Rank two nilpotent co-Higgs sheaves on complex surfaces

Abstract

Let $({{\mathcal {E}}}, \phi )$ be a rank two co-Higgs vector bundles on a Kähler compact surface X with $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$ nilpotent. If $({{\mathcal {E}}}, \phi )$ is semi-stable, then one of the following holds up to finite étale cover: (1) X is uniruled. (2) X is a torus and $({{\mathcal {E}}}, \phi )$ is strictly semi-stable. (3) X is a properly elliptic surface and $({{\mathcal {E}}}, \phi )$ is strictly semi-stable.

1 Introduction

A generalised complex structure on a real manifold X of dimension 2n, as defined by Hitchin [9], is a rank-2n isotropic subbundle $E^{0,1}\subset (T_X\oplus T^*_X)^{{\mathbb {C}}}$ such that

• 1. $E^{0,1}\oplus \overline{ E^{0,1} }= (T_X\oplus T^*_X)^{{\mathbb {C}}}$

• 2. $C^{\infty }(E^{0,1})$ is closed under the Courant bracket.

On a manifold with a generalized complex structure M. Gualtieri in [6] defined the notion of a generalized holomorphic bundle. More precisely, a generalized holomorphic bundle on a generalized complex manifold, is a vector bundle ${{\mathcal {E}}}$ with a differential operator $\overline{D} : C^\infty ({{\mathcal {E}}}) \longrightarrow C^{\infty }({{\mathcal {E}}}\otimes E^{0,1})$ such that for all smooth function f and all section $s\in C^\infty ({{\mathcal {E}}})$ the following holds

• 1. $\overline{D} (fs)=\overline{\partial }(fs)+f \overline{D}(s)$

• 2. $\overline{D}^2=0$.

In the case of an ordinary complex structure and $\overline{D} = \overline{\partial }+\phi$, for operators

$$\begin{aligned} \overline{\partial } : C^\infty ({{\mathcal {E}}}) \longrightarrow C^{\infty }({{\mathcal {E}}}\otimes \overline{T^*_X}) \end{aligned}$$

and

$$\begin{aligned} \phi : C^\infty ({{\mathcal {E}}}) \longrightarrow C^{\infty }({{\mathcal {E}}}\otimes T_X), \end{aligned}$$

the vanishing $\overline{D}^2=0$ means that $\overline{\partial }^2=0$ , $\overline{\partial } \phi =0$ and $\phi \wedge \phi =0$. By a classical result of Malgrange the condition $\overline{\partial }^2=0$ implies that ${{\mathcal {E}}}$ is a holomorphic vector bundle. On the other hand, $\overline{\partial } \phi =0$ implies that $\phi$ is a holomorphic global section

$$\begin{aligned} \phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X) \end{aligned}$$

which satisfies an integrability condition $\phi \wedge \phi =0$. A co-Higgs sheaf on a complex manifold X is a sheaf ${{\mathcal {E}}}$ together with a section $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$(called a Higgs fields) for which $\phi \wedge \phi =0$. General properties of co-Higgs bundles were studied in [8, 15]. There is a motivation in physics for studying co-Higgs bundles, see [7, 10] and [19].

There are no stable co-Higgs bundles with nonzero Higgs field on curves C of genus $g>1$. (When $g = 1$, a co-Higgs bundle is the same thing as a Higgs bundle in the usual sense.) In fact, contracting with a holomorphic differential gives a non-trivial endomorphism of ${{\mathcal {E}}}$ commuting with $\phi$ which is impossible in the stable case [8, 16]. S. Rayan showed in [14] the non-existence of stable co-Higgs bundles with non trivial Higgs field on K3 and general-type surfaces. In this note we prove the following result.

Theorem 1.1

Let $({{\mathcal {E}}}, \phi )$ be a rank two co-Higgs vector bundles on a Kähler compact surface X with $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$ nilpotent. If $({{\mathcal {E}}}, \phi )$ is semi-stable, then one of the following holds up to finite étale cover:

• 1. X is uniruled.

• 2. X is a torus and $({{\mathcal {E}}}, \phi )$ is strictly semi-stable.

