Abstract
A Killing p-form on a Riemannian manifold (M, g) is a p-form whose covariant derivative is totally antisymmetric. If M is a connected, oriented, 4-dimensional manifold admitting a non-parallel Killing 2-form ψ, we show that there exists a dense open subset of M on which one of the following three exclusive situations holds: either ψ is everywhere degenerate and g is locally conformal to a product metric, or g gives rise to an ambikähler structure of Calabi type, or, generically, g gives rise to an ambitoric structure of hyperbolic type, in particular depends locally on two functions of one variable. Compact examples of either types are provided.
Dedicated to Simon Salamon on the occasion of his 60th birthday
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1 Introduction
On any n-dimensional Riemannian manifold (M, g), an exterior p-form ψ is called conformal Killing [13] if its covariant derivative ∇ψ is of the form
for some ( p − 1)-form α and some ( p + 1)-form β, which are then given by
Conformal Killing forms have the following conformal invariance property: if ψ is a conformal Killing p-form with respect to the metric g, then, for any positive function f, \(\tilde{\psi }:= f^{p+1}\,\psi\) is conformal Killing with respect to the conformal metric \(\tilde{g}:= f^{2}\,g\). In other words, if L denotes the real line bundle \(\vert \Lambda ^{n}TM\vert ^{\frac{1} {n} }\) and \(\ell,\tilde{\ell}\) denote the sections of L determined by \(g,\tilde{g}\), then, for any Weyl connection D relative to the conformal class [g], the section \(\boldsymbol{\psi }:=\psi \otimes \ell^{p+1} =\tilde{\psi } \otimes \tilde{\ell}^{p+1}\) of \(\Lambda ^{p}T^{{\ast}}M \otimes L^{p+1}\) satisfies
for some section \(\boldsymbol{\alpha }\) of \(\Lambda ^{p-1}T^{{\ast}}M \otimes L^{p-1}\) and some section \(\boldsymbol{\beta }\) of \(\Lambda ^{p+1}T^{{\ast}}M \otimes L^{p+1}\) (depending on D), cf. e.g. [4, Appendix B].
The p-form ψ is called Killing, resp. ∗-Killing, with respect to g, if ψ satisfies (1) and α = 0, resp. β = 0. In particular, Killing forms are co-closed, ∗-Killing forms are closed, and, if M is oriented and ∗ denotes the induced Hodge star operator, ψ is Killing if and only if ∗ψ is ∗-Killing.
Although the terminology comes from the fact that Killing 1-forms are just metric duals of Killing vector fields, and thus encode infinitesimal symmetries of the metric, no geometric interpretation of Killing p-forms exists in general in terms of symmetries when p ≥ 2, except in the case of Killing 2-forms in dimension 4, which is special for various reasons, the most important being the self-duality phenomenon.
On any oriented four-dimensional manifold (M, g), the Hodge star operator ∗, acting on 2-forms, is an involution and, therefore, induces the well known orthogonal decomposition
where \(\Lambda ^{2}M\) stands for the vector bundle of (real) 2-forms on M and \(\Lambda ^{\pm }M\) for the eigen-subbundle for the eigenvalue ± 1 of ∗. Accordingly, any 2-form ψ splits as
where ψ +, resp. ψ −, is the self-dual, resp. the anti-self-dual part of ψ, defined by \(\psi _{\pm } = \frac{1} {2}(\psi \pm {\ast}\psi )\). Since ∗ acting on 2-forms is conformally invariant, a 2-form ψ is conformal Killing if and only if ψ + and ψ − are separately conformal Killing, meaning that
for some real 1-forms α +, α −, and ψ is Killing, resp. ∗-Killing, if, in addition,
Throughout this paper, (M, g) will denote a connected, oriented, 4-dimensional Riemannian manifold and ψ = ψ + + ψ − a non-trivial ∗-Killing 2-form on M (the choice of the ∗-Killing ψ, instead of the Killing 2-form ∗ψ is of pure convenience). We also discard the non-interesting case when ψ is parallel.
On the open set, M 0 +, resp. M 0 −, where ψ +, resp. ψ −, is non-zero, the associated skew-symmetric operators \(\Psi _{+},\Psi _{-}\), are of the form \(\Psi _{+} = f_{+}\,J_{+}\), resp. \(\Psi _{-} = f_{-}\,J_{-}\), where J +, resp. J −, is an almost complex structure inducing the chosen, resp. the opposite, orientation of M, and f +, resp. f −, is a positive function. It is then easily checked, cf. Sect. 2 below, that the first, resp. the second, condition in (6) is equivalent to the condition that the pair (g +: = f + −2 g, J +), resp. the pair (g −: = f − −2 g, J −), is Kähler. On the open set M 0 = M 0 + ∩ M 0 −, which is actually dense in M, cf. Lemma 2.1 below, we thus get two Kähler structures, whose metrics belong to the same conformal class and whose complex structures induce opposite orientations (in particular, commute), hence an ambikähler structure, as defined in [4]. This actually holds if ψ is simply conformal Killing and had been observed in the twistorial setting by Pontecorvo in [12], cf. also [4, Appendix B2]. The additional coupling condition (7), which, on M 0, reads J + df + = J − df −, cf. Sect. 2, then has strong consequences, that we now explain.
A first main observation, cf. Proposition 3.3, is that the open subset, M S , where ψ is of maximal rank, hence a symplectic 2-form, is either empty or dense in M.
The case when M S is empty is the case when ψ is decomposable, i.e. ψ ∧ψ = 0 everywhere; equivalently, | ψ + | = | ψ − | everywhere; on M 0, we then have f + = f −, hence g + = g − =: g K , and (M 0, g K ) is locally a product of two (real) Kähler surfaces \((\Sigma,g_{\Sigma },\omega _{\Sigma })\) and \((\tilde{\Sigma },g_{\tilde{\Sigma }},\omega _{\tilde{\Sigma }})\), with f + = f − constant on \(\tilde{\Sigma }\), cf. Sect. 6. In this case, no non-trivial Killing vector field shows up in general, but a number of compact examples involving Killing vector fields are provided, coming from [9].
The case when M S is dense is first handled in Proposition 2.4, where we show that the vector field \(K_{1}:= -\frac{1} {2}\,\alpha ^{\sharp }\) is then Killing with respect to g—the chosen normalization is for further convenience—and that each eigenvalue of the Ricci tensor, Ric, of g is of multiplicity at least 2; moreover, on the (dense) open set M 1 = M S ∩ M 0, K 1 is Killing with respect to g +, g − and Hamiltonian with respect to the Kähler forms ω +: = g +(J +⋅ , ⋅ ) and ω −: = g −(J −⋅ , ⋅ ); also, Ric is both J +- and J −-invariant, cf. Proposition 2.4 below. On M 1, the ambikähler structure (g +, J +, ω +), (g −, J −, ω −) is then of the type described in Proposition 11 (iii) of [4].
In Sect. 3, we set the stage for a separation of variables by introducing new functions x, y, defined by \(x = \frac{1} {2}(\,f_{+} + f_{-})\) and \(y = \frac{1} {2}(\,f_{+} - f_{-})\), which, up to a factor 2, are the “eigenvalues” of ψ, and whose gradients are easily shown to be orthogonal. In Proposition 3.1, we show that | dx |2 = A(x) and | dy |2 = B(y), for some positive functions A and B of one variable. In terms of the new functions x, y, the dual 1-form of K 1 with respect to g is simply J + dx + J + dy. Furthermore, in Proposition 3.2 a second Killing vector field, K 2, shows up, whose dual 1-form is y 2 J + dx + x 2 J + dy and which turns out to coincide, up to a constant factor, with the Killing vector field constructed by W. Jelonek in [8, Lemma B], cf. also the proof of Proposition 11 in [4], namely the image of K 1 by the Killing symmetric endomorphism \(S = \Psi _{+} \circ \Psi _{-} + \frac{(\,f_{+}^{2}+f_{ -}^{2})} {2} \mathrm{I}\), cf. Remark 3.1.
In Proposition 3.3, we then show that either K 2 is a (positive) constant multiple of K 1, and we end up with an ambikähler structure of Calabi type, according to Definition 5.1 taken from [1], or K 1, K 2 are independent on a dense open subset of M, determining an ambitoric structure, as defined in [3, 4].
The Calabi case is considered in Sect. 5, where it is shown that, conversely, any ambikähler structure of Calabi type gives rise, up to scaling, to a 1-parameter family of pairs (g (k), ψ (k)), where g (k) is a Riemannian metric in the conformal class and ψ (k) a ∗-Killing 2-form with respect to g (k), cf. Theorem 5.1 and Remark 5.1. The example of Hirzebruch-like ruled surfaces is described in Sect. 8.
The ambitoric case is the case when dx and dy are independent on a dense open subset of M. In Sect. 4, we show that x, y can be locally completed into a full system of coordinates by the addition of two “angular coordinates”, s, t, in such a way that \(K_{1} = \frac{\partial } {\partial s}\) and \(K_{2} = \frac{\partial } {\partial t}\) and giving rise to a general Ansatz, described in Theorem 4.1. As an Ansatz for the underlying ambikähler structure, this turns out to be the same as the ambitoric Ansatz of Proposition 13 in [4] for the “quadratic” polynomial q(z) = 2z, hence in the hyperbolic normal form of [4, Section 5.4], when the functions x, y are identified with the adapted coordinates x, y in [4].
The main observation at this point is that, while the adapted coordinates in [4] are obtained via a quadratic transformation, cf. [4, Section 4.3], the functions x, y are here naturally attached to the ∗-Killing 2-form ψ which determines the ambitoric structure. This is quite reminiscent of the orthotoric situation, described in [1] in dimension 4 and in [2] in all dimensions, where the separation of variables—and the corresponding Ansatz—are similarly obtained via the “eigenvalues” of a Hamiltonian 2-form, which share the same properties as the “eigenvalues” x, y of the ∗-Killing 2-form ψ.
