Rational Morita equivalence for holomorphic Poisson modules

TL;DR

This paper introduces a birational Morita equivalence concept for Poisson modules on complex varieties, classifies their types, and explores the geometric structures they induce, including foliations and projective structures.

Contribution

It defines a weak Morita equivalence for Poisson modules and classifies modules on varieties with mild singularities, linking them to flat sheaves, meromorphic connections, or co-Higgs sheaves.

Findings

Poisson modules are either rationally Morita equivalent to flat, meromorphic, or co-Higgs sheaves.

The study of rank two $rak{sl}_2$-Poisson modules reveals structures analogous to transversally projective foliations.

The geometry of symplectic foliations induced by Poisson connections is characterized.

Abstract

We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an application, we study the geometry of rank two meromorphic rank two $\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson analogous to transversally projective structures for codimension one holomorphic foliations. Moreover, we describe the geometry of the symplectic foliation induced by the Poisson connection on the projectivization of the Poisson module.

Equations

$$(S,\varrho )\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{\nabla }={\delta }_W+\left(\begin{array}{cc}{f}_1& f_2\\ f_0& -f_1\end{array}\right)\otimes v,$$

$$\tilde{\nabla }=\delta +\left(\begin{array}{cc}{f}_1& f_2\\ f_0& -f_1\end{array}\right)\otimes v,$$

$$\begin{array}{c}\delta (v_0)=\text{hfill}{v}_0\wedge v_1\\ \delta (v_1)=2v_0\wedge v_2\\ \delta (v_2)=\text{hfill}{v}_1\wedge v_2,\end{array}$$

$$K_Z=f^{\ast }{K}_X+\sum _{i}a(E_i,X)E_i,$$

$$0\to T\mathcal{F}\to TX\to N\mathcal{F}\to 0.$$

$$\{\cdot ,\cdot \}:{\mathcal{O}}_X\times {\mathcal{O}}_X\to {\mathcal{O}}_X$$

$$\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0$$

$${\sigma }^{\#}:{\Omega }_X^1\to TX$$

$$\nabla (fs)=d(f)\otimes s+f\nabla (s),$$

$$\nabla (fs)=\delta (f)\otimes s+f\nabla (s),$$

$$\{f,s\}:=\nabla (s)(df),$$

$$\nabla (fs)=a^{\ast }(df)\otimes s+f\nabla (s),$$

$${\sigma }^{\#}:{\Omega }_X^1(\log D)\to T_X(-\log D)$$

$${\sigma }^{\#}\circ \nabla :E\to {\Omega }_X^1\otimes E\to T{\mathcal{F}}_{\sigma }\otimes E$$

$$\omega \circ {\rho }^{\ast }\nabla :{\rho }^{\ast }E\to {\Omega }_{Y|X}^1\otimes {\rho }^{\ast }E\to T_{Y|X}\otimes {\rho }^{\ast }E,$$

$$(S,\varrho )\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(X_1,{\sigma }_1)\stackrel{f_1}{⟵}(S,{\varrho }_{12})\stackrel{h_1}{⟶}(X_2,{\sigma }_2)$$

$$(X_3,{\sigma }_3)\stackrel{f_2}{⟵}(Q,{\varrho }_{34})\stackrel{h_2}{⟵}(X_4,{\sigma }_4)$$

$${\mathrm{Bl}}_{\Delta }(S^2)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$Z=W\times Y\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(X,\sigma )⟵(Z,\tilde{\sigma })⟶(Y,{\sigma }_2),$$

$$\varphi \in H^0(W,N{\mathcal{F}}_{\sigma }\otimes End(E))$$

$$\delta (\theta )+\theta \wedge \theta =0.$$

$$\phi_{0}\in H^{0}(Y,N\mbox{${\mathscr{K}}$}\otimes End(E_{0}))$$

$$S:=\{y\in Y;\omega (\xi )(y)=0\},$$

$$\tilde{\phi_{0}}\in H^{0}(Y^{0},(T_{\mbox{${\mathscr{G}}$}}\otimes End(E_{0}))|_{Y^{0}})$$

$$T_{\mbox{${\mathscr{G}}$}}\otimes End(E_{0})(*S)=j_{*}((T_{\mbox{${\mathscr{G}}$}}\otimes End(E_{0}))|_{Y^{0}}).$$

$$(N\mbox{${\mathscr{K}}$}^{*}\otimes End(E_{0}))|_{Y^{0}}\simeq(\Omega^{1}_{Y|B}\otimes End(E_{0}))|_{Y^{0}}$$

Full text

1. Introduction

K. Morita in his celebrated work [40] introduced an equivalence in algebra proving that two rings have equivalent categories of left modules if and only if there exists an equivalence bimodule for the rings. Weinstein [52] and Xu [53] have introduced a geometric Morita equivalence in the context of integrable Poisson real manifolds having as one of the motivations the fact that symplectic realizations of Poisson manifolds is the analogous to representations of associative algebras. For non-integrable Poisson manifolds, an infinitesimal notion of Morita equivalence has been introduced by Crainic in [20] and by Ginzburg in [27] in order to study the invariance, respectively, of Poisson cohomology and Poisson Grothendieck groups.

In this work we introduce a weak concept of Morita equivalence in the birational context. More precisely, we say that two Poisson normal projective varieties $(X,\sigma_{1})$ and $(Y,\sigma_{2})$ are *rationally Morita equivalent * if there exists a normal variety $(S,\varrho)$, with a (possibly meromorphic ) Poisson bivector $\varrho$, and two arrows

[TABLE]

such that $f$ and $h$ are dominant Poisson morphisms. Our main aim is to show that holomorphic Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf.

A. Polishchuk in [41] has studied the algebraic geometry of Poisson modules motived by Bondal’s conjecture about the non-triviality of the degeneration locus of a Poisson structure, see also [7, 29, 22]. Poisson modules also appear in the context of generalized complex geometry introduced by N. Hitchin in [34] and developed by M. Gualtieri in [31]. Gualtieri’s concept of a generalized holomorphic bundle [30] in the Poisson case coincides with the notion of Poisson modules and if the Poisson structure is the trivial one we obtain a co-Higgs bundle [32]. See also [9, 44, 16, 5, 8, 17] for more details on the study of co-Higgs sheaves and their moduli spaces.

A Poisson structure on a projective complex manifold $X$ induces a natural Poisson structure on its minimal model by pushing forward the Poisson bivector. Thus, it is natural to consider Poisson varieties with mild singularities motived by the development of the minimal model program. We say that a Poisson variety $(X,\sigma)$ is *klt * if $X$ is a Kawamata log terminal(klt) variety and the Poisson structure $\sigma$ is either generically symplectic or the associated symplectic foliation $\mathscr{F}_{\sigma}$ has canonical singularities. We say that $(X,\sigma)$ is transcendental, in the spirit of [4, 37], if there is no positive-dimensional algebraic subvariety through a general point of $X$ that is tangent to the symplectic foliation $\mathscr{F}_{\sigma}$.

Theorem 1.1**.**

Let $(E,\nabla)$ be a locally free Poisson module on a klt Poisson projective variety $(X,\sigma)$. Then at least one of the following holds up to rational Morita equivalence.