• 3. X is a properly elliptic surface and $({{\mathcal {E}}}, \phi )$ is strictly semi-stable.

It follows direct of proof of Theorem 1.1 that the we can consider a more general classes of singular projective surfaces.

Theorem 1.2

Let $({{\mathcal {E}}}, \phi )$ be a rank two co-Higgs torsion-free sheaf on a normal projective surface X with $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$ nilpotent. If $({{\mathcal {E}}}, \phi )$ is stable, then X is uniruled.

Finally, in this work we consider a relation between co-Higgs bundles and Poisson geometry on ${\mathbb {P}}^1$-bundles. In [18] Polishchuk associated to each rank-2 co-Higgs bundle $({{\mathcal {E}}}, \phi )$ a Poisson structure on its projectivized bundle ${\mathbb {P}}({{\mathcal {E}}})$. This relation was explained by Rayan in [15] as follows:

let $Y:= {\mathbb {P}}({{\mathcal {E}}})$ and consider the natural projection $\pi :Y \rightarrow X$. The exact sequence

$$\begin{aligned} 0\longrightarrow T_{X|Y}\longrightarrow T_Y\longrightarrow \pi ^*T_X\longrightarrow 0 \end{aligned}$$

implies that $T_{X|Y}\otimes \pi ^*T_X \subset \bigwedge ^2 T_Y$. Since $T_{X|Y}={\mathrm {Aut}}({\mathbb {P}}({{\mathcal {E}}}))={\mathrm {Aut}}({{\mathcal {E}}})/{\mathbb {C}}^*$ we get that

$$\begin{aligned} \pi _*(T_{X|Y}\otimes \pi ^*T_X)=\pi _*T_{X|Y}\otimes T_X=End_0({{\mathcal {E}}})\otimes T_X, \end{aligned}$$

where $End_0({{\mathcal {E}}})$ denotes the trace-free endomorphisms of ${{\mathcal {E}}}$. Therefore, we can associate a trace-free co-Higgs fields $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$ a bi-vector $\pi ^*\phi \in H^0(X,T_{X|Y}\otimes \pi ^*T_X) \subset H^0(X, \bigwedge ^2 T_Y)$ on ${\mathbb {P}}({{\mathcal {E}}})$. The co-Higgs condition $\phi \wedge \phi =0$ implies that bi-vector $\pi ^*\phi$ is integrable, see the introduction of [15]. The codimension one foliation on ${\mathbb {P}}({{\mathcal {E}}})$ is the called foliation by symplectic leaves induced by Poisson struture.

We get an interisting consequence of the proof Theorem 1.2.

Corollary 1.1

If $({{\mathcal {E}}}, \phi )$ is locally free, stable and nilpotent , then the closure of the all leaves of the foliation by symplectic leaves on ${\mathbb {P}}({{\mathcal {E}}})$ are rational surfaces.

2 Semi-stable co-Higgs sheaves

Definition 2.1

A co-Higgs sheaf on a complex manifold X is a sheaf ${{\mathcal {E}}}$ together with a section $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$(called a Higgs fields) for which $\phi \wedge \phi =0$.

Denote by $End_0({{\mathcal {E}}}) := ker(tr : End({{\mathcal {E}}}) \longrightarrow {\mathcal {O}}_X)$ the trace-free part of the endomorphism bundle of ${{\mathcal {E}}}$. Since

$$\begin{aligned} End({{\mathcal {E}}})= End_0({{\mathcal {E}}})\oplus {\mathcal {O}}_X \end{aligned}$$

we have that $End({{\mathcal {E}}})\otimes T_X=(End_0({{\mathcal {E}}})\otimes T_X)\oplus T_X$. Thus, the Higgs field $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$ can be decomposed as $\phi =\phi _1+\phi _2$, where $\phi _1$ is the trace-free part and $\phi _2$ is a global vector field on X. In particular, if the surface X has no global holomorphic vector fields, then every Higgs field is trace-free.

