0_{\alpha ,\beta }$ is the Hessian of a smooth function through the wall $(F_1\cup \cdots \cup F_n)\backslash (F_{m+1}\cup \cdots \cup F_d)$. Writing $u=-(\alpha -\beta x_1)\log x_1+\frac{1}{2}\sum _{j=2}^n x_j\log x_j +w(x)$, with w smooth, we get

$$\begin{aligned} \mathbf {H}^0_{\alpha ,\beta }\mathbf {G}^u = I_n +\begin{pmatrix} \frac{x_1^2}{\alpha x_1+\beta }w_{,11} &{}\quad \cdots &{}\quad \frac{(x_1)^2}{\alpha x_1+\beta }w_{,1n} \\ 2x_2w_{,21} &{}\quad \cdots &{} 2x_2w_{,2n}\\ \vdots &{}\quad &{}\quad \vdots \\ 2x_nw_{,2n} &{}\quad \cdots &{}\quad 2x_nw_{,2n} \end{pmatrix} \end{aligned}$$

which clearly extends smoothly (with positive determinant over the origin). Moreover, on $\mathbf {f}:=(F_1\cap \cdots \cap F_{\ell })\backslash (F_{\ell +1}\cup \cdots \cup F_m)$, for some $\ell \in \{1,\ldots , n\}$

$$\begin{aligned} \mathbf {H}^0_{\alpha ,\beta }\mathbf {G}^u = I_n +\begin{pmatrix} 0 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad &{}\quad \vdots \\ 0 &{}\quad \cdots &{}\quad 0 \\ 2x_{\ell +1}w_{,(\ell +1)1} &{}\quad \cdots &{}\quad 2x_{\ell +1}w_{,(\ell +1) m} \\ \vdots &{}\quad &{}\quad \vdots \\ 2x_nw_{,2n} &{}\quad \cdots &{}\quad 2x_nw_{,2n} \end{pmatrix} = \begin{pmatrix} I_{\ell } &{}\quad 0\\ * &{}\quad \mathbf {H}^0_{\mathbf {f}}\mathbf {G}{u_{\mathbf {f}}} \end{pmatrix} \end{aligned}$$

where $\mathbf {H}^0_{\mathbf {f}}=\mathrm{diag}(2x_{\ell +1}, \ldots , 2x_n)$ and, we recall, $u_{\mathbf {f}}:= \big ( u +(\alpha -\beta x_1)\log x_1)_{|_{\mathbf {f}}}$. Hence $\det [\mathbf {H}^0_{\alpha ,\beta }\mathbf {G}^u]=2^{m-\ell }x_{\ell +1}\cdots x_n \det [\mathbf {G}^{u_{\mathbf {f}}}]>0$ along $\mathbf {f}$, as $\mathbf {G}^{u_\mathbf {f}}$ is positive definite by assumption. Since this holds for all $\mathbf {f}$ and on the vertex $F_1\cap \cdots \cap F_n$, we conclude that $\mathbf {H}^0_{\alpha ,\beta }\mathbf {G}^u$ admits a smooth inverse, i.e. $\mathbf {H}^u\mathbf {G}^0_{\alpha ,\beta }=(\mathbf {H}^0_{\alpha ,\beta }\mathbf {G}^u)^{-1}$ can be extended smoothly through the wall too. Therefore, $(\mathbf {G}^0_{\alpha ,\beta }-\mathbf {G}^u)\mathbf {H}^u\mathbf {G}^0_{\alpha ,\beta }= \mathbf {G}^0_{\alpha ,\beta }\mathbf {H}^u\mathbf {G}^0_{\alpha ,\beta }-\mathbf {G}^0_{\alpha ,\beta }$ extends smoothly as well.