• (a)

$(E,\nabla)$* corresponds to a flat holomorphic sheaf on a transcendental Poisson variety;*

• (b)

$(E,\nabla)$* corresponds to a meromorphic flat connection on a generically symplectic variety.*

• (c)

$(E,\nabla)$* corresponds to a co-Higgs sheaf on a variety with trivial Poisson structure .*

• (d)

$(E,\nabla)$* corresponds to a meromorphic co-Higgs sheaf $(E_{0},\psi)$ on a transcendental Poisson variety $(Y,\sigma_{0})$, there exist a rational map $\zeta:Y\dashrightarrow B$, over a variety $B$ with $\mathrm{dim}\,(B)=\mathrm{dim}\,(\mathscr{F}_{\sigma_{0}})$, and the co-Higgs field $\phi$ is tangent to $T_{Y|B}$ and satisfies $D_{0}(\phi)=0$, where $D_{0}$ is a meromorphic extension of a Poisson connection on $T_{Y|B}\otimes End(E_{0})$.*

As an application of Theorem 1.1, we provide a structure theorem for rank two holomorphic $\mathfrak{sl}_{2}$-Poisson modules.

Corollary 1.2**.**

Let $(E,\nabla)$ be a rank two holomorphic $\mathfrak{sl}_{2}$-Poisson module on a klt Poisson projective variety $(X,\sigma)$. Then there exist projective varieties $Y$ and $Z$ with klt singularities, a quasi-étale Poisson cover $f:W\times Y\to X$ and at least one of the following holds.

• (a)

$(\pi_{2})_{*}f^{*}(E,\nabla)$* is a $\mathfrak{sl}_{2}$ partial holomorphic sheaf on $Y$, where $\pi_{2}$ denotes the projection on $Y$.*

• (b)

If $W$ and $Y$ are generically symplectic, then $(\pi_{2})_{*}f^{*}(E,\nabla)$ is a rank two locally free sheaf with a meromorphic flat connection with poles on the degeneracy Poisson divisor of $Y$.

• (c)

If $W$ is symplectic, then after a birational trivialization of $f^{*}(E,\nabla)$ the Poisson connection on the trivial bundle is defined as

[TABLE]

for some rational vector field $v$ tangent to $(Y,0)$, rational functions $f_{0},f_{1},f_{2}\in K(Y)$, and $\delta_{W}$ denotes the Poisson differential on $W$.

• (d)

There exists a rational map $\zeta:Y\dashrightarrow B$, over a variety $B$ with $\mathrm{dim}\,(B)=\mathrm{dim}\,(\mathscr{F}_{\alpha})$, such that $(\pi_{2})_{*}f^{*}(E,\nabla)$ corresponds to a meromorphic $\mathfrak{sl}_{2}$-Poisson module $(E_{0},\tilde{\nabla})$, such that after a birational trivialization the Poisson connection on the trivial bundle is defined as

[TABLE]

for some rational Poisson vector field $v$ and rational functions $f_{0},f_{1},f_{2}$ on $X$ such that $\{f_{i},f_{j}\}=0$, for all $i,j$.

In the section 5 we point out that the study of rank two $\mathfrak{sl}_{2}$-Poisson modules is equivalent to the understanding of the following objects:

• i)

a triple of rational vector fields $(v_{0},v_{1},v_{2})$ on $X$ such that

[TABLE]

• ii)

the symplectic foliation $\mathscr{F}_{\nabla}$ corresponding to Polishchuk’s Poisson structure induced by $\nabla$ on $\pi:\mathbb{P}(E,\nabla)\to(X,\sigma)$.

We can say that the study of the triples $(v_{0},v_{1},v_{2})$ satisfying (1) is the Poisson analogous to transversely projective holomorphic foliations theory due to B. Scárdua [47]. We refer to the works [19, 18, 38], where the authors have studied transversely projective foliations via meromorphic connections on rank two vector bundles [21].

From Corollary 1.2 we observe that the geometric study of the symplectic foliation $\mathscr{F}_{\nabla}$ reduces, up to a quasi-étale Poisson cover, to the foliation $\mathscr{F}_{\nabla_{0}}$ on $\mathbb{P}(E_{0},\nabla_{0})\to(Y,\sigma_{0})$. Our next result describes the geometry of such foliation.

Corollary 1.3**.**

Under the same assumptions as Corollary 1.2. Let $\mathscr{F}_{\nabla_{0}}$ be the symplectic foliation induced on $\mathbb{P}(E_{0})\to(Y,\sigma_{0})$. Then at least one of the following holds.

• (a)

$\mathscr{F}_{\nabla_{0}}$* is a dimension 2 foliation which is a pull-back of a foliation by curves on $(Y,0)$.*

• (b)

$\mathscr{F}_{\nabla_{0}}$* is a Riccati foliation of codimension one on $\mathbb{P}(E_{0})$, if $(Y,\sigma_{0})$ is generically symplectic.*

• (c)

$\mathscr{F}_{\nabla_{0}}$* is a Riccati foliation of codimension one on $\mathbb{P}(E_{0})$ which is given by a morphism \mbox{{\mathscr{A}}}\to d_{\text{\rm refl}}\pi(\pi^{*}(T\mathscr{F}_{\sigma_{0}}^{*}))\subset\Omega_{\mathbb{P}(E_{0})}^{[1]}, where ${\mathscr{A}}$ is a line bundle and $d_{\text{\rm refl}}\pi:\pi^{*}\Omega_{Y}^{[1]}\to\Omega_{\mathbb{P}(E_{0})}^{[1]}$ is the pull-back morphism of reflexive forms.*

• (d)

There exist a rational Poisson vector field $v$ generically transversal to $\mathscr{F}_{\sigma_{0}}$ such that $\mathscr{F}_{\nabla_{0}}$ has dimension $2k+2$ and it is the pull-back of the foliation induced by $v$ and $\mathscr{F}_{\sigma_{0}}$. In particular, if $\mathrm{dim}\,(Y)=2k+1$, then $\mathbb{P}(E_{0})$ is generically symplectic and there exist a rational Poisson map $\zeta:Y\dashrightarrow B$ generically transversal to $\mathscr{F}_{\sigma_{0}}$, where $B$ is a generically symplectic variety with $\mathrm{dim}\,(B)=2k$ and the induced map $\mathbb{P}(E_{0})\dashrightarrow B$ is Poisson.

Acknowledgments

I would like to express my deep gratitude to Nigel Hitchin for his generosity and patience. I am honoured for the opportunity of having many illuminating discussions with him during my visit to the University of Oxford. I would also like to thank Marcos Jardim, Alex Massarenti, Henrique Bursztyn, Steven Rayan and Brent Pym for useful discussions. I would like to thank the referee for precious comments which improved the presentation of the paper greatly. I am grateful to Mathematical Institute of the University of Oxford and L’Università degli Studi di Ferrara for their hospitality. This work was partially funded by CNPQ grant numbers 202374/2018-1, 302075/2015-1, 400821/2016-8 , and CAPES grant number 2888/2013.

2. Holomorphic foliations

Throughout this paper a variety $X$ is a scheme of finite type over $\mathbb{C}$ and regular in codimension one, with its smooth locus denoted by $X_{reg}$. As usual, consider ${\mathcal{O}}_{X}$ the sheaf of germs of holomorphic functions on $X$. We denote by $TX=\mathcal{H}om(\Omega_{X}^{1},{\mathcal{O}}_{X})$ the tangent sheaf of $X$. Given $p\in\mathbb{N}$, we denote by $\Omega_{X}^{[p]}$ the sheaf $(\Omega_{X}^{p})^{**}$.

Let $X$ be a normal variety and suppose that $K_{X}$ is $\mathbb{Q}$-Cartier, i.e., some nonzero multiple of it is a Cartier divisor. Consider a resolution of singularities $f:Z\to X$. There are uniquely defined rational numbers $a(E_{i},X)$’s such that

[TABLE]

where $E_{i}$ runs through all exceptional prime divisors for $f$. The $a(E_{i},X)$’s do not depend on the resolution $f$, but only on the valuations associated to the $E_{i}$. For more details we refer to [36, Section 2.3].