In spite of this, the ∗-Killing 2-forms considered in this paper are not Hamiltonian 2-forms in general—for a general discussion about Killing or ∗-Killing 2-forms versus Hamiltonian 2-forms, cf. [10], in particular Theorem 4.5 and Proposition 4.8, and, also, [2, Appendix A]—but, in many respects, at least in dimension 4, the role played by Hamiltonian 2-forms in the orthotoric case is played by ∗-Killing 2-forms in the (hyperbolic) ambitoric case.
The three situations described above, namely the decomposable, the Calabi ambikähler and the ambitoric case, cf. Proposition 3.3, are nicely illustrated in the example of the round 4-sphere described in Sect. 7, on which every ∗-Killing form can be written as the restriction of a constant 2-form \(\mathsf{a} \in \mathfrak{s}\mathfrak{o}(5) \simeq \Lambda ^{2}\mathbb{R}^{5}\), which is also the 2-form associated to the covariant derivative of the Killing vector field induced by a. If a has rank 2, the same holds for its restriction on a dense open subset of the sphere, so this corresponds to the decomposable case. Otherwise, a can be expressed as λ e 1 ∧ e 2 + μ e 3 ∧ e 4—cf. Sect. 7 for the notation—with λ, μ both positive, and, depending on whether λ and μ are equal or not, we obtain on a dense subset of the sphere an ambikähler structure of Calabi type or a hyperbolic ambitoric structure respectively. By using the hyperbolic ambitoric Ansatz of Sect. 4, it is eventually shown that the resulting ∗-Killing 2-forms are actually ∗-Killing with respect to infinitely many non-isometric Riemannian metrics on S 4, cf. Remark 7.2.
2 Killing 2-Forms and Ambikähler Structures
In what follows, (M, g) denotes a connected, oriented, 4-dimensional Riemannian manifold admitting a non-parallel Killing 2-form φ, and ψ: = ∗φ denotes the corresponding ∗-Killing 2-form; we then have
for some real, non-zero, 1-form α, where ∇ denotes the Levi-Civita connection of g and X ♭ the dual 1-form of X with respect to g, cf. [13]. By anti-symmetrizing and by contracting (8), it is easily checked that ψ is closed and that
where δ denotes the codifferential with respect to g. Denote by \(\psi _{+} = \frac{1} {2}(\psi +{\ast}\psi )\), resp. \(\psi _{-} = \frac{1} {2}(\psi -{\ast}\psi )\), the self-dual, resp. the anti-self-dual, part of ψ, where ∗ is the Hodge operator induced by the metric g and the chosen orientation. Then, (8) is equivalent to the following two conditions
Here, we used the general identity:
for any vector field X and any p-form ϕ on any oriented Riemannian manifold. In particular, ψ + and ψ − are conformally Killing, cf. [13]. The datum of a (non-parallel) ∗-Killing 2-form ψ on (M, g) is then equivalent to the datum of a pair (ψ +, ψ −) consisting of a self-dual 2-form ψ + and an anti-self-dual 2-form ψ −, both conformally Killing and linked together by
or, equivalently, by
We denote by \(\Psi\), \(\Psi _{+}\), \(\Psi _{-}\) the anti-symmetric endomorphisms of TM associated to ψ, ψ +, ψ − respectively via the metric g, so that \(g(\Psi (X),Y ) =\psi (X,Y )\), \(g(\Psi _{+}(X),Y ) =\psi _{+}(X,Y )\), \(g(\Psi _{-}(X),Y ) =\psi _{-}(X,Y )\). On the open set, M 0, of M where \(\Psi _{+}\) and \(\Psi _{-}\) have no zero, denote by J +, J − the corresponding almost complex structures:
where the positive functions f +, f − are defined by
(here, the norms \(\vert \Psi _{+}\vert\), \(\vert \Psi _{-}\vert\), are relative to the conformally invariant inner product defined on the space of anti-symmetric endomorphisms of TM by \((A,B):= -\frac{1} {2}\mathrm{tr}(A \circ B)\)); the open set M 0 is then defined by the condition
Notice that J + and J − induce opposite orientations, hence commute to each other, so that the endomorphism
is an involution of the tangent bundle of M 0.
From (8), we get
with the following general convention: for any 1-form α and any vector field X, α ∧ X denotes the anti-symmetric endomorphism of TM defined by (α ∧ X)(Y ) = α(Y )X − g(X, Y )α ♯, where α ♯ is the dual vector field to α relative to g (notice that the latter expression is actually independent of g in the conformal class [g] of g). Equivalently:
We infer \((\nabla _{X}\Psi _{+},\Psi _{+}) = \frac{1} {2}(d\vert \Psi _{+}\vert ^{2})(X) = (\Psi _{ +},\alpha \wedge X) =\big (\Psi _{+}(\alpha )\big)(X)\), hence \(\Psi _{+}(\alpha ) = \frac{1} {2}d\vert \Psi _{+}\vert ^{2}\). Similarly, \(\Psi _{-}(\alpha ) = \frac{1} {2}d\vert \Psi _{-}\vert ^{2}\). By using (14), we then get
In particular,
Remark 2.1
For any ∗-Killing 2-form ψ as above, denote by \(\Phi = \Psi _{+} - \Psi _{-}\) the skew-symmetric endomorphism associated to the Killing 2-form φ = ∗ψ and by S the symmetric endomorphism defined by
where I denotes the identity of TM. Then, S is Killing with respect to g, meaning that the symmetric part of ∇S is zero or, equivalently, that g((∇ X S)X, X) = 0 for any vector field X, cf. [11], [4, Appendix B]. This readily follows from the fact that \(\nabla _{X}\Phi (X) = X\lrcorner {\ast} (\alpha \wedge X) = 0\), so that \(g(\nabla _{X}S(X),X) = -2g(\nabla _{X}\Phi (X),\Phi (X)) = 0\), for any vector field X.
Lemma 2.1
The open subset M 0 defined by (16) is dense in M.
Proof
Denote by M 0 ± the open set where f ± ≠ 0, so that M 0 = M 0 + ∩ M 0 −. It is sufficient to show that each M 0 ± is dense. If not, f ± = 0 on some non-empty open set, V, of M, so that ψ ± = 0 on V, hence is identically zero, since ψ ± is conformally Killing, cf. [13]; this, in turn, implies that α, hence also ∇ψ, is identically zero, in contradiction to the hypothesis that ψ is non-parallel. □
In view of the next proposition, we recall the following definition, taken from [4]:
Definition 2.1 ([4])
An ambikähler structure on an oriented 4-manifold M consists of a pair of Kähler structures, \(\big(g_{+},J_{+},\omega _{+} = g_{+}(J_{+}\cdot,\cdot )\big)\) and \(\big(g_{-},J_{-},\omega _{-} = g_{-}(J_{-}\cdot,\cdot )\big)\), where the Riemannian metrics g +, g − belong to the same conformal class, i.e. g − = f 2 g +, for some positive function f, and the complex structure J +, resp. the complex structure J −, induces the chosen orientation, resp. the opposite orientation; equivalently, the Kähler forms ω + and ω − are self-dual and anti-self-dual respectively.
We then have:
Proposition 2.1
Let (M, g) be a connected, oriented, 4-dimensional Riemannian manifold, equipped with a non-parallel ∗-Killing 2-form ψ = ψ + + ψ − as above. Then, on the dense open subset, M 0 , of M defined by (16), the pair (g, ψ) gives rise to an ambikähler structure (g +, J +, ω +), (g −, J −, ω −), with g ± = f ± −2 g and \(J_{\pm } = f_{\pm }^{-1}\,\Psi _{\pm }\) , by setting \(f_{\pm } = \vert \Psi _{\pm }\vert /\sqrt{2}\) . In particular, this ambikähler structure is equipped with two non-constant positive functions f +, f − , satisfying the two conditions
where g − = f 2 g + , and
Conversely, any ambikähler structure (g +, J +, ω +), (g − = f 2 g +, J −, ω −) equipped with two non-constant positive functions f +, f − satisfying (23)–(24) arises from a unique pair (g, ψ), where g is the Riemannian metric in the conformal class [g +] = [g −] defined by
and ψ is the ∗-Killing 2-form relative to g defined by
Proof
Before starting the proof, we recall the following general facts. (i) For any two Riemannian metrics, g and \(\tilde{g} =\varphi ^{-2}\,g\), in a same conformal class, and for any anti-symmetric endomorphism, A, of the tangent bundle with respect to the conformal class \([g] = [\tilde{g}]\), the covariant derivatives \(\nabla ^{\tilde{g}}A\) and ∇g A are related by
by setting \(A\left (\frac{d\varphi } {\varphi } \right ) = -\frac{d\varphi } {\varphi } \circ A\). (ii) For any 1-form β and any vector field X, we have
and
for any orthogonal (almost) complex structures J + and J− inducing the chosen and the opposite orientation respectively.
From (14), (19), (20) and (28), we thus infer
which, by using (27), is equivalent to
where \(\nabla ^{g_{+}}\) denotes the Levi-Civita connection of the conformal metric g + = f + −2 g, meaning that the pair (g +, J +) is Kähler. Similarly, we have
or, equivalently:
where \(\nabla ^{g_{-}}\) denotes the Levi-Civita connection of the conformal metric g − = f − −2 g, meaning that the pair (g −, J −) is Kähler as well. We thus get on M 0 an ambikähler structure in the sense of Definition 2.1. Moreover, because of (21), f + and f − evidently satisfy (23)–(24).
For the converse, define g by
and denote by ∇ the Levi-Civita connection of g. By defining \(\Psi _{+} = f_{+}\,J_{+}\), \(\Psi _{-} = f_{-}\,J_{-}\) and \(\Psi = \Psi _{+} + \Psi _{-}\), we get
Similarly,
By using (21), we obtain
with α: = −2 J + df + = −2 J − df −, meaning that the associated 2-form \(\psi (X,Y ):= g(\Psi (X),Y )\), is ∗-Killing. Finally ψ = f + g(J +⋅ , ⋅ ) + f − g(J −⋅ , ⋅ ) = f + 3 ω + + f − 3 ω −. □
Remark 2.2
The fact that the pair (g + = f + −2 g, J +), resp. the pair (g − = f − −2 g, J −), is Kähler only depends on, in fact is equivalent to, \(\Psi _{+} = f_{+}\,J_{+}\), resp. \(\Psi _{-} = f_{-}\,J_{-}\), being conformal Killing, i.e. ψ being conformally Killing. This was observed in a twistorial setting by Pontecorvo in [12], cf. also Appendix B2 in [4].