Set $\mathbf {Q}^u:=\mathbf {G}^0_{\alpha ,\beta }\mathbf {H}^u\mathbf {G}^0_{\alpha ,\beta }-\mathbf {G}^0_{\alpha ,\beta }$. It follows that $\mathbf {H}^u = \mathbf {H}^0_{\alpha ,\beta }\mathbf {Q}^u\mathbf {H}^0_{\alpha ,\beta }+\mathbf {H}^0_{\alpha ,\beta }$, and hence ${{\mathbf {H}}}^u$ is smooth through the wall. In fact, a direct computation tells us that

$$\begin{aligned} \mathbf {H}^u - \mathbf {H}^0_{\alpha ,\beta }=\begin{pmatrix} x_1^4 f_{11} &{}\quad x_1^2 x_2 f_{12} &{}\quad \cdots &{}\quad \cdots &{}\quad x_1^2x_n f_{1n} \\ x_1^2 x_2 f_{21} &{}\quad x_2^2 f_{22} &{}\quad x_2x_3 f_{23} &{} \cdots &{}\quad x_2x_n f_{2m} \\ \vdots &{}\quad x_2x_3 f_{32} &{}\quad &{}\quad &{}\quad \vdots \\ \vdots &{}\quad \vdots &{}\quad &{}\quad &{}\quad x_{n-1}x_n f_{2n} \\ x_1^2 x_n f_{n1} &{}\quad x_2x_n f_{n2} &{}\quad \cdots &{}\quad x_{n-1}x_n f_{2n} &{}\quad x_n^2 f_{nn} \\ \end{pmatrix} \end{aligned}$$

(118)

with smooth $f_{ab}$, which allows us to see that all the boundary conditions of $\mathbf {H}^u$ are satisfied along a face $\mathbf{f} \subset F_1$.

It remains to check the positivity assertion on the faces ${\mathbf {f}} \subset F_1$; as above, suppose $\mathbf {f} =(F_1\cap \cdots \cap F_{\ell })\backslash (F_{\ell +1}\cup \cdots \cup F_d)$. Along $\mathbf {f}$, $\mathbf {H}^0_{\alpha ,\beta }{\mathbf {G}}^u$ can be written as $\big ({\begin{matrix} I_{\ell } &{} 0 \\ * &{} \mathbf {H}_{\mathbf {f}}^0\mathbf {G}^{u_{\mathbf {f}}} \end{matrix}}\big )$. It thus follows that along $\mathbf {f}$ $(\mathbf {H}^0_{\alpha ,\beta }{\mathbf {G}}^u)^{-1}$ has the shape $\big ({\begin{matrix} I_{\ell } &{} 0 \\ * &{} \mathbf {H}^{u_{\mathbf {f}}}\mathbf {G}_{\mathbf {f}}^0 \end{matrix}}\big )$, with $\mathbf {G}^0_{\mathbf {f}}=(\mathbf {H}_{\mathbf {f}}^0)^{-1}= \mathrm{diag}( \frac{1}{2x_{\ell +1}}, \ldots , \frac{1}{2x_n})$, where $\mathbf{H}^{u_\mathbf{f}} = (\mathbf{G}^{u_\mathbf{f}})^{-1}$. Thus, along $\mathbf f$, ${\mathbf {H}}^u=(\mathbf {H}^0_{\alpha ,\beta }{\mathbf {G}}^u)^{-1}\mathbf {H}^0_{\alpha ,\beta }=\big ({\begin{matrix} 0 &{} 0 \\ * &{} \mathbf {H}^{u_{\mathbf {f}}} \end{matrix}}\big )$ (since $\mathbf {H}^0_{\alpha ,\beta }=\big ({\begin{matrix} 0 &{} 0 \\ 0 &{} \mathbf {H}_{\mathbf {f}}^0 \end{matrix}}\big )$ along $\mathbf {f}$). The desired positivity now readily follows from that of $\mathbf {H}^{u_{\mathbf {f}}}$ along $\mathbf {f}$, which in turn is a direct consequence of the convexity assumption of $u_\mathbf{f}$.