We say that $X$ is * Kawamata log terminal (klt) * if all $a(E_{i},X)>-1$ for every $f$-exceptional prime divisor $E_{i}$, for some resolution $f:Z\to X$.

Definition 2.1**.**

A foliation $\mathscr{F}$ on a normal variety $X$ is a coherent subsheaf $T\mathscr{F}\subset TX$ such that $\mathscr{F}$ is closed under the Lie bracket, and the dimension of $\mathscr{F}$ is the generic rank of $T\mathscr{F}$. The *canonical bundle * is defined by $K_{\mathscr{F}}=\det(T\mathscr{F})^{**}$.

We will denote by $a:T\mathscr{F}\to TX$ an injective morphism (anchor map) defining the foliation $\mathscr{F}$.

The singular set of $\mathscr{F}$ is defined by $Sing(\mathscr{F})=Sing(N\mathscr{F})$, where $N\mathscr{F}=T_{X}/T\mathscr{F}$ is the normal sheaf of the foliation. Hereafter we will suppose that $cod(Sing(\mathscr{F}))\geq 2$. We have a exact sequence of sheaves

[TABLE]

Definition 2.2**.**

[39] Let $\mathscr{F}$ be a holomorphic foliation on a projective variety $X$ and $f:Y\to X$ a projective birational morphism. We say that $\mathscr{F}$ has canonical singularities if the divisor $K_{f^{-1}\mathscr{F}}-f^{*}K_{\mathscr{F}}$ is effective.

3. Poisson modules

A Poisson structure on a variety $X$ is a $\mathbb{C}$-linear Lie bracket

[TABLE]

which satisfies the Leibniz rule $\{f,gh\}=h\{f,g\}+f\{g,h\}$ and Jacobi identities

[TABLE]

for all germs of holomorphic functions $f,g,h$. The bracket corresponds to a bivector field $\sigma\in H^{0}(X,\wedge^{2}TX)$ given by $\sigma(df\wedge dg)=\{f,g\}$, for all germs of holomorphic functions $f,g$.

We will denote a Poisson structure on $X$ as the pair $(X,\sigma)$, where $\sigma\in H^{0}(X,\wedge^{2}TX)$ is the corresponding Poisson bivector field. The bivector induces a morphism

[TABLE]

which is called the anchor map and it is defined by $\sigma^{\#}(\theta)=\sigma(\theta,\cdot)$, where $\theta$ is a germ of holomorphic $1$-form.

Definition 3.1**.**

The symplectic foliation associated to $\sigma$ is the foliation given by $\mathscr{F}_{\sigma}:=\mathrm{Ker}(\sigma^{\#})$, whose dimension is the rank the anchor map $\sigma^{\#}:\Omega_{X}^{1}\to TX$. A Poisson variety $(X,\sigma)$ is called generically symplectic if the anchor map $\sigma^{\#}:\Omega_{X}^{1}\to TX$ is generically an isomorphism. Then, the degeneracy loci of $\sigma^{\#}$ is an effective anti-canonical divisor $D(\sigma)\in|-K_{X}|$.

A meromorphic Poisson bivector is a meromorphic section $\sigma$ of $\wedge^{2}TX$ such that $[\sigma,\sigma]=0$, where $[\ ,\ ]$ denotes the Schouten bracket. Observe that in this case we have a Poisson structure outside of the poles divisor of $\sigma$.

A rational vector field is a section v\in H^{0}(X,TX\otimes\mbox{{\mathscr{L}}}) for some invertible sheaf ${\mathscr{L}}$. We say that $v$ is a Poisson rational vector field with respect to $\sigma$ if it is an infinitesimal symmetry of $\sigma$, ie., $L_{v}(\sigma)=0$, where $L_{v}(\cdot)$ denotes the Lie derivative.

We denote by $\delta$ the corresponding Poisson differential. For instance, $\delta(f)=-\sigma^{\#}(df)$ and $\delta(v)=L_{v}(\sigma)$, for all germs of holomorphic function $f$ and germs of vector field $v$.

Definition 3.2**.**

We say that a Poisson variety $(X,\sigma)$ is *klt * if $X$ is a klt variety and the Poisson structure $\sigma$ is either generically symplectic or the associated symplectic foliation $\mathscr{F}_{\sigma}$ has canonical singularities.

Definition 3.3**.**

[2] Let $X$ be a projective variety. A holomorphic connection on a sheaf of ${\mathcal{O}}_{X}$-modules $E$ is a $\mathbb{C}$-linear morphism of sheaves $\nabla:E\to\Omega_{X}^{1}\otimes E$ satisfying the Leibniz rule

[TABLE]

where $f$ is germ of holomorphic function on $X$ and $s$ is a germ of holomorphic section of $E$. For a holomorphic connection $\nabla$, one defines as usual its extension(curvature) $\nabla^{2}:E\to\Omega_{X}^{2}\otimes E$. We say that $\nabla$ is flat if $\nabla^{2}=0$. We say that $E$ is a flat holomorphic sheaf if it admits a flat holomorphic connection.

Definition 3.4**.**

Let $(X,\sigma)$ be a Poisson projective variety. A Poisson connection on a sheaf of ${\mathcal{O}}_{X}$-modules $E$ is a $\mathbb{C}$-linear morphism of sheaves $\nabla:E\to T_{X}\otimes E$ satisfying the Leibniz rule

[TABLE]

where $f$ is a germ of holomorphic function on $X$ and $s$ is a germ of holomorphic section of $E$. We say that $E$ is a Poisson module if it admits a Poisson flat connection, i.e, if its curvature $\nabla^{2}:E\to\Omega_{X}^{2}\otimes E$ vanishes. Equivalently, a Poisson connection defines a $\mathbb{C}$-linear bracket $\{\ ,\ \}:{\mathcal{O}}_{X}\times E\to E$ by

[TABLE]

where $f$ is a germ of holomorphic function on $X$ and $s$ is a germ of holomorphic section of $E$.

Definition 3.5**.**

Let a:\mbox{{\mathscr{G}}}\to TX be a holomorphic foliation. A * holomorphic partial connection* on a sheaf of ${\mathcal{O}}_{X}$-module $E$ is a $\mathbb{C}$-linear morphism of sheaves \nabla:E\to\mbox{{\mathscr{G}}}^{*}\otimes E satisfying the Leibniz rule

[TABLE]

where $f$ is a germ of holomorphic function on $X$, $s$ is a germ of holomorphic section of $E$ and a^{*}:\Omega_{X}^{[1]}\to\mbox{{\mathscr{G}}}^{*} denotes the dual map. If $E$ admits a flat holomorphic partial connection with respect a foliation ${\mathscr{G}}$ we say that $E$ is a flat holomorphic sheaf along ${\mathscr{G}}$.

Definition 3.6**.**

[45, 32] A co-Higgs sheaf on a variety $X$ is a sheaf $E$ together with a section $\phi\in H^{0}(X,TX\otimes End(E))$ (called a co-Higgs fields) for which $\phi\wedge\phi=0$. If $E$ is a locally free sheaf we say that it is a co-Higgs bundle.

Definition 3.7**.**

[9, section 2] Let $G$ be a connected, reductive, affine algebraic group defined over $\mathbb{C}$. A holomorphic principal $G$-bundle $E_{G}$ is a $G$-co-Higgs bundle if there exist a holomorphic section $\phi$ of $\mbox{ad}(E_{G})\otimes TX$, where $\mbox{ad}(E_{G})$ denotes the adjoint vector bundle associated to $E_{G}$. If $\phi$ is a meromorphic section of $\mbox{ad}(E_{G})\otimes TX$ we say that $E_{G}$ is a meromorphic $G$-co-Higgs bundle.