We now explain under which circumstances an ambikähler structure satisfies the conditions (23)–(24).
Proposition 2.2
Let M be an oriented 4-manifold equipped with an ambikähler structure (g +, J +, ω +), (g − = f 2 g +, J −, ω −). Assume moreover that f is not constant. Then, on the open set where f ≠ 1, there exist non-constant positive functions f +, f − satisfying (23)–(24) of Proposition 2.1 if and only if the 1-form
is exact.
Proof
For any ambikähler structure (g +, J +, ω +), (g − = f 2 g +, J −, ω −) and any positive functions f +, f − satisfying (23)–(24), we have
On the open set where f ≠ 1, this can be rewritten as
in particular, κ is exact on this open set. Conversely, if κ is exact, but not identically zero, then \(\kappa = \frac{d\varphi } {\varphi }\), for some, non-constant, positive function, φ, and we then define f +, f − by \(\frac{df_{+}} {f_{+}} = \frac{d\varphi } {\varphi } + \frac{df} {f(1-f^{2})}\) and \(\frac{df_{-}} {f_{-}} = \frac{d\varphi } {\varphi } + \frac{f\,df} {(1-f^{2})}\), hence by \(f_{+}:= \frac{f\,\varphi } {\vert 1-f^{2}\vert ^{\frac{1} {2} }}\) and \(f_{-}:= \frac{\varphi } { \vert 1-f^{2}\vert ^{\frac{1} {2} }}\), which clearly satisfy (23)–(24). □
Remark 2.3
It follows from (39) that if f = k, where k is a constant different from 1, then f + and f − are constant and the corresponding ∗-Killing 2-form ψ is then parallel. More generally, the existence of a pair (g, ψ) inducing an ambikähler structure depends on the chosen relative scaling of the Kähler metrics. More precisely, if the ambikähler structure (g +, J +, ω +), (g − = f 2 g +, J −, ω −) arises from a ∗-Killing 2-form in the conformal class, in the sense of Proposition 2.1, then for any positive constant k ≠ 1, the ambikähler structure (g +, J +, ω +), \((\tilde{g}_{-} = k^{2}\,g_{-},J_{-},k^{2}\omega _{-})\) does not arise from a ∗-Killing 2-form, unless τ(df) = ±df. This is because the 1-forms \(\frac{\tau (df)} {(1-f^{2})}\) and \(\frac{\tau (df)} {(1-k^{2}\,f^{2})}\) would then be both closed, implying that τ(df) = ϕ df for some function ϕ; since | τ(df) | = | df |, we would then have ϕ = ±1.
The 1-form κ in Proposition 2.2 is clearly exact on the open set where f ≠ 1 whenever τ(df) = df or τ(df) = −df, and it readily follows from (40) that f +, f − are then given by
if τ(df) = df, or by
if τ(df) = −df, for some positive constant c. If
denotes the orthogonal splitting determined by τ, where τ is the identity on T + and minus the identity on T −—equivalently, J +, J − coincide on T + and are opposite on T −—then τ(df) = ±df if and only if \(df_{\vert T^{\mp }} = 0\) and we also have:
Proposition 2.3
The distribution T ± is involutive if and only if τ(df) = ±df.
Proof
For a general ambikähler structure (g +, J +, ω +) and (g − = f 2 g +, J −, ω −), with g − = f 2 g +, we have
for any X, Y in T + and any Z in T −, and
for any X, Y in T − and any Z in T +. This can be shown as follows. Suppose that X, Y are in T + and Z is in T −. Then, since the Kähler form ω +(⋅ , ⋅ ) = g +(J +⋅ , ⋅ ) and ω −(⋅ , ⋅ ) = g(J −⋅ , ⋅ ) are closed and T +, T − are ω +- and ω −-orthogonal, we have
and
which can be rewritten as
or else:
Comparing (46) and (49), we readily deduce the first identity in (44); the other three identities are checked similarly. Proposition 2.3 then readily follows from (44)–(45). □
In the following statement, M 0 stills denotes the (dense) open subset of M defined by (16); we also denote by M S the open subset of M defined by
on which ψ is a symplectic 2-form, and by M 1 the intersection M 1: = M 0 ∩ M S .
Proposition 2.4
Let (M, g) be an oriented Riemannian 4-dimensional manifold admitting a non-parallel ∗-Killing 2-form ψ. Denote by (g + = f + 2 g, J +, ω +), (g − = f − 2 g, J −, ω −) the induced ambikähler structure on M 0 as explained above. Then, on the open set M 1 , the Ricci endomorphism, Ric, of g is J + - and J − -invariant, hence of the form
for some functions a, b, where I denotes the identity of TM 1 and τ is defined by (17). Moreover, the vector field
is Killing with respect to g and preserves the whole ambikähler structure.
Proof
Let R be the curvature tensor of g, defined by
for any vector field X, Y, Z. We denote by Scal its scalar curvature, by Ric0 the trace-free part of Ric, by W the Weyl tensor of g, and by W+ and W− its self-dual and anti-self-dual part respectively. As in the previous section, \(\Psi\) denotes the skew-symmetric endomorphism of TM determined by ψ, \(\Psi _{+}\) its self-dual part, \(\Psi _{-}\) its anti-self-dual part, with \(\Psi _{+} = f_{+}\,J_{+}\) and \(\Psi _{-} = f_{-}\,J_{-}\) on M 0. Since g = f + 2 g + = f − 2 g −, where g + and g − are Kähler with respect to J + and J − respectively, W+ and W− are both degenerate and \(\mathrm{W}^{+}(\Psi _{+}) =\lambda _{+}\,\Psi _{+}\), \(\mathrm{W}^{-}(\Psi _{-}) =\lambda _{-}\Psi _{-}\), for some functions λ +, λ −. For any vector fields X, Y on M, the usual decomposition of the curvature tensor reads:
by setting {Ric0, X ♭ ∧Y }: = Ric0 ∘ (X ♭ ∧Y ) + (X ♭ ∧Y ) ∘ Ric0 = Ric0(X) ∧Y + X ∧ Ric0(Y ), cf. e.g. [5, Chapter 1, Section G]. On M 0 we then have:
and
We thus get
Similarly,
and
On the other hand, from (18), we get
hence
whereas, from (19), we obtain
hence
where \(\left (\nabla \alpha \right )^{s}\) denotes the symmetric part of ∇α. Indeed, we have
as δα = 0 and \(e_{i}\lrcorner {\ast} (\nabla _{Y }\alpha \wedge e_{i})\) is clearly equal to zero thanks to the general identity (11). We obtain similarly:
From the above, we infer
It follows that
and that the vector field \(\alpha ^{\sharp _{g}}\) is Killing with respect to g if and only if \([\mathrm{Ric},\Psi _{+}] = [\mathrm{Ric},\Psi _{-}] = 0\). We now show that (68) actually implies \([\mathrm{Ric},\Psi _{+}] = [\mathrm{Ric},\Psi _{-}] = 0\) at each point where f + ≠ f −. Indeed, in terms of the decomposition (4), Ric, J +, J − can be written in the following matricial form
where J denotes the restriction of J + on T + and on T −, so that:
Then (68) can be expanded as
Since f + > 0 and f − > 0 on M 0, from (71) we readily infer [R, J] = 0 and Q = 0, meaning that
Moreover, on the open subset M 1 = M 0 ∩ M S , where f + − f − ≠ 0, we also infer from (71) that [P, J] = 0, hence that [Ric, J +] = [Ric, J −] = 0. By (67), (∇α)s = 0, meaning that the vector field \(K_{1}:= -\frac{1} {2}\alpha ^{\sharp } = J_{ +}\mathrm{grad}_{g}f_{+}\) is Killing with respect to g. Notice that
In particular, K 1 is also Killing with respect to g + and g − and is (real) holomorphic with respect to J + and J −. □
3 Separation of Variables
In this section we restrict our attention to the open subset M 1: = M 0 ∩ M S , defined by the conditions (16) and (50). Recall that since ψ ∧ψ = ψ + ∧ψ + + ψ −∧ψ − = 2( f + − f −) v g , where v g denotes the volume form of g relative to the chosen orientation, M S is the open subset of M where ψ is non-degenerate, hence a symplectic 2-form. According to Proposition 2.4, on M 1 the Ricci tensor Ric is of the form (51), for some functions a, b and the vector field α ♯ is Killing; we then infer from (67) that ∇α ♯ can be written as:
with
We then introduce the functions x, y defined by
Notice that (2x, 2y), resp. (2x, −2y), are the eigenvalues of the Hermitian operator \(-J_{+} \circ \Psi = f_{+}\,\mathrm{I} + f_{-}\,\tau\), resp. \(-J_{-}\circ \Psi = f_{+}\,\tau + f_{-}\,\mathrm{I}\), relative to the eigen-subbundle T + and T − respectively. From (16) and (50) we deduce that x, y are subject to the conditions
whereas, from (21), we infer
In particular, dx, J + dx = J − dx, dy and J + dy = −J − dy are pairwise orthogonal and
We then have:
Proposition 3.1
On each connected component of the open subset of M 1 where dx ≠ 0 and dy ≠ 0, the square norm of dx, dy and the Laplacians of x, y relative to g are given by
where A, B are functions of one variable.