We now deal with the converse direction of Proposition 4.19, i.e. given a strictly convex $u\in C^{\infty }(\Delta ^0)$ such that the associated ${\mathbf {H}}^u$ verifies the conditions of Proposition 4.19, we have to show that $u\in \mathcal {S}_{\alpha , \beta }(\Delta , \mathbf{L}, F_1)$. Again, as this is local and already known far from the Poincaré face $F_1$, we focus on the same region as above. Arguments analogous to those in [4, pp. 290–291] allow one to show that the boundary conditions for $\mathbf{H}^u$ yield that $\mathbf {G}^u-\mathbf {G}^0_{\alpha , \beta }$ and $\mathbf {H}^u\mathbf {G}^0_{\alpha , \beta }$ extend smoothly through $(F_1\cup \cdots \cup F_n)\backslash (F_{n+1}\cup \cdots \cup F_d)$, the latter having positive determinant on $F_1$; the smooth extension of $\mathbf {G}^u-\mathbf {G}^0_{\alpha , \beta }$ ensures that u can be written as

$$\begin{aligned} -(\alpha -\beta L_1)\log \ L_1+\frac{1}{2}\sum _{j=2}^d L_j\log L_j + w(x) \end{aligned}$$

for some $w \in C^{\infty }\big ({\Delta }, {\mathbb {R}}\big )$.

The boundary conditions for $\mathbf{H}^u$ also tell us that ${\mathbf {H}}^u=\mathbf {H}^0_{\alpha ,\beta }+\mathbf {R}^u$, with $\mathbf {R}^u$ a smooth matrix of shape given by (118) (the third-order boundary condition on the Poincaré face gives precisely the ${O}\big (x_1^4\big )$ estimate for the upper left coefficient of $\mathbf {R}^{u}$). Hence, ${\mathbf {H}}^u\mathbf {G}^0_{\alpha ,\beta }=I_n+\mathbf {R}^u\mathbf {G}^0_{\alpha ,\beta }$ with

$$\begin{aligned} \mathbf {R}^u\mathbf {G}^0_{\alpha ,\beta }= \begin{pmatrix} (x_1)^2 g_{11} &{}\quad (x_1)^2 g_{12} &{}\quad \cdots &{}\quad \cdots &{}\quad (x_1)^2 g_{1n} \\ x_2 g_{21} &{}\quad x_2 g_{22} &{}\quad x_2 g _{23} &{}\quad \cdots &{}\quad x_2 g_{2n} \\ \vdots &{}\quad \vdots &{}\quad &{}\quad &{}\quad \vdots \\ \vdots &{}\quad \vdots &{}\quad &{}\quad &{}\quad x_{n-1} g_{(n-1)n}\\ x_n g_{n1} &{}\quad x_n g_{n2} &{} \quad \cdots &{}\quad x_n g_{n,(n-1)} &{}\quad x_n g_{nn} \end{pmatrix}, \end{aligned}$$

(119)

for smooth $g_{ab}$. It follows that along $\mathbf {f}=(F_1\cap \cdots \cap F_{\ell })\backslash (F_{\ell +1}\cup \cdots \cup F_d)$ ($1\le \ell \le n$), $\mathbf {H}^0_{\alpha ,\beta }{\mathbf {G}}^u=({\mathbf {H}}^u\mathbf {G}^0_{\alpha ,\beta })^{-1}=\big ({\begin{matrix} I_{\ell } &{} 0 \\ * &{} \mathbf {H}_{\mathbf {f}}^0\mathbf {G}^{u_{\mathbf {f}}} \end{matrix}}\big )$. As ${\mathbf {H}}^u\mathbf {G}^0_{\alpha ,\beta }$ extends as a smooth and positive definite matrix over $\mathbf{f}$, we conclude that $\mathbf {G}^{u_{\mathbf {f}}}>0$, i.e. $u_\mathbf{f}$ is strictly convex in the relative interior of $\mathbf f$.