In this work we are interested in the case where $G=\mbox{SL}(2,\mathbb{C})$, see section 5.

Example 3.8**.**

Let $(X,\sigma)$ be a generically symplectic variety of dimension $2n$ with degeneracy divisor $D(\sigma)=D$. It follows from [42, Proposition 4.4.1] that $\sigma$ induces a skewsymmetric morphism

[TABLE]

which is an isomorphism if and only if $D$ is reduced [42, Proposition 4.4.2]. Therefore, if $D$ is reduced then the isomorphism $\sigma^{\#}$ gives a one-to-one correspondence between Poisson flat connections and logarithmic flat connections . If $D$ is not reduced, then a Poisson connection corresponds to a meromorphic flat connection with poles along $D$.

Example 3.9**.**

Let $(X,\sigma)$ be a Poisson variety which is not generically symplectic. If $E$ admites a holomorphic flat connection $\nabla:E\to\Omega_{X}^{1}\otimes E$. Then

[TABLE]

is a Poisson flat connection on $E$ tangent to the symplectic foliation $\mathscr{F}_{\sigma}$.

Example 3.10**.**

Let $(X,\sigma)$ be a Poisson projective variety whose the leaves of the symplectic foliation $\mathscr{F}_{\sigma}$ are the fibers of a rational map $\rho:X\dashrightarrow Y$. If $\nabla:E\to\Omega_{Y}^{1}\otimes E$ is a holomorphic flat connection, then $\rho^{*}E$ is a Poisson module on $X$, with Poisson connection given by

[TABLE]

where $\omega:\Omega^{1}_{Y|X}\to T_{Y|X}$ denotes the induced isomorphism given by $(\sigma^{\#})_{|T\mathscr{F}_{\sigma}}$.

Example 3.11**.**

Let $(X,0)$ be a projective variety with the trivial Poisson structure. Let $(E,\phi)$ be a co-Higgs bundle on $X$. Then $E$ is a Poisson module on $X$ whose Poisson connection is given by $\nabla(fs):=\phi(df)s$, where $f$ is a germ of holomorphic function on $X$, $s$ is a germ of holomorphic section of $E$.

Example 3.12**.**

Let $(X,\sigma)$ be a Poisson projective manifold and $\rho:X\dashrightarrow Y$ a rational map with connected fibers such that $\rho_{*}\sigma=0$, then $\rho_{*}E$ is a co-Higgs sheaf on $Y$. In general, $\rho_{*}E$ is a Poisson module on $(Y,\rho_{*}\sigma)$, since $f_{*}{\mathcal{O}}_{X}\simeq{\mathcal{O}}_{Y}$. See in the next section Proposition 4.3 due to Polishchuk.

4. Rational Morita equivalence

Le us recall the notion of Morita equivalence of real Poisson manifolds which was developed by Weinstein [52] and Xu [53] and it works verbatim in the complex context: we say that two Poisson manifolds $(X,\sigma_{1})$ and $(Y,\sigma_{2})$ are Morita equivalence if there exists a symplectic manifold $(S,\varrho)$ and with two arrows

[TABLE]

such that are Poisson submersions. This equivalence has the following important properties :

• •

there is a bijection between the leaves of the sympectic foliations of $(X_{1},\sigma_{1})$ and $(X_{2},\sigma_{2})$.

• •

the space of Casimir functions of $(X_{1},\sigma_{1})$ and $(X_{2},\sigma_{2})$ are isomorphic.

Definition 4.1**.**

We say that two Poisson normal projective varieties $(X,\sigma_{1})$ and $(Y,\sigma_{2})$ are rationally Morita equivalence if there exists a projective normal variety $(S,\varrho)$, with a (possibly meromorphic ) Poisson bivector $\varrho$, and two diagrams

[TABLE]

such that $f$ and $h$ are dominant Poisson morphisms, i.e, $f_{*}\varrho=\sigma_{1}$ and $h_{*}\varrho=\sigma_{1}$. We say that two Poisson modules $E_{1}\to(X,\sigma_{1})$ and $E_{2}\to(Y,\sigma_{2})$ are rationally Morita equivalence if there is an equivalence $(X,\sigma_{1})\stackrel{{\scriptstyle f}}{{\longleftarrow}}(S,\varrho_{12})\stackrel{{\scriptstyle h}}{{\longrightarrow}}(Y,\sigma_{2})$ such that $h_{*}f^{*}E_{1}$ and $E_{2}$ are isomorphic as Poisson modules.

We say that

[TABLE]

is isomorphic to

[TABLE]

if there is a birational map $\zeta:(S,\varrho_{12})\longrightarrow(Q,\varrho_{34})$ such that $\zeta_{*}\varrho_{12}=\varrho_{34}$. Therefore, for each $(X_{1},\sigma_{1})\stackrel{{\scriptstyle f}}{{\longleftarrow}}(S,\varrho_{12})\stackrel{{\scriptstyle h}}{{\longrightarrow}}(X_{2},\sigma_{2})$ we can aways take the resolution of singularities $\zeta:\tilde{S}\to S$ and the lifting of the meromorphic Poisson bivector field $\tilde{\varrho_{12}}$ will give us that $\zeta_{*}{\varrho_{12}}=\varrho_{12}$. Thus, we obtain an equivalence

\textstyle{(\tilde{S},\tilde{\varrho_{12}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{(S,\varrho_{12})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{(X,\sigma_{1})}$$\textstyle{(Y,\sigma_{2}),}

with $\tilde{S}$ smooth.

Since $f^{-1}\mathscr{F}_{\sigma_{1}}=h^{-1}\mathscr{F}_{\sigma_{2}}$ , we obtain the following.

Proposition 4.2**.**

The following holds:

there is a bijection between the leaves of the symplectic foliations of $(X_{1},\sigma_{1})$ and $(X_{2},\sigma_{2})$,

• 2)

the spaces of Casimir rational functions of $(X_{1},\sigma_{1})$ and $(X_{2},\sigma_{2})$ are isomorphic,

• 3)

Morita equivalence implies the rational Morita equivalence.

Let us list some known results:

Proposition 4.3**.**

(Polishchuk [41]) Let $X$ be a Poisson variety. Let $f:X\to Y$ be a morphism such that $f_{*}{\mathcal{O}}_{X}\simeq{\mathcal{O}}_{Y}$. Then the Poisson structure on $X$ induces canonically a Poisson structure on $Y$ such that $f$ is a Poisson morphism. Furthermore, if ${\mathscr{E}}$ is a Poisson module on $X$, then f_{*}\mbox{{\mathscr{E}}} is a Poisson module on $Y$.

Theorem 4.4**.**

(Kaledin [35]) Let $X$ be a Poisson variety. The reduction, any completion and the normalization of $X$ are again Poisson varieties.

Therefore, we can assume that the Poisson variety $X$ is reduced and normal. Moreover, if there exist a morphism $f:X\to Y$ with connected fibers, then by Proposition 4.3 we have that $X$ is rationally Morita equivalent to $Y$, since $f_{*}{\mathcal{O}}_{X}\simeq{\mathcal{O}}_{Y}$ [24, Chapter 9]. This also says us that a Poisson structure on a projective complex manifold $X$ induces a natural Poisson structure on its minimal model by pushing forward the Poisson bivector.