Proof
By using (30) and (32) and setting g τ (X, Y ): = g(τ(X), Y ), we infer from (20) and (74) that
In terms of the functions x, y, this can be rewritten as
In particular:
To simplify the notation, we temporarily put
By contracting ∇dx by dx and ∇dy by dy in (82), and taking (83) into account, we obtain:
From (85), we get
It follows that (x 2 − y 2) F = A(x), for some (smooth) function A of one variable and that \(A'(x) = -(x^{2} - y^{2})\,\Delta x\); likewise, (x 2 − y 2) G = B(y) and \(B'(y) = -(x^{2} - y^{2})\,\Delta y\). □
A simple computation using (83) shows that in terms of A, B, the functions h +, h − appearing in (74) and their derivatives dh +, dh − have the following expressions:
and
In particular:
Proposition 3.2
The vector fields
(which is equal to the vector field \(K_{1} = -\frac{1} {2}\alpha ^{\sharp }\) appearing in Proposition 2.4 ), and
are Killing with respect to g, g +, g − and Hamiltonian with respect to ω + and ω − . The momenta, μ 1 + , μ 2 + of K 1, K 2 with respect to ω + , and the momenta, μ 1 − , μ 2 − , of K 1, K 2 with respect to ω − , are given by
and Poisson commute with respect to ω + and ω − , meaning that ω ±(K 1, K 2) = 0, so that [K 1, K 2] = 0 as well. In particular, on the open set M 1 , the ambikähler structure (g +, J +, ω +), (g −, J −, ω −) is ambitoric in the sense of [4, Definition 3].
Proof
In terms of A, B, (82) can be rewritten as
By taking (30)–(32) into account, we infer
and
In particular, the symmetric parts of ∇(J + dx) and ∇(J + dy) are opposite and given by
The symmetric parts of ∇(J + dx + J + dy) and of ∇(y 2 J + dx + x 2 J + dy) = y 2∇(J + dx) + x 2∇(J + dy) + 2dy ⊗J + dx + 2xdx⊗J + dy then clearly vanish, meaning that K 1 and K 2 are Killing with respect to g. In view of the expressions of K 1, K 2 as symplectic gradients in (91)–(92), K 1 and K 2 are Hamiltonian with respect to ω + and ω −, their momenta are those given by (93) and their Poisson bracket with respect to ω ± is equal to ω ±(K 1, K 2), which is zero, since dx lives in the dual of T + and dy in the dual of T −. This, in turn, implies that K 1 and K 2 commute. □
Remark 3.1
As already observed, the Killing vector field K 1 appearing in Proposition 3.2 is the restriction to M 1 of the smooth vector field, also denoted by K 1, appearing in Proposition 2.4, which is defined on the whole manifold M by
Similarly, it is easily checked that K 2 is the restriction to M 1 of the smooth vector field, still denoted by K 2, defined on M by
(recall that the Killing 2-form φ = ψ + −ψ − = ∗ψ is co-closed). It is also easily checked that K 2 and K 1 are related by
where, we recall, S denotes the Killing symmetric endomorphism defined by (22) in Remark 2.1; this is because, on the dense open subset M 0, S can be rewritten as
whereas K 1 ♭ = J +(dx + dy), so that S(K 1 ♭) = 2y 2 J + dx + 2x 2 J + dy = 2K 2 ♭; we thus get (100) on M 0, hence on M. In view of (100), the fact that K 2 is Killing can then be alternatively deduced from [8, Lemma B], cf. also the proof of [4, Proposition 11 (iii)].
In view of the above, we eventually get the following rough classification:
Proposition 3.3
For any connected, oriented, 4-dimensional Riemannian manifold (M, g) admitting a non-parallel ∗-Killing 2-form ψ, the open subset M S defined by (50)is either empty or dense and we have one of the following three mutually exclusive cases:
-
(1)
M S is dense; the vector fields K 1, K 2 are Killing and linearly independent on a dense open set of M, or
-
(2)
M S is dense; the vector fields K 1, K 2 are Killing and K 2 = c K 1 , for some non-zero real number c, or
-
(3)
M S is empty, i.e. ψ is decomposable everywhere; then, K 2 is identically zero, whereas K 1 is non-identically zero and is not a Killing vector field in general.
Proof
Being Killing on M 0 ∩ M S and zero on any open set where f + = f −, K 2 is Killing everywhere on M. We next observe that, for any x in M S , K 2(x) = 0 if and only if K 1(x) = 0, as readily follows from (100) and from the fact that S is invertible if and only if x belongs to M S , as the eigenvalues of S are equal to \(\frac{(\,f_{+}+f_{-})^{2}} {2}\) and \(\frac{(\,f_{+}-f_{-})^{2}} {2}\).
Suppose now that M S is not dense in M, i.e. that M∖M S contains some non-empty open subset V; then, K 2 vanishes on V, hence vanishes identically on M, as K 2 is Killing; from (99), we then infer \(0 = \Psi (K_{2}) = \frac{1} {8}(\,f_{+}^{2} - f_{ -}^{2})\mathrm{grad}_{ g}(\,f_{+}^{2} - f_{ -}^{2})\), which implies that the (smooth) function ( f + 2 − f − 2)2 is constant on M, hence identically zero, meaning that M S is empty. If M S is empty, then f + = f − everywhere (equivalently, ψ ∧ψ is identically zero); it follows that K 2 is identically zero, whereas K 1, which is not identically zero since ψ is not parallel, is not Killing in general, cf. Sect. 6.
If M S is dense, then K 1 and K 2 are both Killing vector fields on M, hence either linearly independent on some dense open subset of M or dependent everywhere and, by the above discussion, K 2 is then a constant, non-zero multiple of K 1. □
In the next sections we consider in turn the three cases listed in Proposition 3.3.
4 The Ambitoric Ansatz
In this section, we assume that M S is dense and that K 1, K 2 are linearly independent on some dense open set \(\mathcal{U}\). In the remainder of this section, we focus our attention on \(\mathcal{U}\), i.e. we assume that dx and dy are linearly independent everywhere—equivalently, τ(df) ≠ ± df everywhere—so that {dx, J + dx = J − dx, dy, J + dy = −J − dy} form a direct orthogonal coframe. By Proposition 3.1, the metric g and the Kähler forms ω +, ω − can then be written as
and we also have:
Proposition 4.1
The functions Scal = 4a and b appearing in the expression (51) of the Ricci tensor of g are given by:
and
Proof
Since α ♯ is Killing, the Bochner formula reads:
whereas, by (51),
By using
which easily follows from (30)–(32), we infer from (74) that
By putting together (106), (109) and (90), we get
hence
We thus get
By using (87), we obtain (104) and (105). □
Recall that a function φ is called J +-pluriharmonic if d(J + dφ) = 0 and J −-pluriharmonic if d(J − dφ) = 0.
Proposition 4.2
-
(i)
The space of real J + -pluriharmonic functions, modulo additive constants, of the form φ + = φ +(x, y) is generated by φ 1 + , φ 2 + defined by:
$$\displaystyle{ \varphi _{1}^{+}(x,y) =\int ^{x} \frac{\zeta ^{2}d\zeta } {A(\zeta )} -\int ^{y} \frac{\zeta ^{2}d\zeta } {B(\zeta )},\quad \varphi _{2}^{+}(x,y) =\int ^{x} \frac{d\zeta } {A(\zeta )} -\int ^{y} \frac{d\zeta } {B(\zeta )}, }$$(113)where ∫ x , resp. ∫ y , stands for any primitive of the variable x, resp. y.
-
(ii)
The space of real J − -pluriharmonic functions, modulo additive constants, of the form φ − = φ −(x, y) is generated by φ 1 − , φ 2 − defined by:
$$\displaystyle{ \varphi _{1}^{-}(x,y) =\int ^{x} \frac{\zeta ^{2}d\zeta } {A(\zeta )} +\int ^{y} \frac{\zeta ^{2}d\zeta } {B(\zeta )},\quad \varphi _{2}^{-}(x,y) =\int ^{x} \frac{d\zeta } {A(\zeta )} +\int ^{y} \frac{d\zeta } {B(\zeta )}. }$$(114)
Proof
From (95)–(96), we readily infer the following expression of d(J ± dx) and d(J ± dy):
and
Let φ = φ(x, y) be any function of x, y and denote by φ x , φ y , φ xx , etc. … its derivative with respect to x, y. Then
By (115)–(116), φ is J +-pluriharmonic if and only if φ xy = 0—meaning that φ is of the form φ(x, y) = C(x) + D(y)—and C, D satisfy
or, equivalently, by multiplying the first equation by A(x) and the second by B(y), which are both positive, and by setting F(x): = A(x)C′(x), G(y): = B(y)D′(y):
It is easily checked that the general solution of this system is given by:
for real constants k 1, k 2. We thus get Part (i) of Proposition 4.2. Part (ii) is obtained similarly. □
In view of Proposition 4.2, we (locally) define s and t, up to additive constants, by
Equivalently:
Notice that \(ds \wedge dt = \frac{(x^{2}-y^{2})} {A(x)B(y)}\,J_{+}dx \wedge J_{+}dy\); it then follows from Proposition 3.1 that
where v g denotes the volume form of g with respect to the orientation induced by J +, showing that dx, dy, ds, dt form a direct coframe. In view of (102), (103), (122), on the open set where x, y, s, t form a coordinate system, the metrics g, g +, g −, the complex structures J +, J −, the involution τ and the Kähler forms ω +, ω − have the following expressions:
while it follows from (26) that the ∗-Killing 2-form ψ is given by
Notice that, in view of (124), the (local) vector fields \(\frac{\partial } {\partial s}\) and \(\frac{\partial } {\partial t}\) are Killing with respect to g and respectively coincide with the Killing vector fields K 1 and K 2 appearing in Proposition 3.2 on their domain of definition.
It turns out that the expressions of (g + = (x + y)−2 g, J +, ω +) and (g − = (x − y)−2 g, J −, ω −) just obtained coincide with the ambitoric Ansatz described in [4, Theorem 3], in the case where the quadratic polynomial is q(z) = 2z, which is the normal form of the ambitoric Ansatz in the hyperbolic case considered in [4, Paragraph 5.4].