Appendix B: Proof of Proposition 5.8 and Corollary 5.9

For the proof of Proposition 5.8 and Corollary 5.9, we will change point of view slightly and use explicit computations for quadrilaterals in $\mathbb {R}^2$ with the standard lattice $\mathbb {Z}^2$. The moment polytope corresponding to a Hirzebruch surface has at least one pair of parallel edges. Therefore after possibly scaling the polytope, we can take it to be given as the intersection $\cap _{i=1}^4 L_{i}^{-1}([0, \infty ))$ of

$$\begin{aligned} L_1(x,y)&= y,\nonumber \\ L_2(x,y)&= (1-x),\nonumber \\ L_3(x,y)&= (q-k)x- y + k,\nonumber \\ L_4(x,y)&= x , \end{aligned}$$

(120)

for some positive real numbers q and k. This will then correspond to a Hirzebruch surface exactly when $q - k \in \mathbb {Z}$.

We begin with the proof of Proposition 5.8 in the case of two edges, and then show that the case of one edge is a corollary of this, using the convexity of the set of stable weights. We end by proving Corollary 5.9.

1.1 Appendix B.1: The Case of Two Edges

For this we will use a criterion for stability found in [40] which used the ambitoric framework of [5, 6] described above. We begin by recalling this result.

For a pair of edges $F_1,F_2$ of a general 2-dimensional convex polytope $\Delta$, one can parametrize the lines that meet both $F_1$ and $F_2$ by $[0,1] \times [0,1]$. Let the vertices of $F_1$ be $v_0$ and $v_1$ and let the vertices of $F_2$ be $w_0$ and $w_1$. Then let $v_s = (1-s) v_0 + s v_1$ and $w_t = (1-t)w_0 + t w_1$. Picking an affine linear function $h_{s,t}$ whose zero set is the line containing $v_s$ and $w_t$, one then obtains a corresponding simple piecewise linear function $f_{s,t} = \max \{ 0, h_{s,t} \}$. We can parametrize the Donaldson–Futaki invariant of these functions as a map

$$\begin{aligned} \phi : [0,1]\times [0,1] \longrightarrow \mathbb {R}, \\ \qquad \qquad (s,t) \longmapsto \mathcal {L}_{(\Delta , \mathbf{L}, F)} ( f_{s,t} ). \end{aligned}$$

The positivity of this is independent of the scaling of $h_{s,t}$ chosen. Since all lines meeting $F_1$ and $F_2$ are traced out as (s, t) takes all values in $[0,1] \times [0,1]$, it suffices to check the positivity of $\phi$ in order to check whether or not there are any simple piecewise linear functions with crease meeting $F_1$ and $F_2$ violating stability. By choosing an appropriate scaling of $h_{s,t}$, $\phi$ can be taken to be polynomial in (s, t) of bidegree (3, 3). This was essentially shown in [22], see also [40, Lemma 2.10].

If $F_1$ and $F_2$ are adjacent to a common edge $\tilde{F}$, then each of the $F_i$ has a vertex lying on $\tilde{F}$. Thus, up to reordering $v_i$ and $w_i$, we have that $v_0$ and $w_0$ lie on $\tilde{F}$. In the domain $[0,1] \times [0,1]$ of $\phi$ the point (0, 0) will then correspond to the simple piecewise linear function $f_{0,0}$ whose crease is $\tilde{F}$; $f_{0,0}$ is then actually affine linear on P and so $\phi \big (( 0,0) \big ) =0$. Moreover, if $\tilde{F}$ is one of the components of F, the facets along which we let the boundary measure vanish, then the point will be a critical point of $\phi$.

In the case when $\Delta = Q$ is a quadrilateral, there are exactly two pairs of such edges that have common adjacent edges, namely the two pairs of opposite edges. As above, we then have functions $\phi _1, \phi _2$ parametrizing the Donaldson–Futaki invariant of simple piecewise linear functions meeting opposite edges of Q. In the case when F consists of two edges of Q, there will then be exactly two critical points corresponding to two of the vertices of $\phi _1$ and/or $\phi _2$.