Example 4.5**.**

Let $S$ be a smooth Poisson surface with a Poisson structure $\sigma\in H^{0}(S,{\mathcal{O}}_{S}(-K_{S}))$. Let $S^{[r]}=\hbox{\rm Hilb}^{r}(S)$ the Hilbert scheme parametrizing 0-dimensional subschemes of $S$ of length $r$. Bottacin in [11] extended Beauville’s construction of a symplectic structure on the Hilbert scheme $S^{[r]}$, let us say $\sigma^{[r]}$, of a symplectic surface $(S,\sigma)$ [6]. Consider $\hbox{\rm Bl}_{\Delta}(S^{2})$, the blowup along the diagonal $\Delta\subset S^{2}=S\times S$. Ran in [43] showed that there is a diagram

[TABLE]

such that $q^{*}(\sigma\times\sigma)$ is a meromorphic Poisson bivector on $\hbox{\rm Bl}_{\Delta}(S^{2})$ with simple pole on $\mathbb{P}(\Omega^{1}_{S})$ and $p_{*}q^{*}(\sigma\times\sigma)$ is the Poisson structure $\sigma^{[2]}$ on $S^{[2]}$ . Therefore, $S^{[2]}$ is rationally Morita equivalent to $S^{2}$. In the general, we can conclude by Ran’s induction construction that $S^{[r]}$ is rationally Morita equivalent to $S^{[r-1]}\times S$, for all $r\geq 2$, see [43, Section 1.6]. Since $S^{[r-1]}\times S$ is clearly Morita equivalent to $S$, we conclude that $S^{[r]}$ is Morita equivalent to $S$, for all $r\geq 2$.

Definition 4.6**.**

Let $(X,\sigma)$ be a Poisson variety which is not generically symplectic. We say that $(X,\sigma)$ is transcendental, in the spirit of [4, 37], if there is no positive-dimensional algebraic subvariety through a general point of $X$ that is tangent to the symplectic foliation $\mathscr{F}_{\sigma}$.

Definition 4.7**.**

A morphism $f:Z\to X$ between normal varieties is called a quasi-étale morphism if $f$ is finite and étale in codimension one.

Now, we will prove our main result.

Theorem 4.8**.**

Let $(E,\nabla)$ be a locally free Poisson module on a klt Poisson projective variety $(X,\sigma)$. Then at least one of the following holds up to rational Morita equivalence.

• (a)

$(E,\nabla)$* corresponds to a flat holomorphic sheaf on a transcendental Poisson variety;*

• (b)

$(E,\nabla)$* corresponds to a meromorphic flat connection on a generically symplectic variety.*

• (c)

$(E,\nabla)$* corresponds to a co-Higgs sheaf on a variety with trivial Poisson structure .*

• (d)

$(E,\nabla)$* corresponds to a meromorphic co-Higgs sheaf $(E_{0},\psi)$ on a transcendental Poisson variety $(Y,\sigma_{0})$, there exist a rational map $\zeta:Y\dashrightarrow B$, over a variety $B$ with $\mathrm{dim}\,(B)=\mathrm{dim}\,(\mathscr{F}_{\sigma_{0}})$, and the co-Higgs field $\phi$ is tangent to $T_{Y|B}$ and satisfies $D_{0}(\phi)=0$, where $D_{0}$ is a meromorphic extension of a Poisson connection on $T_{Y|B}\otimes End(E_{0})$.*

Proof.

Let $(X,\sigma)$ be a generically symplectic variety. Consider the degeneracy Poisson divisor $D=\{\sigma^{n}=0\}\in|-K_{X}|$. Then, it follows from Example 3.8 that $\nabla$ induces a meromorphic connection with poles along $D$ .

Suppose that $(X,\sigma)$ is not generically symplectic and consider the associated symplectic foliation $\mathscr{F}_{\sigma}$. Since $K\mathscr{F}_{\sigma}\simeq{\mathcal{O}}_{X}$ and $\mathscr{F}_{\sigma}$ has canonical singularities, it follows from [23, Proposition 8.14] that there exist two arrows

[TABLE]

such that $f:Z\to X$ is a quasi-étale cover, $\pi_{2}:Z\to Y$ is the natural projection, $Y$ and $Z$ are normal klt projective varieties, and it there is a transcendental foliation ${\mathscr{K}}$ on $Y$ such that \pi_{2}^{-1}\mbox{{\mathscr{K}}}=f^{-1}\mathscr{F}_{\sigma}.

Since $f:Z\to X$ is a quasi-étale cover there is a Poisson bivector $\tilde{\sigma}\in H^{0}(Z,\wedge^{2}TZ)$ such that $f_{*}\tilde{\sigma}=\sigma.$ That is, $f$ is a Poisson morphism. Indeed, the map $f:f^{-1}(X_{reg})\to X_{reg}$ is a map between complex manifolds which is a local biholomorphism, so there is a well-defined pull back Poisson bivector field $(f^{-1})^{*}(\sigma|_{X_{reg}})$ which extends to a section $\tilde{\sigma}\in H^{0}(Z,\wedge^{2}TZ)$. Similarly, one can also see that $f^{*}E$ is a Poisson module, with respect to $\tilde{\sigma}$, by lifting the local matrices of vector fields which represent the Poisson connection $\nabla$. Let $\tilde{\nabla}$ denote such induced Poisson connection on $f^{*}E$. Thus, we have a Morita equivalence

[TABLE]

where $\sigma_{2}=:(\pi_{2})_{*}\tilde{\sigma}$. In particular, we have that $E_{1}$ is rationally Morita equivalent to $(\pi_{2})_{*}(f^{*}E):=E_{0}$. Denote by $\nabla_{0}$ the Poisson connection induced on $E_{0}$.

Let us make the following simple but important observation:

Let $(E,\nabla)\to(W,\sigma)$ be a Poisson module with a flat Poisson connection $\nabla$ on a Poisson variety $(W,\sigma)$ such that $Sing(\mathscr{F}_{\sigma})\cup Sing(W)$ has codimension $\geq 2$. Then, we have that either:

• (i)

$E$ corresponds to a flat holomorphic sheaf along the symplectic foliation $\mathscr{F}_{\sigma}$, or

• (ii)

$\nabla$ induces a non-trivial section $\phi\in H^{0}(W,N\mathscr{F}_{\sigma}\otimes End(E))$ such that $\phi\wedge\phi=0$.

In fact, if $(E,\nabla)$ is such that $\nabla:E\to T\mathscr{F}_{\sigma}\otimes E$, then $E$ corresponds to a flat holomorphic sheaf along the symplectic foliation. Indeed, since $cod(Sing(\mathscr{F}_{\sigma}))\geq 2$, then the map

\textstyle{T\mathscr{F}_{\sigma}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\vartheta}$$\textstyle{\Omega_{W}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{2}^{\#}}$$\textstyle{T\mathscr{F}_{\sigma}}

is an isomorphism, where $i:T\mathscr{F}_{\sigma}^{*}\to\Omega_{W}^{1}$ is the inclusion and $\sigma_{2}^{\#}:\Omega_{W}^{1}\to T\mathscr{F}_{\sigma}$ denotes the anchor map. Therefore, the Poisson connection

\textstyle{\vartheta^{-1}\circ\nabla_{0}:E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{F}_{\sigma}^{*}\otimes E}

is a partial flat connection on $E$. If the Poisson connection $\nabla_{0}:E\to TW\otimes E$ does not factor through $T\mathscr{F}_{\sigma}$, then it induces a non-trivial section

[TABLE]

such that $\phi\wedge\phi=0$. In fact, we have the commutative diagram

\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\nabla}$$\textstyle{TW\otimes E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\otimes Id}$$\textstyle{N\mathscr{F}_{\sigma}\otimes E.}

where $\phi=(\pi\otimes Id)\circ\nabla$ and $\pi:TW\to N\mathscr{F}_{\sigma}$ denotes the projection. Now, let $W^{0}:=W-Sing(\mathscr{F}_{\sigma})\cup Sing(W)$ and consider $\phi_{0}$ the restriction of $\phi$ on $W^{0}$. In order to show that $\phi\wedge\phi=0$ we follow the argument in [49] and next we extend $\phi_{0}$. Indeed, let $\theta$ be a local matrix representing the connection $\nabla_{0}$. By flatness we have that

[TABLE]

Then $\pi(\delta(\theta))=0$, since $\delta(\theta)$ is tangent to $\mathscr{F}_{\sigma}|_{W^{0}}$ and $0=\pi(\theta\wedge\theta)=\pi(\theta)\wedge\pi(\theta)$. Since $\pi(\theta)$ is the local matrix of $\phi_{0}$, we conclude that $\phi_{0}\wedge\phi_{0}=0$, i.e, $\phi\wedge\phi=0$, since $Sing(\mathscr{F}_{\sigma})\cup Sing(W)$ has codimension $\geq 2$.