The discussion in this section can then be summarized as follows:
Theorem 4.1
Let (M, g) be a connected, oriented, 4-dimensional manifold admitting a non-parallel, ∗-Killing 2-form ψ = ψ + + ψ − and assume that the open set, M S , where | ψ + | ≠ | ψ − | is dense, cf. Proposition 3.3 . On the open subset, \(\mathcal{U}\) , of M S where ψ + and ψ − have no zero and d | ψ + | ∧ d | ψ − | ≠ 0, the pair (g, ψ) gives rise to an ambitoric structure of hyperbolic type, in the sense of [4], relative to the conformal class of g, which, on any simply-connected open subset of \(\mathcal{U}\) , is described by (124)–(125)–(127), where the Hermitian structures (g + = (x + y)−2 g, J +, ω +) and (g − = (x − y)−2 g, J −, ω −) are Kähler, while ψ is described by (128).
Conversely, on the open set, \(\mathcal{U}\) , of \(\mathbb{R}^{4}\) , of coordinates x, y, s, t, with x > | y | > 0, the two almost Hermitian structures (g + = (x + y)−2 g, J +, ω +), (g − = (x − y)−2 g, J −, ω −) defined by (124)–(125)–(127), with A(x) > 0 and B(y) > 0, are Kähler and, together with the Killing vector fields \(K_{1} = \frac{\partial } {\partial s}\) and \(K_{2} = \frac{\partial } {\partial t}\) , constitute an ambitoric structure of hyperbolic type, while the 2-form ψ defined by (128) is ∗-Killing with respect to g.
Proof
The first part follows from the preceding discussion. For the converse, we first observe that the 2-forms ω + and ω − defined by (127) are clearly closed and not degenerate. To test the integrability of the almost complex structures J + and J − defined by (125), we consider the complex 1-forms:
which generate the space of (1, 0)-forms with respect to J +. We then have:
which shows that J + is integrable. For J −, we likewise consider the complex 1-forms:
which generate the space of (1, 0)-forms with respect to J +. We then get
which, again, shows that J − is integrable. It follows that the almost Hermitian structures (g + = (x + y)−2 g, J +, ω +) and (g − = (x − y)−2 g, J −, ω −) are both Kähler and thus determine an ambikähler structure on \(\mathcal{U}\). Moreover, the vector fields \(\frac{\partial } {\partial s}\) and \(\frac{\partial } {\partial t}\) are clearly Killing with respect to g, g +, g −, and satisfy:
meaning that they are both Hamiltonian with respect to ω + and ω −, with momenta given by (93) in Proposition 3.2. This implies that \(\frac{\partial } {\partial s}\) and \(\frac{\partial } {\partial t}\) preserve the two Kähler structures (g +, J +, ω +) and (g −, J −, ω −) and actually coincide with the vector field K 1 and K 2 respectively defined in a more general context in Proposition 3.2. We thus end up with an ambitoric structure, as defined in [4]. According to Theorem 3 in [4], it is an ambitoric structure of hyperbolic type, with “quadratic polynomial” q(z) = 2z. To check that the 2-form ψ defined by (128)—which is evidently closed—is ∗-Killing with respect to g, denote by f +, f − the positive functions on \(\mathcal{U}\) defined by f + = x + y, f − = x − y, so that g + = f + −2 g, g − = f − −2 g and ψ = f + 3 ω + + f − 3 ω −; it then follows from (126) that τ(df +) = df −, hence that ψ is ∗-Killing by Proposition 2.1. □
5 Ambikähler Structures of Calabi Type
The second case listed in Proposition 3.3, which is considered in this section, can be made more explicit via the following proposition:
Proposition 5.1
Let (M, g) be a connected, oriented, Riemannian 4-manifold admitting a non-parallel ∗-Killing 2-form ψ = ψ + + ψ − . In view of Proposition 3.3 , assume that the open set M S —where ψ is non-degenerate—is dense in M and that the Killing vector fields K 1, K 2 defined by (98)–(99) are related by K 2 = c K 1 , for some non-zero real number c. Then, c is positive and one of the following three cases occurs:
-
(1)
\(f_{+}(\mathsf{x}) + f_{-}(\mathsf{x}) = 2\sqrt{c}\) , for any x in M, or
-
(2)
\(f_{+}(\mathsf{x}) - f_{-}(\mathsf{x}) = 2\sqrt{c}\) , for any x in M, or
-
(3)
\(f_{-}(\mathsf{x}) - f_{+}(\mathsf{x}) = 2\sqrt{c}\) , for any x in M,
with the usual notation: \(f_{+} = \vert \psi _{+}\vert /\sqrt{2}\) and \(f_{-} = \vert \psi _{-}\vert /\sqrt{2}\) .
Proof
First recall that \((\Psi _{+} + \Psi _{-}) \circ (\Psi _{+} - \Psi _{-}) = -(\,f_{+}^{2} - f_{-}^{2})\,\mathrm{I}\). From (99) and K 1 = J +grad g f + = J −grad g f −, we then infer
On M S , where \(\Psi\) is invertible, the identity K 2 = c K 1 then reads:
or, else:
Since | df + | = | df − | on M 0, on M 1 = M 0 ∩ M S we then get:
Since M 1 is dense this identity actually holds on the whole manifold M. It can be rewritten as
this forces c to be positive—if not, f + 2 − f − 2 would be identically zero—and we eventually get the identity:
Denote by \(\tilde{M}\) the open subset of M obtained by removing the zero locus K 1 −1(0) of K 1 from M (notice that \(\tilde{M}\) is a connected, dense open subset of M, as K 1 −1(0) is a disjoint union of totally geodesic submanifolds of codimension a least 2). It readily follows from (139) that \(\tilde{M}\) is the union of the following four closed subsets \(\tilde{F}_{0}:= F_{0} \cap \tilde{ M}\), \(\tilde{F}_{+}:= F_{+} \cap \tilde{ M}\), \(\tilde{F}_{-}:= F_{-}\cap \tilde{ M}\) and \(\tilde{F}_{S}:= F_{S} \cap \tilde{ M}\) of \(\tilde{M}\), where F 0, F +, F −, F S denote the four closed subsets of M defined by:
We now show that if the interior, V, of \(\tilde{F}_{0}\) is non-empty then \(\tilde{F}_{0} =\tilde{ M}\) (and thus F 0 = M by density); this amounts to showing that the boundary \(B:=\bar{ V }\setminus V\) of V in \(\tilde{M}\) is empty. If not, let x be any element of B; then, x belongs to \(\tilde{F}_{0}\), as \(\tilde{F}_{0}\) is closed, and it also belongs to \(\tilde{F}_{+}\) or \(\tilde{F}_{-}\): otherwise, there would exist an open neighbourhood of x disjoint from \(\tilde{F}_{+} \cup \tilde{ F}_{-}\), hence contained in \(\tilde{F}_{0} \cup \tilde{ F}_{S}\); as \(\tilde{F}_{S}\) has no interior, this neighbourhood would be contained in contained in \(\tilde{F}_{0}\), which contradicts the fact that x sits on the boundary of V. Without loss, we may thus assume that x belongs to \(\tilde{F}_{+}\), so that \(f_{+}(\mathsf{x}) = 2\sqrt{c}\) and f −(x) = 0; since K 1(x) ≠ 0—by the very definition of \(\tilde{M}\)—f + is regular at x, implying that the locus of \(f_{+} = 2\sqrt{c}\) is a smooth hypersurface, S, of \(\tilde{M}\) near x; moreover, since \(\tilde{F}_{+}\) and \(\tilde{F}_{-}\) are disjoint, f − = 0 on S, meaning that \(\Psi _{-} = 0\) on S; for any X in T x S we then have \(\nabla _{X}\Psi _{-} = 0\). On the other hand, \(\nabla _{X}\Psi = (\alpha (\mathsf{x}) \wedge X)_{-}\), for any X in T x M, cf. (19), and we can then choose X in T x S in such a way that (α(x) ∧ X)− be non-zero, hence ∇ X ψ − ≠ 0, contradicting the previous assertion. We similarly show that M = F + or M = F − whenever the interior of \(\tilde{F}_{+}\) or of \(\tilde{F}_{-}\) is non-empty. □
A direct consequence of Proposition 5.1 is that on the (dense) open subset M 0, the associated ambikähler structure \((g_{+} = f_{+}^{-2}\,g,J_{+} = f_{+}^{-1}\,\Psi _{+},\omega _{+})\), \((g_{-} = f_{-}^{-2}\,g = f^{2}\,g_{+},J_{-} = f_{-}^{-1}\,\Psi _{-},\omega _{-})\), with f = f +∕f −, satisfies
in the first case listed in Proposition 5.1, and
in the remaining two cases. The ambikähler structure is then of Calabi type, according to the following definition, taken from [1]:
Definition 5.1
An ambikähler structure (g +, J +, ω +), (g −, J −, ω −), with g + = f −2 g −, is said to be of Calabi type if df ≠ 0 everywhere, and if there exists a non-vanishing vector field K, Killing with respect to g + and g − and Hamiltonian with respect to ω + and ω −, which satisfies
with τ = −J + J − = −J − J +.
By replacing the pair (J +, J −) by the pair (J +, −J −) if needed, we can assume, without loss of generality, that τ(K) = −K. In the following proposition, we recall some general facts concerning this class of ambikähler structures, cf. e.g. [1, Section 3]:
Proposition 5.2
For any ambikähler structure of Calabi type, with τ(K) = −K:
-
(i)
The Killing vector field K is an eigenvector of the Ricci tensor, \(\mathrm{Ric}^{g_{+}}\) , of g + and of the Ricci tensor, \(\mathrm{Ric}^{g_{-}}\) , of g − ; in particular, \(\mathrm{Ric}^{g_{+}}\) and \(\mathrm{Ric}^{g_{-}}\) are both J + - and J − -invariant;
-
(ii)
the Killing vector field K is a constant multiple of \(J_{-}\mathrm{grad}_{g_{-}}f = J_{+}\mathrm{grad}_{g_{+}} \frac{1} {f}\) .