Definition 2.2

Let $(X,\omega )$ be a polarized Kähler compact manifold. We say that $({{\mathcal {E}}},\phi )$ is semi-stable if

$$\begin{aligned} \frac{c_1(\mathcal {F})\cdot [\omega ]}{rank(\mathcal {F})}\le \frac{c_1({{\mathcal {E}}})\cdot [\omega ]}{rank({{\mathcal {E}}})} \end{aligned}$$

for all coherent subsheaves $0\ne \mathcal {F}\subsetneq {{\mathcal {E}}}$ satisfying $\Phi (\mathcal {F})\subseteq \mathcal {F}\otimes T_X$, and stable if the inequality is strict for all such $\mathcal {F}$. We say that $({{\mathcal {E}}},\phi )$ is strictly semi-stable if $({{\mathcal {E}}},\phi )$ is semi-stable but non-stable.

3 Holomorphic foliations

Definition 3.1

Let X be a connected complex manifold. A one-dimensional holomorphic foliation is given by the following data:

• 1. an open covering ${\mathcal {U}}=\{U_{\alpha }\}$ of X;

• 2. for each $U_{\alpha }$ an holomorphic vector field $\zeta _\alpha$ ;

• 3. for every non-empty intersection, $U_{\alpha }\cap U_{\beta } \ne \emptyset$, a holomorphic function

  $$\begin{aligned} f_{\alpha \beta } \in {\mathcal {O}}_X^*(U_\alpha \cap U_\beta ); \end{aligned}$$

such that $\zeta _\alpha = f_{\alpha \beta }\zeta _\beta$ in $U_\alpha \cap U_\beta$ and $f_{\alpha \beta }f_{\beta \gamma } = f_{\alpha \gamma }$ in $U_\alpha \cap U_\beta \cap U_\gamma$.

We denote by $K_{\mathcal {F}}$ the line bundle defined by the cocycle $\{f_{\alpha \beta }\}\in {\mathrm {H}}^1(X, {\mathcal {O}}^*)$. Thus, a one-dimensional holomorphic foliation $\mathcal {F}$ on X induces a global holomorphic section $\zeta _{\mathcal {F}}\in {\mathrm {H}}^0(X,T_X\otimes K_{\mathcal {F}})$. The line bundle $K_{\mathcal {F}}$ is called the canonical bundle of $\mathcal {F}$. Two sections $\zeta _{\mathcal {F}}$ and $\eta _{\mathcal {F}}$ of ${\mathrm {H}}^0(X,T_X\otimes K_{\mathcal {F}})$ are equivalent, if there exists a never vanishing holomorphic function $\varphi \in {\mathrm {H}}^0(X,{\mathcal {O}}^*)$, such that $\zeta _{\mathcal {F}}= \varphi \cdot \eta _{\mathcal {F}}$. It is clear that $\zeta _{\mathcal {F}}$ and $\eta _{\mathcal {F}}$ define the same foliation. Thus, a holomorphic foliation $\mathcal {F}$ on X is an equivalence of sections of $H^0(X, T_X\otimes K_{\mathcal {F}})$.

4 Examples

4.1 Canonical example of split co-Higgs bundles

Here we will give an example which naturally generalizes the canonical example given by Rayan on [15, Chapter 6]. Let $(X, \omega )$ be a polarized Kähler compact manifold. Suppose that there exists a global section $\zeta \in H^0(X , Hom(N,TX\otimes L))\simeq H^0(X ,TX\otimes L \otimes N^*)$. Now, consider the vector bundle

$$\begin{aligned} {{\mathcal {E}}}=L \oplus N. \end{aligned}$$

Define the following co-Higgs fields

$$\begin{aligned} \phi : L \oplus N \longrightarrow (T_X \otimes L)\oplus (T_X \otimes N)\in H^0(X,End({{\mathcal {E}}})\otimes T_X) \end{aligned}$$

by $\phi (s,t)=(\zeta (t),0)$. Since $\phi \circ \phi =0 \in H^0 (End({{\mathcal {E}}})\otimes T_X\otimes T_X)$ we get that $\phi \wedge \phi =0$. Moreover, observe that the kernel of $\phi$ is the $\phi$-invariant line bundle L. On the other hand, the line bundle L is destabilising only when

$$\begin{aligned}{}[2c_1(L)- c_1({{\mathcal {E}}})]\cdot [\omega ]=[c_1(L)-c_1(N)]\cdot [\omega ]> 0. \end{aligned}$$