If $\sigma_{2}=0$, then $(\pi_{2})_{*}[\tilde{\nabla}]\in H^{0}(Y,T_{Y}\otimes End(E_{0}))$ is a co-Higgs field.

If $(Y,\sigma_{2})$ is generically symplectic, then the Poisson module $(E_{0},\nabla_{0})$ corresponds to a meromorphic flat connection.

From now on we suppose that $\sigma_{2}\neq 0$ and $(Y,\sigma_{2})$ is not generically symplectic.

If $\mathscr{F}_{\sigma}$ is algebraic, i.e., the symplectic foliation $f^{*}\mathscr{F}_{\sigma}$ is given by the fibration $\pi_{2}:Z=W\times Y\to Y$. Then, the connection $\tilde{\nabla}$ corresponds either to a relative flat connection $\tilde{\nabla}:f^{*}E\to\Omega^{1}_{Z|Y}\otimes f^{*}E$, since we have the isomorphism $\Omega^{1}_{Z|Y}\simeq T_{Z|Y}$, or the induced co-Higgs field $[\tilde{\nabla}]\in H^{0}(Z,\pi_{2}^{*}T_{Y}\otimes End(f^{*}E))$ is such that $(\pi_{2})_{*}[\tilde{\nabla}]\in H^{0}(Y,T_{Y}\otimes End(E_{0}))$.

If the symplectic foliation $\mathscr{F}_{\tilde{\sigma}}$ is not algebraic, then ${\mathscr{K}}$ corresponds to the symplectic foliation of the Poisson structure $\sigma_{2}$, since \pi_{2}^{-1}\mathscr{F}_{\sigma_{2}}=\mathscr{F}_{\tilde{\sigma}}=f^{-1}\mathscr{F}_{\sigma}=\pi_{2}^{-1}\mbox{{\mathscr{K}}}. Therefore, as we have seen above, it is either:

• (i)

$(E_{0},\nabla_{0})$ corresponds to a flat holomorphic sheaf along the symplectic foliation, or

• (ii)

$\nabla_{0}$ induces a non-trivial section

[TABLE]

such that $\phi_{0}\wedge\phi_{0}=0$.

Consider an embedding $Y\subset\mathbb{P}^{N}$ and let k:=\mathrm{dim}\,(\mbox{{\mathscr{K}}}). We can take a generic projection $q:\mathbb{P}^{N}\dashrightarrow\mathbb{P}^{n}$, such that restricted to $Y$ we produce a finite surjective morphism $q_{|Y}:Y\to\mathbb{P}^{n}$. We also can take a generic rational linear projection $p:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{k}$ in such way that the fibration $p\circ(q_{|Y}):Y\dashrightarrow\mathbb{P}^{k}$ is generically transversal to ${\mathscr{K}}$. After we take a Stein factorization we obtain a rational fibration $\zeta:Y\dashrightarrow B$, with connected fibers, which induces on $Y$ an algebraic foliation ${\mathscr{G}}$ of dimension equal to $n-k$. Consider the tangency loci between ${\mathscr{K}}$ and ${\mathscr{G}}$ given by

[TABLE]

where \omega\in H^{0}(Y,\Omega_{Y}^{[n-k]}\otimes\det(N\mbox{{\mathscr{K}}})) and \xi\in H^{0}(Y,\wedge^{k}TY\otimes\det(T_{\mbox{{\mathscr{G}}}})^{*}) are the tensors inducing ${\mathscr{K}}$ and ${\mathscr{G}}$, respectively. Observe that $S$ contain the singular sets of ${\mathscr{K}}$ and ${\mathscr{G}}$. By transversality the induced map

\textstyle{T_{\mbox{{\mathscr{G}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\beta}$$\textstyle{TY\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{N\mbox{{\mathscr{K}}}}

is an isomorphism on $Y^{0}:=Y-S$. This gives us a co-Higgs field

[TABLE]

given by $\tilde{\phi_{0}}=\phi_{0}\circ\beta^{-1}$. Denote by $j:Y^{0}\to Y$ the inclusion map and consider the sheaf of meromorphic sections with poles of arbitrary order on $S$ given by

[TABLE]

Thus, we obtain a meromorphic co-Higgs field \phi\in H^{0}(Y,T_{\mbox{{\mathscr{G}}}}\otimes End(E_{0})\otimes{\mathcal{O}}_{Y}(*S)) which is a meromorphic extension of $\tilde{\phi_{0}}$. Recall that ${\mathscr{K}}$ and ${\mathscr{G}}$ are regular on $Y^{0}$. Thus, it follows from [49, Corollary 3.3] that the Partial connection on N\mbox{{\mathscr{K}}}|_{Y^{0}}\simeq T_{\mbox{{\mathscr{G}}}}|_{Y^{0}} and the Poisson connection $\nabla_{0}$ induces a Poisson connection $\tilde{D_{0}}$ on

[TABLE]

such that $\tilde{D_{0}}(\tilde{\phi_{0}})=0$. Denoting by $D_{0}$ its meromorphic extension we have that $D_{0}(\phi)=0$.

∎

Remark 4.9**.**

If $\phi$ denotes a local representation of $\phi_{0}$, then from [49, Proposition 3.2, equation 3.2 ] we conclude that $D_{0}(\phi)=0$ implies that

[TABLE]

Fixed a klt Poisson structure $(X,\sigma)$, consider the associated category of Poisson modules ${\text{\rm Rep}}(X,\sigma)$. Also, denote by ${\text{\rm Co-Higgs}}(X)$ the category of co-Higgs bundles and $\text{\rm Conn}(X,D)$ the category of meromorphic connections along $D$. It follows from the proof of Theorem 4.8 that via the Morita equivalence between $(X,\sigma)$ and $(Y,\sigma_{2})$ we obtain the following induced functors:

• •

$\pi_{*}f^{*}:{\text{\rm Rep}}(X,\sigma)\to{\text{\rm Co-Higgs}}(Y)$, if $\pi_{*}f^{*}(\sigma)=0$.

• •

$\pi_{*}f^{*}:{\text{\rm Rep}}(X,\sigma)\to{\text{\rm Conn}}(Y,D)$, if $\pi_{*}f^{*}(\sigma)$ is generically symplectic and $D$ is the degeneracy Poisson Divisor.

5. Rank two $\mathfrak{sl}_{2}$-Poisson modules

Let $(X,\sigma)$ be a Poisson projective variety with Poisson bivector $\sigma$ and denote by $\delta$ the Lie algebroid derivation induced by $\sigma$. As we have seen in the Theorem 1.1, the presence of singularities of the Poisson structure forces in certain situations that the Poisson connections have poles along a divisor. In this section we will study the geometry of rank two meromorphic Poisson modules which are trace free.