Proof
By hypothesis, \(K = J_{+}\mathrm{grad}_{g_{+}}z_{+} = J_{-}\mathrm{grad}_{g_{-}}z_{-}\), for some real functions z + and z −. Since J − K = −J + K, we infer \(\mathrm{grad}_{g_{+}}z_{+} = -\mathrm{grad}_{g_{-}}z_{-}\), hence
Since df ≠ 0 everywhere, this, in turn, implies that
for some real (smooth) functions F, G defined on \(\mathbb{R}^{>0}\) up to an additive constant and satisfying:
Moreover,
Since K has no zero and satisfies τ(K) = −K, we have
so that
In (148)–(149), the dual 1-form K ♭ and the square norm | K |2 are relative to any metric in [g +] = [g −]. For definiteness however, we agree that they are both relative to g +. Since g + = f −2 g −, we have:
By using (24), we then infer from (149):
By contracting with K, and by using K ♭ = F′ J + df and \(J_{+}\nabla _{X}^{g_{+}}K = \nabla _{J_{ +}X}K\) (as K is J ±-holomorphic), we obtain
Since K is Killing with respect to g +, \(\nabla ^{g_{+}}K\) is anti-symmetric; in view of (152), this forces | K |2 to be of the form
for some (smooth) function H from \(\mathbb{R}^{>0}\) to \(\mathbb{R}^{>0}\), hence
By substituting (154) in (152), we eventually get the following expression of \(\nabla ^{g_{+}}K\):
with
Since K is Killing with respect to g +, it follows from the Bochner formula that
whereas, from (155) we get
and, from \(\nabla _{X}^{g_{+}}J_{-} = [J_{-}, \frac{df} {f} \wedge X]\):
By putting together (155), (157)–(159), we get
with
Since the metric g + is Kähler with respect to J +, in particular is J +-invariant, (160) implies that the two eigenspaces of \(\mathrm{Ric}^{g_{+}}\) are the space {K, J + K} generated by K and J + K (where J − = J +) and its orthogonal complement, {K, J + K}⊥ (where J − = −J +). It follows that \(\mathrm{Ric}^{g_{+}}\) is both J +- and J −-invariant. This establishes the part (i) of the proposition (it is similarly shown that \(\mathrm{Ric}^{g_{-}}\) is J +- and J −-invariant).
Before proving part (ii), we first recall the general transformation rules of the curvature under a conformal change of the metric. If g and \(\tilde{g} =\phi ^{-2}\,g\) are two Riemannian metrics in a same conformal class [g] in any n-dimensional Riemannian manifold (M, g), n > 2, then the scalar curvature, \(\mathrm{Scal}^{\tilde{g}}\), and the trace-free part, \(\mathrm{Ric}_{0}^{\tilde{g}}\), of \(\tilde{g}\) are related to the scalar curvature, Scalg, and the trace-free part, Ric0 g, of g by
and
where \(\left (\nabla ^{g}d\phi \right )_{0}\) is the trace-free part of the Hessian ∇g dϕ of ϕ with respect of g, cf. e.g. [5, Chapter 1, Section J]. Applying (163) to the conformal pair (g −, g + = f −2 g −), we get
Since \(\mathrm{Ric}^{g_{+}}\) and \(\mathrm{Ric}^{g_{+}}\) are both J +- and J −-invariant, it follows that \(\left (\nabla ^{g_{-}}df\right )_{0}\) is J −-invariant, as well as \(\nabla ^{g_{-}}df\), since all metrics in [g +] = [g −] are J +- and J −-invariant. This means that the vector field \(\mathrm{grad}_{g_{-}}f\) is J −-holomorphic, hence that \(J_{-}\mathrm{grad}_{g_{-}}f\) is Hamiltonian with respect to ω −, hence Killing with respect to g −; since \(J_{-}\mathrm{grad}_{g_{-}}f = \frac{1} {G'(\,f)}\,K\), we conclude that G′( f) is constant, hence, by using (146), that F( f) and G( f) are of the form
for a non-zero real constant a and arbitrary real constants b, c. This, together with (24), establishes part (ii) of the proposition. □
Theorem 5.1
Let (M, g) be a connected, oriented 4-manifold admitting a non-parallel ∗-Killing 2-form ψ = ψ + + ψ − , satisfying the hypothesis of Proposition 5.1 , corresponding to Case (2) of Proposition 3.3 . Then, on the dense open set M 0 ∖K 1 −1(0) the associated ambikähler structure is of Calabi type, with respect to the Killing vector field K = K 1 , with τ(K) = −K in the first case of Proposition 5.1 and τ(K) = K in the two remaining cases.
Conversely, let (g +, J +, ω +), (g − = f 2 g +, J −, ω −) be any ambikähler structure of Calabi type with non-vanishing Killing vector field K, defined on some oriented 4-dimensional manifold M. If τ(K) = −K, there exist, up to scaling, a unique metric g in the conformal class [g +] = [g −] and a unique non-parallel ∗-Killing 2-form ψ with respect to g, inducing the given ambikähler structure. If τ(K) = K, such a pair (g, ψ) exists and is unique outside the locus { f = 1}.
Proof
The first part of the proposition has already been discussed in the preceding part of this section. Conversely, let (g +, J +, ω +), (g − = f 2 g +, J −, ω −) be an ambikähler structure of Calabi type, with respect to some non-vanishing Killing vector field K, with τ(K) = −K or τ(K) = K. Then, according to Proposition 5.2, K can be chosen equal to
if τ(K) = −K, or
if τ(K) = K. According to Proposition 2.2 and (42), if τ(K) = −K, hence τ(df) = −df, the ambikähler structure is then induced by the metric g, in the conformal class [g +] = [g −], defined by g = f + −2 g + = f − −2 g −, with
for some positive constant c, and the ∗-Killing 2-form ψ defined by
If τ(K) = K, hence τ(df) = df, it similarly follows from Proposition 2.2 and (41) that the ambikähler structure is induced by the metric g = f + 2 g + = f − 2 g −, with
for some constant c, positive if f < 1, negative if f > 1, and the ∗-Killing 2-form
but the pair (g, ψ) is only defined outside the locus { f = 1}. □
Remark 5.1
Any ambikähler structure (g +, J +, ω +), (g −, J −, ω −) generates, up to global scaling, a 1-parameter family of ambikähler structures, parametrized by a non-zero real number k, obtained by, say, fixing the first Kähler structure (g +, J +, ω +) and substituting (g − (k) = k −2 g − = f k 2 g +, J − (k) = ε(k) J −, ω − (k) = ε(k) k −2 ω −) to the second one, with \(\epsilon (k) = \frac{k} {\vert k\vert }\) and \(f_{k} = \frac{f} {\vert k\vert }\). Assume that the ambikähler structure (g +, J +, ω +), (g −, J −, ω −) is of Calabi type, with τ(df) = −df. For any k in \(\mathbb{R}\setminus \{0\}\), we then have τ (k)(df k ) = −ε(k) df k , by setting τ (k) = −J + J − (k) = −J − (k) J + = ε(k) τ, whereas, from (40) we infer:
up to global scaling; the ambikähler structure (g +, J +, ω +), (g − (k), J − (k), ω − (k)) is then induced by the pair (g (k), ψ (k)), where g (k) is defined in the conformal class by
and ψ (k) is the ∗-Killing 2-form with respect to g (k) defined by
both defined outside the locus { f + k = 0}.
Remark 5.2
As observed in [1, Section 3.1], any ambikähler structure of Calabi type (g +, J +, ω +), (g − = f 2 g +, J −, ω −), with τ(df) = df, admits a Hamiltonian 2-form, ϕ +, with respect to the Kähler structure (g +, J +, ω +) and a Hamiltonian 2-form, ϕ −, with respect to the (g −, J −, ω −), given by
6 The Decomposable Case
Assume now that (M, g, ψ) is as in Case (3) in Proposition 3.3, that is, that the ∗-Killing 2-form ψ = ψ + + ψ − is degenerate (or decomposable). This latter condition holds if and only if ψ ∧ψ = 0, if and only if | ψ + | = | ψ − |, i.e. f + = f − =: φ, or f = 1, meaning that g + = g − =: g K , whereas g = φ 2 g K . Denote by ∇K the Levi-Civita connection of g K . Then from (31)– (33) we get ∇K J + = ∇K J − = ∇K τ = 0, which implies that (M, g K ) is locally a Kähler product of two Kähler curves of the form \(M = (\Sigma,g_{\Sigma },J_{\Sigma },\omega _{\Sigma }) \times (\tilde{\Sigma },g_{\tilde{\Sigma }},J_{\tilde{\Sigma }},\omega _{\tilde{\Sigma }})\), with
Moreover, from (21) we readily infer τ(dφ) = dφ, meaning that φ is the pull-back to M of a function defined on \(\Sigma\). Conversely, for any Kähler product \(M = (\Sigma,g_{\Sigma },J_{\Sigma },\omega _{\Sigma }) \times (\tilde{\Sigma },g_{\tilde{\Sigma }},J_{\tilde{\Sigma }},\omega _{\tilde{\Sigma }})\) as above and for any positive function φ defined on \(\Sigma\), regarded as a function defined on M, the metric \(g:=\varphi ^{2}\,(g_{\Sigma } + g_{\tilde{\Sigma }})\) admits a ∗-Killing 2-form ψ, given by
whose corresponding Killing 2-form ∗ψ is given by
Note that by (9) \(\alpha = \frac{1} {3}\delta ^{g}\psi = \frac{1} {\varphi ^{2}} {\ast}_{\Sigma }d\varphi\), so \(K_{1} = -\frac{1} {2}\alpha ^{\sharp }\) is not a Killing vector field in general.
The above considerations completely describe the local structure of 4-manifolds with decomposable ∗-Killing 2-forms. They also provide compact examples, simply by taking \(\Sigma\) and \(\tilde{\Sigma }\) to be compact Riemann surfaces. We will show, however, that there are compact 4-manifolds with decomposable ∗-Killing 2-forms which are not products of Riemann surfaces (in fact not even of Kähler type). They arise as special cases (for n = 4) of the classification, in [9], of compact Riemannian manifolds (M n, g) carrying a Killing vector fields with conformal Killing covariant derivative.