That is, if

$$\begin{aligned}{}[c_1(L)]\cdot [\omega ]> [c_1(N)]\cdot [\omega ]. \end{aligned}$$

4.2 Co-Higgs bundles on ruled surfaces

Let C be a curve of genus $g >1$. Now, consider the ruled surface $X:={\mathbb {P}}(K_C\oplus {\mathcal {O}}_C)$ and

$$\begin{aligned} \pi :{\mathbb {P}}(K_C\oplus {\mathcal {O}}_C)\longrightarrow C \end{aligned}$$

the natural projection. Consider a Poisson structure on X given by a bivector $\sigma \in H^0(X, \wedge ^2T_X)$. Let $({{\mathcal {E}}},\phi )$ be a nilpotent Higgs bundle on C. S. Rayan showed in [15] that $(\pi ^*{{\mathcal {E}}},\sigma (\pi ^*\phi ))$ is a stable co-Higgs bundle on X.

4.3 Co-Higgs orbibundles on weighted projective spaces

Let $w_{0},w_1,w_{2}$ be positive integers, set $|w|:= w_0+ w_1 + w_2$. Assume that $w_{0},w_1,w_{2}$ are relatively prime. Define an action of ${{\mathbb {C}}}^*$ in ${{\mathbb {C}}}^{3} \setminus \{0\}$ by

$$\begin{aligned} \begin{array}{ccc} {{\mathbb {C}}}^*\times ({{\mathbb {C}}}^{3} \setminus \{0\} )&{} \longrightarrow &{}( {{\mathbb {C}}}^{3} \setminus \{0\}) \\ \lambda . (z_0, z_1, z_2) &{} \longmapsto &{} (\lambda ^{w_0} z_0, \lambda ^{w_1} z_1, \lambda ^{w_2} z_2)\\ \end{array} \end{aligned}$$

(4.1)

and consider the weighted projective plane

$$\begin{aligned} {\mathbb {P}}(w_{0},w_1,w_{2}) := ({{\mathbb {C}}}^{3} \setminus \{0\} )/ \sim \end{aligned}$$

induced by the action above. We will denote this space by ${\mathbb {P}}(\omega )$. On ${\mathbb {P}}(\omega )$ we have an Euler sequence

$$\begin{aligned} 0\longrightarrow {\mathcal {O}}_{{\mathbb {P}}(\omega )} \mathop {\longrightarrow }\limits ^{\varsigma }\bigoplus _{i=0}^{2 }{\mathcal {O}}_{{\mathbb {P}}(\omega )}(\omega _i) \longrightarrow T{\mathbb {P}}(\omega ) \longrightarrow 0, \end{aligned}$$

where ${\mathcal {O}}_{{\mathbb {P}}(\omega )}$ is the trivial line orbibundle and $T{\mathbb {P}}(\omega ) = {\mathrm {Hom}} (\Omega _{{\mathbb {P}}(\omega )}^ 1, {\mathcal {O}}_{{\mathbb {P}}(\omega )} )$ is the tangent orbibundle of ${\mathbb {P}}(\omega )$. The map $\varsigma$ is given explicitly by $\varsigma (1)=(\omega _0z_0,\omega _1z_1,\omega _2z_2).$ Now, let $({{\mathcal {E}}},\phi )$ be a co-Higgs orbibundle on ${\mathbb {P}}(\omega )$. Tensoring the Euler sequence by $End({{\mathcal {E}}})$, we obtain

$$\begin{aligned} 0 \longrightarrow End({{\mathcal {E}}}) \longrightarrow \bigoplus \limits _{i=0}^2 End({{\mathcal {E}}}) ( \omega _i ) \longrightarrow End({{\mathcal {E}}}) \otimes T {\mathbb {P}}(\omega ) \longrightarrow 0. \end{aligned}$$