Definition 5.1**.**

A rank two holomorphic vector bundle $E$ on a projective Poisson variety $(X,\sigma)$ is called a meromorphic Poisson module if there exists a connection

[TABLE]

with effective polar divisor $D$, such that $\nabla^{2}=0$. If $\mathrm{tr}(\nabla)=0$ we say that $(E,\nabla)$ is a meromorphic $\mathfrak{sl}_{2}$-Poisson module. If $D=\emptyset$ , then $(E,\nabla)$ is a holomorphic Poisson module.

Given two different $\mathfrak{sl}_{2}$-Poisson structures $(E,\nabla_{1})$ and $(E,\nabla_{2})$, with effective polar divisor $D$, then $(E,\nabla_{1}-\nabla_{2})$ is a meromorphic $\mbox{SL}(2,\mathbb{C})$-co-Higgs bundle.

First, we prove the following Polishchuk’s result [41] in our context.

Proposition 5.2**.**

Let $(E,\nabla)$ be a rank two meromorphic $\mathfrak{sl}_{2}$-Poisson module on a projective Poisson manifold $X$. Then, there exists a triple of rational vector fields $(v_{0},v_{1},v_{2})$ on $X$ such that

[TABLE]

where $\delta(v)=[v,\sigma]$, $\sigma$ is the Poisson bivector of $X$ and $[\ ,\ ]$ denotes the Schouten bracket. Moreover, $\mathbb{P}(E,\nabla)$ has a meromorphic Poisson structure induced by $\nabla$.

Proof.

Since $X$ is projective we have that $\mathbb{P}(E)$ birationally equivalent to $X\times\mathbb{P}^{1}$. In the vector bundle $X\times\mathbb{C}^{2}$ we have a trace free Poisson connection given by

[TABLE]

where $Z=(z_{1},z_{2})\in\mathbb{C}^{2}$. The flatness condition is equivalent to

[TABLE]

Consider the meromorphic bivector

[TABLE]

where $[1:z]\in\mathbb{P}^{1}$ denotes the affine coordinate. Then $\Sigma$ is Poisson if and only if

[TABLE]

∎

We observe that if the Poisson connection $(E,\nabla)$ is such that

[TABLE]

then the associated triple $(v_{0},v_{1},v_{2})$ satisfying (2) induces on each leaf of $\mathscr{F}$ a transversely projective holomorphic foliation [47, Chapter II]. Let $p\in X\setminus Sing(\mathscr{F})$ and $F_{p}$ the symplectic leaf of $\mathscr{F}$ passing through $p$. Since (\sigma^{\#})^{-1}\circ\delta|_{F_{p}}=d\circ(\sigma^{\#})^{-1}|_{F_{p}}\, where $d$ denotes the de Rham differential, we have that

[TABLE]

Now, defining $(\sigma^{\#})^{-1}|_{F_{p}}(v_{i})=\omega_{i}$, we obtain the transversely projective structure on $F_{p}$ given by the triple $(\omega_{0},\omega_{1},\omega_{2})$, where $\omega_{0}$ is the $1$-form inducing the transversely projective holomorphic foliation on $F_{p}$. As we already have seen, a natural way to produce such Poisson structure is by considering a meromorphic connection $\tilde{\nabla}:E\to E\otimes\Omega^{1}_{X}(D)$ and composing with the anchor map $\sigma^{\#}:\Omega_{X}^{1}\to TX$ we get $\nabla=\sigma^{\#}\circ\tilde{\nabla}:E\to E\otimes T\mathscr{F}(D)$. See [25, Proposition 1.8.2] for a more general consideration for principal bundles in the real category.

Remark 5.3**.**

There is a connection between transversely projective holomorphic foliation and quantization of symplectic foliation. Biswas in [10] showed that for any regular transversely projective foliation $\mathscr{F}$ there is a regular transversely symplectic foliation $\tilde{\mathscr{F}}$ on its conormal bundle $N\mathscr{F}^{*}$. Moreover, he proves that the restriction of $\tilde{\mathscr{F}}$ to the complement of the zero section admits a canonical quantization.

Example 5.4**.**

Consider a Poisson structure in $\mathbb{P}^{3}$ induced in homogeneous coordinates by $\sigma=v_{0}\wedge v_{1}$, where $v_{0}$ and $v_{1}$ are degree one polynomial vector fields satisfying $[v_{0},v_{1}]=0$. In this case we have that

[TABLE]

for $i=1,2$.

Example 5.5**.**

Consider the Poisson structure in $\mathbb{P}^{3}$ induced in homogeneous coordinates by $\sigma=v_{0}\wedge v_{1}$, where

[TABLE]

This Poisson structure corresponds to the exceptional foliation appearing in the Cerveau–Lins Neto’s classification [15]. A computation gives us

[TABLE]

since $[v_{1},v_{0}]=-v_{1}$. We refer the reader to [42, Proposition 8.9.2] for a more conceptual construction.

Remark 5.6**.**

Let $(E,\nabla)$ be a rank two holomorphic $\mathfrak{sl}_{2}$-Poisson module on a normal projective Poisson variety $(X,\sigma)$. Then $\mathbb{P}(E,\nabla)$ is rationally Morita equivalence to $(X,\sigma)$, since $\pi_{*}\Sigma_{\sigma}=\sigma$. In particular, if $(E,\phi)$ is a rank two $\mbox{SL}(2,\mathbb{C})$-co-Higgs bundle, then $(\mathbb{P}(E),\Sigma_{\phi})$ is rationally Morita equivalence to $X$ with zero Poisson structure, since $\pi_{*}\Sigma_{\phi}=0$.

Proposition 5.7**.**

Let $(E,\phi)$ be a rank two meromorphic $\mbox{SL}(2,\mathbb{C})$-co-Higgs bundle on a normal projective Poisson variety $X$. Then after a birational trivialization of $(E,\phi)$ the co-Higgs field is of the form

[TABLE]

for some rational vector field v_{\phi}\in H^{0}(X,TX\otimes\mbox{{\mathscr{L}}}) and rational functions $f_{0},f_{1},f_{2}\in K(X)$. Moreover, the symplectic foliation induced on $\mathbb{P}(E)$ has dimension two and it is the pull-back of the one-dimensional foliation $\mathscr{H}_{v_{0}}$ on $X$ induced by $v_{\phi}$. In particular, if $\mathscr{H}_{v_{0}}$ has canonical singularities and ${\mathscr{L}}$ is not pseudo-effective, then symplectic foliation on $\mathbb{P}(E)$ is a foliation by rational surfaces.

Proof.

We have that

[TABLE]

We assume without loss of generality that the rational vector field $v_{0}$ is not identically zero. Then, there exist rational functions $f,g,h$ such that $v_{1}=fv_{0}$, $v_{2}=gv_{0}$, and $v_{2}=hv_{1}=hfv_{0}$, so $g=hf_{1}$. Therefore, we get the rational co-Higgs fields

[TABLE]

Now, since the induced Poisson bivector is given by

[TABLE]

we conclude that the symplectic $\mathscr{F}_{\Sigma}$ has dimension two and it is the pull-back of the foliation $\mathscr{H}_{v_{0}}$ tangent to $v_{0}$. Now, if $\mathscr{H}_{v_{0}}$ has canonical singularities and ${\mathscr{L}}$ is not pseudo-effective, then it follows from [13, 23] that $\mathscr{H}_{v_{0}}$ is a foliation by rational curves. Hence, the leaves of $\mathscr{F}_{\Sigma}=\pi^{*}\mathscr{H}_{v_{0}}$ are rational surfaces. ∎

The author showed in [16] that if $(E,\phi)$ is a stable and nilpotent co-Higgs holomorphic bundle on a compact Kähler surface, then the symplectic foliation induced on $\mathbb{P}(E)$ is algebraic with rational leaves.