It turns out that if ψ is a non-trivial ∗-Killing 2-form which can be written as ψ = dξ ♭ for some Killing vector field ξ on M, then either ψ has rank 2 on M, or M is Sasakian or has positive constant sectional curvature (Proposition 4.1 and Theorem 5.1 in [9]). For n = 4, the Sasakian situation does not occur, and the case when M has constant sectional curvature will be treated in detail in the next section. The remaining case—when ψ is decomposable—is the one which we are interested in, and is described by cases 3. and 4. in Theorem 8.9 in [9]. We obtain the following two classes of examples:
-
1.
(M, g) is a warped mapping torus
$$\displaystyle{M = (\mathbb{R} \times N)/_{(t,x)\sim (t+1,\varphi (x))},\qquad g =\lambda ^{2}d\theta ^{2} + g_{ N},}$$where (N, g N ) is a compact 3-dimensional Riemannian manifold carrying a function λ, such that dλ ♯ is a conformal vector field, φ is an isometry of N preserving λ, \(\xi = \frac{\partial } {\partial \theta }\) and ψ = dξ ♭ = 2λdλ ∧ dθ. One can take for instance \((N,g_{N}) = \mathbb{S}^{3}\) and λ a first spherical harmonic. Further examples of manifolds N with this property are given in Section 7 in [9].
-
2.
(M, g) is a Riemannian join \(\mathbb{S}^{2} {\ast}_{\gamma,\lambda }\mathbb{S}^{1}\), defined as the smooth extension to S 4 of the metric \(g = ds^{2} +\gamma ^{2}(s)g_{\mathbb{S}^{2}} +\lambda ^{2}(s)d\theta ^{2}\) on (0, l) × S 2 × S 1, where l > 0 is a positive real number, \(\gamma: (0,l) \rightarrow \mathbb{R}^{+}\) is a smooth function satisfying the boundary conditions
$$\displaystyle{\gamma (t) = t(1 + t^{2}a(t^{2}))\quad \mbox{ and}\quad \gamma (l - t) = \frac{1} {c} + t^{2}b(t^{2}),\quad \forall \ \vert t\vert <\epsilon,}$$for some smooth functions a and b defined on some interval (−ε, ε), λ(s): = ∫ s l γ(t)dt, \(\xi = \frac{\partial } {\partial \theta }\) and ψ = 2λ(s)λ′(s)ds ∧ dθ.
In particular, we obtain infinite-dimensional families of metrics on S 3 × S 1 and on S 4 carrying decomposable ∗-Killing 2-forms.
7 Example: The Sphere \(\mathbb{S}^{4}\) and Its Deformations
We denote by \(\mathbb{S}^{4}:= (S^{4},g)\) the 4-dimensional sphere, embedded in the standard way in the Euclidean space \(\mathbb{R}^{5}\), equipped with the standard induced Riemannian metric, g, of constant sectional curvature 1, namely the restriction to \(\mathbb{S}^{4}\) of the standard inner product (⋅ , ⋅ ) of \(\mathbb{R}^{5}\). We first recall the following well-known facts, cf. e.g. [13]. Let ψ = ψ + + ψ − be any ∗-Killing 2-form with respect to g, so that \(\nabla _{X}\Psi =\alpha \wedge X\), cf. (8). Since g is Einstein, the vector field α ♯ is Killing and it follows from (74)–(75) that ∇α = ψ. Conversely, for any Killing vector field Z on \(\mathbb{S}^{4}\), it readily follows from the general Kostant formula
that, in the current case, ∇ X (∇Z) = Z ∧ X, so that the 2-form ψ: = ∇Z ♭ is ∗-Killing with respect to g. The map Z ↦ ∇Z ♭ is then an isomorphism from the space of Killing vector fields on \(\mathbb{S}^{4}\) to the space of ∗-Killing 2-forms.
It is also well-known that there is a natural 1 − 1-correspondence between the Lie algebra \(\mathfrak{s}\mathfrak{o}(5)\) of anti-symmetric endomorphisms of \(\mathbb{R}^{5}\) and the space of Killing vector fields on \(\mathbb{S}^{4}\): for any a in \(\mathfrak{s}\mathfrak{o}(5)\), the corresponding Killing vector field, Z a , is defined by
for any u in \(\mathbb{S}^{4}\), where a(u) is viewed as an element of the tangent space \(T_{u}\mathbb{S}^{4}\), via the natural identification \(T_{u}\mathbb{S}^{4} = u^{\perp }\).
By combining the above two isomorphisms, we eventually obtained a natural identification of \(\mathfrak{s}\mathfrak{o}(5)\) with the space of ∗-Killing 2-forms on \(\mathbb{S}^{4}\) and it is easy to check that, for any a in \(\mathfrak{s}\mathfrak{o}(5)\), the corresponding ∗-Killing 2-form, ψ a , is given by
for any u in \(\mathbb{S}^{4}\) and any X, Y in \(T_{u}\mathbb{S}^{4} = u^{\perp }\); alternatively, the corresponding endomorphism \(\Psi _{\mathsf{a}}\) is given by
for any X in \(T_{u}\mathbb{S}^{4} = u^{\perp }\).
Since, for any u in \(\mathbb{S}^{4}\), the volume form of \(\mathbb{S}^{4}\) is the restriction to \(T_{u}\mathbb{S}^{4}\) of the 4-form \(u\lrcorner \mathrm{v}_{0}\), where v0 stands for the standard volume form of \(\mathbb{R}^{5}\), namely v0 = e 0 ∧ e 1 ∧ e 2 ∧ e 3 ∧ e 4, for any direct frame of \(\mathbb{R}^{5}\) (here identified with a coframe via the standard metric), we easily check that, for any a in \(\mathfrak{s}\mathfrak{o}(5)\), the corresponding Killing 2-form ∗ψ a has the following expression
for any u in \(\mathbb{S}^{4}\) and any X, Y in \(T_{u}\mathbb{S}^{4} = u^{\perp }\); here, ∗5 denotes the Hodge operator on \(\mathbb{R}^{5}\) and we keep identifying vector and covectors via the Euclidean inner product.
From (182), we easily infer
at any u in \(\mathbb{S}^{4}\), where the norm is the usual Euclidean norm of endomorphisms, whereas the Pfaffian of ψ a is given by:
On the other hand, when f +, f − are defined by (15), we have
and
For any a in \(\mathfrak{s}\mathfrak{o}(5)\), we may choose a direct orthonormal basis e 0, e 1, e 2, e 3, e 4 of \(\mathbb{R}^{5}\), with respect to which a has the following form
for some real numbers λ, μ, with 0 ≤ λ ≤ μ. Then,
for any u = ∑ i = 0 4 u i e i in \(\mathbb{S}^{4}\). We thus get
hence
From (190)–(191), we easily obtain the following three cases, corresponding, in the same order, to the three cases listed in Proposition 3.3:
- Case 1::
-
a is of rank 4—i.e. λ and μ are both non-zero—and λ < μ. Then:
-
(i)
f +(u) = f −(u) if and only if u belongs to the equatorial sphere \(\mathbb{S}^{3}\) defined by u 0 = 0;
-
(ii)
f +(u) = 0 if and only u belongs to the circle \(\mathrm{C}_{+} =\{ u_{0} = -\frac{\lambda }{\mu },u_{1} = u_{2} = 0\}\), and we then have \(f_{-}(u) = \frac{\lambda } {2}\);
-
(iii)
f −(u) = 0 if and only if u belongs to the circle \(\mathrm{C}_{-} =\{ u_{0} = \frac{\lambda }{\mu },u_{1} = u_{2} = 0\}\), and we then have \(f_{+}(u) = \frac{\lambda } {2}\);
-
(iv)
the 2-form df + 2 ∧ df − 2 is non-zero outside the 2-spheres S+ 2 = {u 1 = u 2 = 0} and S− 2 = {u 3 = u 4 = 0}; this is because
$$\displaystyle{ \begin{array}{rl} df_{+}^{2} \wedge df_{-}^{2} & = \frac{\lambda \mu (\lambda ^{2}-\mu ^{2})} {2} \,du_{0} \wedge (u_{1}\,du_{1} + u_{2}\,du_{2}) \\ \qquad & = \frac{\lambda \mu (\mu ^{2}-\lambda ^{2})} {2} \,du_{0} \wedge (u_{3}\,du_{3} + u_{4}\,du_{4}),\end{array} }$$(192)which readily follows from (190).
-
(i)
- Case 2::
-
a is of rank 4 and λ = μ. Then
$$\displaystyle{ f_{+}(u) = \frac{\lambda } {2}(1 + u_{0}),\quad f_{-}(u) = \frac{\lambda } {2}(1 - u_{0}); }$$(193)in particular,
$$\displaystyle{ f_{+} + f_{-} =\lambda; }$$(194)moreover, f +(u) = 0 if and only if u = −e 0 and f −(u) = 0 if and only if u = e 0.
- Case 3::
-
a is of rank 2, i.e. λ = 0. Then, f + − f − is identically zero and f +(u) = f −(u) vanishes if and only if u belongs to the circle C0 = {u 0 = u 1 = u 2 = 0}.
Remark 7.1
Consider the functions \(x = \frac{f_{+}+f_{-}} {2},y = \frac{f_{+}-f_{-}} {2}\) defined in Sect. 3, as well as the functions of one variable, A and B, appearing in Proposition 3.1. If a is of rank 4, with 0 < λ < μ, corresponding to Case 1 in the above list, we easily infer from (190) that
Since x ≥ | y |, the above identities imply that the image of (x, y) in \(\mathbb{R}^{2}\) is the rectangle \(R:= \left [ \frac{\lambda }{2}, \frac{\mu } {2}\right ] \times \left [-\frac{\lambda }{2}, \frac{\lambda } {2}\right ]\). A simple calculation then shows that A and B are given by
Notice that A(x) and B(y) are positive in the interior of R, corresponding to the open set of \(\mathbb{S}^{4}\) where dx, dy are linearly independent, and vanish on its boundary. Also notice that the above expressions of A, B fit with the identities (104)–(105), with Scal = 12 and b = 0.