Proposition A.3

Let Q be a quadrilateral and pick two edges $F_1, F_2$ of Q. Then $(Q, \mathbf{L}, F_1 \cup F_2)$ is stable if and only if the determinant of the Hessian of the functions $\phi _1,\phi _2$ at the points corresponding to an affine linear function is

• non-negative if $F_1, F_2$ are adjacent,

• positive if $F_1, F_2$ are opposite.

Moreover, the positivity of the determinant implies that the relative Székelyhidi numerical constraint is satisfied.

Note that the converse of the final statement is not true. If the determinant vanishes, the Hessian is positive semi-definite but not positive definite at the critical point of $\phi _i$. This means that there is a family $f_c$ of simple piecewise linear functions, with $f_0$ corresponding to the critical point of the domain of $\phi _i$, such that $\frac{d^2}{dc^2}\big |_{c=0} \big ( \mathcal {L}_{(Q, \mathbf{L}, F)}(f_c) \big ) = 0$. However, it is not necessarily the case that the crease of this family can be taken to be parallel to relevant edge of Q. This would have to be the case if the positivity of the determinant was equivalent to the relative Székelyhidi numerical constraint.

For Q being the moment polytope of a Hirzebruch surface given by Eq. (120) and $\mathbf{L}$ the canonical scaling of the normals to Q, one can then compute the functions $\phi _1$ and $\phi _2$ of Proposition A.3 and hence their determinants directly in terms of q and k. The result of this computation is given in Lemmas A.4 and A.5.

Lemma A.4

Suppose F consists of two adjacent edges, which without loss of generality can be assumed to be the two edges not lying on the coordinate axes. Then the determinants of the Hessians of $\phi _1, \phi _2$ at the two critical points are up to a positive constant given by

$$\begin{aligned} k^4 + 2 k^2 q^2 + q^4 - k^3 + 3k^2 q + 3 kq^2 - q^3 \end{aligned}$$

(121)

and

$$\begin{aligned} \begin{aligned}&3k^6 q + 3k^5 q^2 + 6k^4q^3 + 6 k^3 q^4 + 3k^2 q^5 + 3k q^6 + 2k^6 + 2k^5q + 6k^4 q^2 \\&\quad + 4 k^3 q^3 + 6 k^2 q^4 + 2 k q^5 + 2 q^6 - 2 k^5 - 2 k^4 q + 4 k^3 q^2 + 4k^2 q^3 - 2 q^4 k - 2 q^5. \end{aligned} \end{aligned}$$

(122)

Thus to complete the proof of Proposition 5.8 in the case when F consists of two adjacent edges, we have to show that both these numbers are always positive. This is not true for arbitrary positive q and k, but we will use that $q-k \in \mathbb {Z}$. In fact, having $|q-k| \ge 1$ ensures that both (121) and (122) are positive. Note that we do not have to consider the case $q=k$ as this corresponds to a product.

So assume first that $q>k$, so that $q \ge k+1$. We then use the substitution $q = k+1+\lambda$, where $\lambda \ge 0$ by assumption. The expression (121) is then given by

$$\begin{aligned}&\lambda ^4 + 4 k \lambda ^3 + 8 k^2 \lambda ^2 + 8 k^3 \lambda + 4 k^4 + 3 \lambda ^3 + 12 k \lambda ^2 \\&\quad + 22 k^2 \lambda + 12 k^3 + 3 \lambda ^2 + 12 k \lambda +14 k^2 + \lambda + 4 k . \end{aligned}$$

Since $k>0$ and $\lambda \ge 0$, it therefore follows that the term in Eq. (121) is always positive.

Similarly, using the substitution $k=q+1+\lambda$ instead, one can show that (121) is always positive when $q<k$. The same technique also works to show that (122) is positive whenever $q-k \in \mathbb {Z}$. This completes the proof of Proposition 5.8 for the case of adjacent edges.