Thus, the co-Higgs fields $\phi$ can be represented, in homogeneous coordinates, by

$$\begin{aligned} \phi =\phi _0\otimes \frac{\partial }{\partial z_0}+\phi _1\otimes \frac{\partial }{\partial z_1}+\phi _2\otimes \frac{\partial }{\partial z_2} , \end{aligned}$$

where $\phi _i\in H^0({\mathbb {P}}(\omega ), End({{\mathcal {E}}})(w_i))$, for all $i=0,1,2$, and $\phi +\theta \otimes R_\omega$ define the same co-Higgs field as $\phi$, where $R_\omega$ is the adapted radial vector field

$$\begin{aligned} R_\omega = \omega _0 z_0 \frac{\partial }{\partial z_0} +\omega _1 z_1 \frac{\partial }{\partial z_1}+ + \omega _2 z_2 \frac{\partial }{\partial z_2}, \end{aligned}$$

with $\theta$ a endomorphism of ${{\mathcal {E}}}$. Suppose that

$$\begin{aligned} {{\mathcal {E}}}= {\mathcal {O}}(m_1) \oplus {\mathcal {O}}(m_2) \end{aligned}$$

and that there exists a stable $\phi$ for ${{\mathcal {E}}}$ such that $m_1 \ge m_2$. Then

$$\begin{aligned} |m_1- m_2| \le \max \limits _{0 \le i\ne j \le 2}\{\omega _{i}+\omega _{j} \}. \end{aligned}$$

In fact, this is a consequence of Bott’s Formulae for weighted projective spaces. It follows from (see [5]) that

$$\begin{aligned} {\mathrm {H}}^0({\mathbb {P}}(\omega ),T{\mathbb {P}}(\omega )\otimes {\mathcal {O}}_{\omega }(k))\simeq {\mathrm {H}}^0({\mathbb {P}}(\omega ), \Omega ^{1}_{{\mathbb {P}}(\omega )}\left( \sum \nolimits _{i=0}^{2} \omega _{i}+k)\right) \ne \emptyset \end{aligned}$$

if and only if $k>-\max \limits _{0 \le i\ne j \le 2}\{\omega _{i}+\omega _{j} \}$. This generalize the example given by S. Rayan in [14].

4.4 Co-Higgs bundles on two dimensional complex tori

Let X be a two dimensional complex torus and a co-Higgs bundle $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$. Then $\phi$ is equivalente to a pair of commutative endomorphism of ${{\mathcal {E}}}$. In fact, since the tangent bundle $T_X$ is holomorphically trivial, we can take a trivialization by choosing two linearly independent global vector fields $v_1,v_2\in H^0(X,T_X)$. Then, we can write

$$\begin{aligned} \phi = \phi _1 \otimes v_1 +\phi _2 \otimes v_2. \end{aligned}$$

The condition $\phi \wedge \phi =0$ implies that

$$\begin{aligned} \phi _1\circ \phi _2=\phi _2\circ \phi _1. \end{aligned}$$

We have a canonical nilpotent co-Higgs bundle $({{\mathcal {E}}},\phi )$, where ${{\mathcal {E}}}=T_X={\mathcal {O}}\oplus {\mathcal {O}}$ and

$$\begin{aligned} \left( {\begin{array}{*{20}c} 0 &{} v \\ 0 &{} 0 \\ \end{array} } \right) , \end{aligned}$$

where v is a global vector field on X.

5 Proof of Theorem

By using that the Higgs field $\phi \in H^0(X,End({{\mathcal {E}}})\otimes T_X)$ is nilpotent we have that $Ker(\phi )=:L$ is a well defined line bundle on X. Thus , we have a exact sequence

$$\begin{aligned} 0\rightarrow L\longrightarrow {{\mathcal {E}}}\longrightarrow {\mathcal {I}}_Z\otimes N \longrightarrow 0, \end{aligned}$$

where the nilpotent Higgs field $\phi$ factors as

$$\begin{aligned} \begin{array}{ccc} {{\mathcal {E}}}&{} \longrightarrow &{}{{\mathcal {E}}}\otimes T_X \\ \\ \downarrow &{} &{} \uparrow \\ \\ {\mathcal {I}}_Z\otimes N&{} \longrightarrow &{}L\otimes T_X. \end{array} \end{aligned}$$