Remark 7.2
By using the ambitoric Ansatz in Theorem 4.1, the above situation can easily be deformed in Case 1, where a is of rank 4, with 0 < λ < μ, and the 2-form ψ a defined by (181) is ∗-Killing with respect to the round metric (We warmly thank Vestislav Apostolov for this suggestion.) On the open set \(\mathcal{U} = \mathbb{S}^{4}\setminus \left (S_{+}^{2} \cup S_{-}^{2}\right )\), where f + ≠ 0, f − ≠ 0 and df + ∧ df − ≠ 0, the round metric of \(\mathbb{S}^{4}\) takes the form (124), where A and B are given by (196), \(x \in \left ( \frac{\lambda }{2}, \frac{\mu } {2}\right )\), \(y \in \left (-\frac{\lambda }{2}, \frac{\lambda } {2}\right )\) are determined by (195) and ds, dt are explicit exact 1-forms determined by the last two equations of (125). It can actually be shown that outside the 2-spheres S + 2 and S − 2, ds and dt are given by:
Moreover, ψ a is given by (128) with respect to these coordinates.
Consider now a small perturbation \(\tilde{A}\), \(\tilde{B}\) of the functions A and B such that \(\tilde{A}(x) = A(x)\) near \(x = \frac{\lambda } {2}\) and \(x = \frac{\mu } {2}\) and \(\tilde{B}(y) = B(y)\) near \(y = \pm \frac{\lambda }{2}\). If the perturbation is small enough, the expression analogue to (124)
is still positive definite so defines a Riemannian metric on \(\mathcal{U}\), which coincides with the canonical metric on an open neighbourhood of \(\mathbb{S}^{4}\setminus \mathcal{U} = S_{+}^{2} \cup S_{-}^{2}\), and thus has a smooth extension to \(\mathbb{S}^{4}\) which we still call \(\tilde{g}\). Since the expression (128) of the ∗-Killing form in the Ansatz of Sect. 4 does not depend on A and B, the 2-form ψ a is still ∗-Killing with respect to the new metric \(\tilde{g}\). We thus get an infinite-dimensional family (depending on two functions of one variable) of Riemannian metrics on S 4 which all carry the same non-parallel ∗-Killing form.
8 Example: Complex Ruled Surfaces
In general, a (geometric) complex ruled surface is a compact, connected, complex manifold of the form \(M = \mathbb{P}(E)\), where E denotes a rank 2 holomorphic vector bundle over some (compact, connected) Riemann surface, \(\Sigma\), and \(\mathbb{P}(E)\) is then the corresponding projective line bundle, i.e. the holomorphic bundle over \(\Sigma\), whose fiber at each point y of \(\Sigma\) is the complex projective line \(\mathbb{P}(E_{y})\), where E y denotes the fiber of E at y. A complex ruled surface is said to be of genus g if \(\Sigma\) is of genus g.
In this section, we restrict our attention to complex ruled surfaces \(\mathbb{P}(E)\) as above, when \(E = L \oplus \mathbb{C}\) is the Whitney sum of some holomorphic line bundle, L, over \(\Sigma\) and of the trivial complex line bundle \(\Sigma \times \mathbb{C}\), here simply denoted \(\mathbb{C}\): M is then the compactification of the total space of L obtained by adding the point at infinity \([L_{y}]:= \mathbb{P}(L_{y} \oplus \{ 0\})\) to each fiber of M over y. The union of the points at infinity is a divisor of M, denoted by \(\Sigma _{\infty }\), whereas the (image of) the zero section of L, viewed as a divisor of M, is denoted \(\Sigma _{0}\); both \(\Sigma _{0}\) and \(\Sigma _{\infty }\) are identified with \(\Sigma\) by the natural projection, π, from M to \(\Sigma\). The open set \(M\setminus (\Sigma _{0} \cup \Sigma _{\infty })\), denoted M 0, is naturally identified with \(L\setminus \Sigma _{0}\). We moreover assume that the degree, d(L), of L is negative and we set: d(L) = −k, where k is a positive integer.
Complex ruled surfaces of this form will be called Hirzebruch-like ruled surfaces. When g = 0, these are exactly those complex ruled surfaces introduced by F. Hirzebruch in [7]. When g ≥ 2, they were named pseudo-Hirzebruch in [14].
In general, the Kähler cone of a complex ruled surface \(\mathbb{P}(E)\) was described by A. Fujiki in [6]. In the special case considered in this section, when \(M = \mathbb{P}(L \oplus \mathbb{C})\) is a Hirzebruch-like ruled surface, if \([\Sigma _{0}]\), \([\Sigma _{\infty }]\) and [F] denote the Poincaré duals of the (homology class of) \(\Sigma _{0}\), \(\Sigma _{\infty }\) and of any fiber F of π in \(\mathrm{H}^{2}(M, \mathbb{Z})\), the latter is freely generated by \([\Sigma _{0}]\) and [F] or by \([\Sigma _{\infty }]\) and [F], with \([\Sigma _{0}] = [\Sigma _{\infty }] - k\,[F]\), and the Kähler cone is the set of those elements, \(\Omega _{a_{0},a_{\infty }}\), of \(\mathrm{H}(M, \mathbb{R})\) which are of the form \(\Omega _{a_{0},a_{\infty }} = 2\pi \,\big(-a_{0}\,[\Sigma _{0}] + a_{\infty }\,[\Sigma _{\infty }]\big)\), for any two real numbers a 0, a ∞ such that 0 < a 0 < a ∞ .
We assume that \(\Sigma\) comes equipped with a Kähler metric \((g_{\Sigma },\omega _{\Sigma })\) polarized by L, in the sense that L is endowed with a Hermitian (fiberwise) inner product, h, in such a way that the curvature, R∇, of the associated Chern connection, ∇, is related to the Kähler form \(\omega _{\Sigma }\) by R∇ = i ω; in particular, \([\omega _{\Sigma }] = 2\pi \,c_{1}(L^{{\ast}})\), where \([\omega _{\Sigma }]\) denotes the de Rham class of \(\omega _{\Sigma }\), L ∗ the dual line bundle to L and c 1(L ∗) the (de Rham) Chern class of L ∗. The natural action of \(\mathbb{C}^{{\ast}}\) extends to a holomorphic \(\mathbb{C}^{{\ast}}\)-action on M, trivial on \(\Sigma _{0}\) and \(\Sigma _{\infty }\); we denote by K the generator of the restriction of this action on \(S^{1} \subset \mathbb{C}^{{\ast}}\). On \(M^{0} = L\setminus \Sigma _{0}\), we denote by t the function defined by
where r stands for the distance to the origin in each fiber of L determined by h; on M 0, we then have
(beware: the function t defined by (199) has nothing to do with the local coordinate t appearing in Sect. 4). Any (smooth) function F = F(t) of t will be regarded as function defined on M 0, which is evidently K-invariant; moreover:
-
1.
F = F(t) smoothly extends to \(\Sigma _{0}\) if and only if \(F(t) = \Phi _{+}(e^{2t})\) near t = −∞, for some smooth function \(\Phi _{+}\) defined on some neighbourhood of 0 in \(\mathbb{R}^{\geq 0}\), and
-
2.
F = F(t) smoothly extends to \(\Sigma _{\infty }\) if and only if \(F(t) = \Phi _{-}(e^{-2t})\) near t = ∞, for some smooth function \(\Phi _{-}\) defined on some neighbourhood of 0 in \(\mathbb{R}^{\geq 0}\), cf. e.g. [14], [1, Section 3.3].
For any (smooth) real function φ = φ(t), denote by ω φ the real, J-invariant 2-form defined on M 0 by
where φ′ denotes the derivative of φ with respect to t. Then, ω φ is a Kähler form on M 0, with respect to the natural complex structure J = J +, of M, if and only if φ is positive and increasing as a function of t; moreover, ω ϕ extends to a smooth Kähler form on M, in the Kähler class \(\Omega _{a_{0},a_{\infty }}\), if and only if φ satisfies the above asymptotic conditions (1)–(2), with \(\Phi _{+}(0)\,=\,a_{0}> 0\), \(\Phi _{+}'(0)> 0\), \(\Phi _{-}(0)\,=\,a_{\infty }> 0\), \(\Phi _{-}'(0) <0\). Kähler forms of this form on M, as well as the corresponding Kähler metrics
are called admissible.
Denote by J − the complex structure, first defined on the total space of L by keeping J on the horizontal distribution determined by the Chern connection and by substituting − J on the fibers, then smoothly extended to M. The new complex structure induces the opposite orientation, hence commutes with J + = J.
Any admissible Kähler form ω φ is both J +- and J −-invariant, as well as the associated 2-form \(\tilde{\omega }_{\varphi }\) defined by
which is moreover Kähler with respect to J −, with metric
We thus obtain an ambikähler structure of Calabi-type, as defined in Sect. 5, with \(f = \frac{1} {\varphi }\) and τ(K) = −K. According to Theorem 5.1 and Remark 5.1, for any k in \(\mathbb{R}\setminus \{0\}\), the metric g (k) defined, outside the locus {1 + k φ = 0}, by
there admits a non-parallel ∗-Killing 2-form ψ φ (k), namely
Notice that the pair (g φ (k), ψ φ (k)) smoothly extends to M for any \(k \in \mathbb{R}\setminus [-\frac{1} {a_{0}},- \frac{1} {a_{\infty }}]\), including k = 0 for which we simply get the Kähler pair (g φ , ω +).
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Acknowledgements
We warmly thank Vestislav Apostolov and David Calderbank for their interest in this work and for many useful suggestions. We also thank the anonymous referee, whose valuable observations allowed us to correct a mistake and significantly improve a part of the paper. This work was partially supported by the Procope Project No. 32977YJ.
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Gauduchon, P., Moroianu, A. (2017). Killing 2-Forms in Dimension 4. In: Chiossi, S., Fino, A., Musso, E., Podestà, F., Vezzoni, L. (eds) Special Metrics and Group Actions in Geometry. Springer INdAM Series, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-67519-0_7
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