Though we have already proved this by different means, the above technique also works when F consists of two opposite edges. Székelyhidi showed in [46, Prop. 15] that if the two edges that are not in F are parallel, then $(Q,\mathbf{L} , F)$ is always strictly semistable. It can also be verified directly that the determinant of the Hessian as in Proposition A.3 vanishes in this situation. With Q determined by positive numbers k, q as above, we can therefore assume that F consists of the two edges of Q contained in $L_2 =0$ and $L_4=0$, respectively, since the edges lying in $L_1=0$ and $L_3=0$ are the only opposite edges that may not be parallel.

In the case of opposite edges, the two determinant conditions turn out to be equivalent. Thus we need to determine that this single number is non-negative and vanishes precisely if $q=k$. This is a consequence of the lemma below.

Lemma A.5

Suppose F consists of the two opposite edges lying on $L_2=0$ and $L_4=0$. Then the determinant of the Hessian of the function corresponding to the Donaldson–Futaki invariant of simple piecewise linear functions with crease meeting $L_1=0$ and $L_3=0$ at the critical point corresponding to $L_2=0$ is given by

$$\begin{aligned} \frac{(k-q)^2 (k+q)^2k^2}{2(k^2+4kq+q^2)^2}. \end{aligned}$$

The determinant is thus always non-negative and since k and q are positive it vanishes if and only if $k=q$, i.e. if and only if Q is a rectangle, as expected.

Remark A.6

The determinant condition Proposition A.3 holds regardless of the normals we use for the remaining two edges in F. In particular, it applies when we have a third facet in F. In this case similar formulae to the ones given above show that the determinant condition is violated, and so (Q, F) is always unstable when F consists of three edges of Q.

1.2 Appendix B.2: The Case of One Edge

To prove that $(Q,\mathbf{L} , F)$ is stable when Q corresponds to a Hirzebruch surface and F is a single edge of Q, we will use the notion of weighted stability and the convexity of the set of stable weights. In general, for a Delzant polytope $\Delta$, we let $r_i \in \mathbb {R}_{\ge 0}$ be the reciprocal of the scaling of the $i^{ th }$ defining function of $\Delta$ given by the data $\mathbf{L}$ if this facet is not in F and $r_i =0$ if it is. Then we can identify the triple with $(\Delta , \mathbf{L}, F)$ with $(\Delta , \underline{r})$ where $\underline{r} \in \mathbb {R}^d_{\ge 0}$ is the weight of $(\Delta , \mathbf{L}, F)$. The weight $\underline{r}$ is stable if the corresponding triple $(\Delta , \mathbf{L}, F)$ is.

For us the key property of weighted stability is that the set of non-zero stable weights, thought of as a subset of $\mathbb {R}_{\ge 0}^d \setminus \{ 0 \}$, is a convex cone. Thus a positive linear combination of semistable weights is semistable, and moreover, if at least one of the weights is stable, then the linear combination is stable too.

Going back to the case when $\Delta = Q$ corresponds to a Hirzebruch surface, the notion of the stability of $(Q,\mathbf{L} , F)$ where F is a single edge of Q is exactly the same as the stability of the weight (0, 1, 1, 1), where $F_1$ is the edge in F and $F_2, F_3, F_4$ are the remaining three edges of P.

We now note that this weight can be written as

$$\begin{aligned} (0,1,1,1) = \frac{1}{2} (0,0,1,1) + \frac{1}{2} (0,1,0,1) + \frac{1}{2} (0,1,1,0) . \end{aligned}$$

The stability of the weights (0, 0, 1, 1) , (0, 1, 0, 1) and (0, 1, 1, 0) each correspond to the stability of some $(Q,\mathbf{L} , F')$ where $F'$ consists of exactly two edges of Q. Thus (0, 1, 1, 1) is a positive combination of semistable weights, and hence is semistable. Moreover, at least two of the weights are then in fact stable, since two of these weights corresponds to the case when $F'$ consists of two adjacent sides. Hence (0, 1, 1, 1) must be stable as well. This means that $(Q,\mathbf{L} , F)$ where Q is a Hirzebruch surface, $\mathbf{L}$ are the canonical defining functions of Q and F is a single edge of Q, is always stable, and this completes the proof of Proposition 5.8.