(5.1)

The morphism ${\mathcal {I}}_Z\otimes N\rightarrow L\otimes T_X$ induces a holomorphic foliation on X which induces a global section $\zeta _{\phi } \in H^0(X, T_X\otimes L\otimes N^*)$. Since $\det ({{\mathcal {E}}})=L\otimes N$ we conclude that $L\otimes N^*=L^2\otimes \det ({{\mathcal {E}}}^*)$. Then

$$\begin{aligned} \zeta _{\phi }\in H^0(X, T_X\otimes L^2\otimes \det ({{\mathcal {E}}}^*) ). \end{aligned}$$

Let $K:=L^2\otimes \det ({{\mathcal {E}}}^*)$ the canonical bundle of the foliation $\mathcal {F}$ associated to the co-Higgs fields $\phi$. If ${{\mathcal {E}}}$ is semi-stable then

$$\begin{aligned}{}[c_1(K)]\cdot [\omega ]=[2c_1(L)- c_1({{\mathcal {E}}})]\cdot [\omega ]\le 0 \end{aligned}$$

for some Kähler class $\omega$. If $K\cdot [\omega ]<0$, it follows from [11] that K is not pseudo-efective [4]. It follows from Brunella’s Theorem [1] that X is uniruled.

Now, suppose X is a normal projective surface and that $K\cdot H=0$, for some H ample By Hodge index theorem we have that $K^2\cdot H^2\le (K\cdot H)^2=0$, then $K^2\le 0$. Suppose that $K^2<0$. We have that $D=H+\epsilon K$ is a ${\mathbb {Q}}$-divisor ample for $0<\epsilon <<1$, see [12, proposition 1.3.6]. Thus, we have that

$$\begin{aligned} K\cdot D=K\cdot H+\epsilon ^2K^2=\epsilon ^2 K^2<0. \end{aligned}$$

By Bogomolov-McQuillan-Miyaoka’s theorem [3] we conclude that X is uniruled. If $K^2=0$, then K is numerically trivial. This fact is well known, but for convenience of the reader we give a proof. Suppose that there exists $C\subset X$ such that $K\cdot C>0$. Now, Consider the divisor $B=(H^2)C-(H\cdot C)H$. Then $B\cdot H=0$ and $K\cdot B=(H^2)K\cdot C<0.$ Define $F=mK+B$ for $0<m<<1$. Therefore $F\cdot H=0$ and $F^2>0$. This is a contradiction by the Hodge index Theorem. In this case ${{\mathcal {E}}}$ is strictly semi-stable.

Now, we apply the classification, up to finite étale cover, of holomorphic foliations on projective surfaces with canonical bundle numerically trivial [2, 13, 17]. Therefore, up to finite étale cover, either:

• (i) X is uniruled;

• (ii) X is a torus;

• (iii) $k(X)=1$ and $X=B\times C$ with $g(B)\ge 2$, C is elliptic. That is, X is a sesquielliptic surface.

If X is Kähler and non-algebraic it follows from [2] that, up to finite étale cover, either X has a unique elliptic fibration or X is a torus.

6 Proof of Corollary 1.1

Since $({{\mathcal {E}}},\phi )$ is nilpotent and stable the co-Higgs fields induces a foliation $\mathcal {F}$ by rational curves on X. Now, consider the projective bundle $\pi :{\mathbb {P}}({{\mathcal {E}}}) \rightarrow X$. Then the foliation by symplectic leaves ${\mathcal {G}}$ on ${\mathbb {P}}({{\mathcal {E}}})$ is the pull-back of $\mathcal {F}$ by $\pi$. In particular, a closure of the a leaf of the foliation by symplectic leaves ${\mathcal {G}}$ is of type $\pi ^{-1}(f({\mathbb {P}}^1))$, where $f:{\mathbb {P}}^1\rightarrow X$ is the uniformization of a rational leaf of $\mathcal {F}$. Clearly $\pi ^{-1}(f({\mathbb {P}}^1))$ is a rational surface.