1.3 Appendix B.3: The Proof of Corollary 5.9

The missing component in the proof is to show that the extremal metrics obtained cannot be of Poincaré type. For this we will show that in the above situation, the necessary condition in Eq. (42) for the existence of a toric extremal metric of Poincaré type is violated. In the Hirzebruch surface cases we are interested in, this condition becomes the following:

Lemma A.7

Let Q be the moment polytope of a Hirzebruch surface X and Kähler class $\Omega$ given by the data in equations (120). If X admits an extremal Poincaré type metric on the complement of a divisor Z corresponding to the a union F of facets of Q, then the associated affine linear function A satisfies that

• A is constant along F, and so in particular at its vertices, if F is a single edge,

• A is constant along the line with vertices (0, k) and $(\frac{1}{k},0)$, if F consists of the two edges lying in the x and y-axes.

Proof

The only part that needs clarification is the last statement regarding the case of two adjacent edges. Let A be the associated affine linear function to Q given by the equations (120). We are taking the edges in F to be $F_1$ and $F_4$.

Recall that for the moment polytope $I = [0,\lambda ]$ with coordinate z and F being the end-point 0, the associated affine linear function is

$$\begin{aligned} B_{\lambda } (z) = \frac{6}{\lambda ^2} z - \frac{2}{\lambda }. \end{aligned}$$

Since F is $F_1 \cup F_4$ for Q, the condition of Eq. (42) then becomes that

$$\begin{aligned} A_{| y = 0}&= B_1 (x) + c_1 \\ A_{| x = 0}&= B_k (y ) + c_2 , \end{aligned}$$

where the $c_i$ are constants. Thus if $A = ax + b y +c$, we have that

$$\begin{aligned} a&= 6 \\ b&= \frac{6}{k^2}. \end{aligned}$$

Thus $A (0,k) = c+ \frac{6}{k}$ and $A(\frac{1}{k},0) = c + \frac{6}{k}$, too. $\square$

We will change our parametrization of Q slightly from the beginning of this Appendix and the above proof, and instead take Q to have vertices

$$\begin{aligned} v_1&= (-d, 0), \\ v_2&=(k,0) ,\\ v_3&=(0,1) ,\\ v_4&=(-d, 1). \end{aligned}$$

Here $d>0$ and $k \in \mathbb {Z}_{\ge 0}$. We can always take Q to be of this form up to scaling. We will let $F_1$ be the edge connecting $v_1$ and $v_4$, $F_2$ to be the edge connecting $v_1$ and $v_2$, $F_3$ to be edge connecting $v_2$ and $v_3$ and $F_4$ to be the edge connecting $v_3$ and $v_4$.

First, we let $A_i$ be the associated affine linear function the case when F consists of all edges but $F_i$. Then

$$\begin{aligned} A_1 =&- \frac{12}{2d^2 + 2dk + k^2} x - \frac{24 kd (k+d)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} y \\&- \frac{6(4d^3 -2d^2k -6k^2d - k^3)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} , \\ A_2 =&- \frac{12 (3d^2 + 4dk + k^2)}{6 d^2 + 6 dk + k^2} y - \frac{6(4d^2 + 5 dk + k^2)}{6 d^2 + 6 dk + k^2} , \\ A_3 =&\frac{12}{2d^2 + 2dk + k^2} x + \frac{12 k (4d^2 + 4dk + k^2)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} y \\&+ \frac{6(8d^3 + 2d^2k -4k^2d - k^3)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} , \\ A_4 =&- \frac{12 (3d + 2k)}{6 d^2 + 6 dk + k^2} y - \frac{6d (k+2d)}{6 d^2 + 6 dk + k^2} . \end{aligned}$$

Since the associated linear functions depend linearly on the inverse normals, the associated linear function $B_i$ to when F consists of a single edge $F_i$ is therefore given by $B_i = \sum _{j \ne i} A_i$, which is

$$\begin{aligned} B_1 =&\frac{12}{2d^2 + 2dk + k^2} x - \frac{12 k (4 d^3 + 6d^2 k + 4 dk^2 + k^3 - 4d^2 -4 dk - k^2)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} y \\&+ \frac{6(4d^4 +12d^3 k + 12 d^2 k^2 + 6 dk^3 + k^4 + 8 d^3 + 2d^2k - 4dk^2 - k^3)}{(2d^2 + 2 dk + k^2)(6 d^2 + 6 dk + k^2)} , \\ B_2 =&\frac{12 (3d^2 + 2 dk + k)}{6 d^2 + 6 dk + k^2} y - \frac{6d (2d + k- 2)}{6 d^2 + 6 dk + k^2} , \\ B_3 =&- \frac{12}{2d^2 + 2dk + k^2} x - \frac{12 k (4d^3 + 6d^2 k+ 4dk^2 + k^3 + 2d^2 + 2dk)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} y \\&+ \frac{6 ( 4d^4 + 12 d^3 k + 12 d^2 k^2 + 6dk^3 + k^4 - 4d^3 + 2d^2 k + 6dk^2 + k^3)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} , \\ B_4 =&- \frac{12 (3d^2 + 4 dk + k^2 - k)}{6 d^2 + 6 dk + k^2} y + \frac{6(4 d^2 + 5dk + k^2 + 2d)}{6 d^2 + 6 dk + k^2} . \end{aligned}$$

Using Lemma A.7, it suffices to verify whether or not the linear part of $B_i$ is equal at the two vertices of $F_i$. Let $K_i$ be the difference of these two numbers. Then

$$\begin{aligned} K_1&= \frac{12k (4 d^3 + 6 d^2 k + 4 dk^2 + k^3 -4 d^2 - 4dk - k^2)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)}, \\ K_2&= 0, \\ K_3&= - \frac{12k (4 d^3 + 6 d^2 k + 4 dk^2 + k^3 - 4 d^2 - 4 dk - k^2)}{(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} ,\\ K_4&= 0. \end{aligned}$$

The cases of $K_2$ and $K_4$ are the cases when Z is the zero or infinity section in X. This is already treated in Corollary 5.5 where we know that there exists Poincaré type extremal metrics. The above computations then confirm that the condition of equation (42) holds, which we also know by general theory regarding extremal Poincaré type metrics.

To prove Corollary 5.9 in the case when F consists of a single edge using Lemma A.7, we need to show that $K_1$ and $K_3$ can never be 0 if $d>0$ and $k \ge 1$. The requirement $k \ge 1$ comes from the fact that if $k=0$, then $X = \mathbb {CP}^1 \times \mathbb {CP}^1$, which is the case we are not considering.

First note that denominator of $K_1$ is always positive, so it will have the same sign as

$$\begin{aligned} 4 d^3 + 6 d^2 k + 4 dk^2 + k^3 -4 d^2 - 4dk - k^2 \end{aligned}$$

which equals

$$\begin{aligned} 4 d^3 + 2 d^2 k + 4 dk (k-1) + k^2(k-1) + 4 d^2 (k-1) . \end{aligned}$$

This is always positive as $d>0$ and $k \ge 1$. For $K_3$, note that it equals $-K_1$, and hence is always negative.

The remaining case is that of when F consists of two adjacent edges of Q, which we take to be $F_1$ and $F_2$. The associated affine linear function A is then $A_3 + A_4$ and by Lemma A.7 we need to verify that $K = A(k,0) - A(-d, \frac{1}{k+d})$ can never be 0. A computation shows that this quantity is given by

$$\begin{aligned} K&= \frac{12(6d^4+18d^3k+19d^2k^2+8dk^3+k^4+6d^3+6d^2k+3dk^2+k^3)}{(k+d)(2d^2 + 2dk + k^2)(6 d^2 + 6 dk + k^2)} \end{aligned}$$

which is clearly positive. In particular, it can never be 0.