Example 5.8**.**

Rayan in [44] gave a complete description for the locus of the moduli space of stable holomorphic $\mbox{SL}(2,\mathbb{C})$-co-Higgs bundles whose underlying bundle is Schwarzenberger. Let $Q$ be an irreducible element of $H^{0}(\mathbb{P}^{2},\mathcal{O}_{\mathbb{P}^{2}}(2))$ such that the curve $C=\{Q=0\}\subset\mathbb{P}^{2}$ is a nonsingular conic. Consider a degree two covering $f^{Q}:\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{2}$ branched over $C=\{Q=0\}$. The rank two vector bundle $f^{Q}_{*}{\mathcal{O}}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(0,k)=V^{Q}_{k}$, for each $k\geq 0$ is called Schwarzenberger bundle [48]. Rayan in [44] proved that Schwarzenberger bundles are $\mbox{SL}(2,\mathbb{C})$-co-Higgs bundles with co-Higgs fields of the form

[TABLE]

where $\Phi\in H^{0}(\mathbb{P}^{2},End_{0}(V^{Q}_{k})(-1))$ and $v\in H^{0}(\mathbb{P}^{2},T\mathbb{P}^{2}(-1))$. It follows from Proposition 5.7 that the dimension two symplectic foliation on $\mathbb{P}(V^{Q}_{k})$ has rational leaves which are pull-back of the lines of the pencil tangent to $v\in H^{0}(\mathbb{P}^{2},T\mathbb{P}^{2}(-1))$.

Example 5.9**.**

We consider in this example a Poisson interpretation due to Pym [42, Section 8.7] for degree two pull-back foliations on $\mathbb{P}^{3}$. Let $E={\mathcal{O}}_{\mathbb{P}^{2}}\oplus{\mathcal{O}}_{\mathbb{P}^{2}}(1)$ and a rational vector field $v_{0}:{\mathcal{O}}_{\mathbb{P}^{2}}\to T\mathbb{P}^{2}(-1)$. Then the nilpotent co-Higgs field induces on $\mathbb{P}(E)$ a symplectic foliation such that the contraction of

[TABLE]

correspond a symplectic foliation given by $v\wedge v_{0}$, where $v_{0}\in H^{0}(\mathbb{P}^{3},T\mathbb{P}^{3}(-1))$ is a rational vector field which is tangent to a linear projection $\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{2}$ and $v$ is a global holomorphic vector field on $\mathbb{P}^{2}$.

Definition 5.10**.**

Let $Z\to X$ be a $\mathbb{P}^{1}$-bundle. A codimension one holomorphic foliation ${\mathscr{G}}$ on $Z$ is called by Riccati foliation if it is generically transversal to the $\mathbb{P}^{1}$-bundle $Z\to X$.

In the next result, as a consequence of Theorem 1.1, we will give a geometric description for rank two holomorphic $\mathfrak{sl}_{2}$-Poisson modules.

Corollary 5.11**.**

Let $(E,\nabla)$ be a rank two holomorphic $\mathfrak{sl}_{2}$-Poisson module on a klt Poisson projective variety $(X,\sigma)$. Then there exist projective varieties $Y$ and $Z$ with klt singularities, a quasi-étale Poisson cover $f:W\times Y\to X$ and at least one of the following holds.

• (a)

$(\pi_{2})_{*}f^{*}(E,\nabla)$* is a $\mathfrak{sl}_{2}$ partial holomorphic sheaf on $Y$, where $\pi_{2}$ denotes the projection on $Y$.*

• (b)

If $W$ and $Y$ are generically symplectic, then $(\pi_{2})_{*}f^{*}(E,\nabla)$ is a rank two locally free sheaf with a meromorphic flat connection with poles on the degeneracy Poisson divisor of $Y$.

• (c)

If $W$ is symplectic, then after a birational trivialization of $f^{*}(E,\nabla)$ the Poisson connection on the trivial bundle is defined as

[TABLE]

for some rational vector field $v$ tangent to $(Y,0)$, rational functions $f_{0},f_{1},f_{2}\in K(Y)$, and $\delta_{W}$ denotes the Poisson differential on $W$.

• (d)

There exists a rational map $\zeta:Y\dashrightarrow B$, over a variety $B$ with $\mathrm{dim}\,(B)=\mathrm{dim}\,(\mathscr{F}_{\alpha})$, such that $(\pi_{2})_{*}f^{*}(E,\nabla)$ corresponds to a meromorphic $\mathfrak{sl}_{2}$-Poisson module $(E_{0},\tilde{\nabla})$, such that after a birational trivialization the Poisson connection on the trivial bundle is defined as

[TABLE]

for some rational Poisson vector field $v$ and rational functions $f_{0},f_{1},f_{2}$ on $X$ such that $\{f_{i},f_{j}\}=0$, for all $i,j$.

Proof.

From the proof of Theorem 1.1 we have that there exist projective varieties $Y$ and $Z$ with klt singularities and a quasi-étale Poisson cover $f:(W\times Y,\tilde{\sigma})\to(X,\sigma)$ such that $f^{*}(E,\nabla)$ is a $\mathfrak{sl}_{2}$-Poisson module on $(W\times Y,\tilde{\sigma})$. Then, the result follows from Theorem 1.1. In fact, if $(Y,(\pi_{2})_{*}f^{*}(\sigma))$ is generically symplectic, then $(\pi_{2})_{*}f^{*}(E,\nabla)$ is a rank two locally free sheaf with a meromorphic flat connection with poles on the degeneracy Poisson divisor of $Y$. If $(\pi_{2})_{*}f^{*}(\sigma)=0$, then $(\pi_{2})_{*}f^{*}(E,\nabla)$ is a co-Higgs sheaf with co-Higgs field $(\pi_{2})_{*}f^{*}(\nabla)=\phi$ and after we take a birational trivialization of $f^{*}(E,\nabla)$ the Poisson connection is defined as $\tilde{\nabla}=\delta_{W}+(\pi_{2})^{*}\phi.$ Hence, we conclude the part (c) from proposition 5.7. Finally, for the item (d), after we take a birational trivialization of $f^{*}(E,\nabla)$ we use the remark 4.9 and Theorem 4.8, part (d), in order to conclude that

[TABLE]

with $\phi$ tangent a rational map $\zeta:Y\dashrightarrow B$. This implies that

[TABLE]

Then, there exist rational functions $f_{0},f_{1},f_{2}$ such that $v_{1}=f_{1}v_{0}$, $v_{2}=f_{2}v_{0}$, and $v_{1}=f_{0}v_{2}$. We may assume without loss of generality that $v_{0}\neq 0$ and $f_{2}\neq 0$, since the other cases follow similarly. By taking the Poisson derivation $\delta$ in $v_{2}=f_{2}v_{0}$, we get that

[TABLE]

which implies that $v_{0}=h_{2}\delta(f_{2})$, for some invertible rational function $h_{2}$. We also can conclude that $v_{0}=h_{1}\delta(f_{1})$. Then

[TABLE]

Thus,

[TABLE]

Now, on the one hand, by using that $v_{1}=f_{0}v_{2}$, we obtain that $v_{1}=h_{0}\delta(f_{0})$. On the other hand, since $v_{1}=f_{1}h_{1}\delta(f_{1})$ and $v_{2}=f_{2}h_{2}\delta(f_{2})$ we have that

[TABLE]

and also $\{f_{0},f_{2}\}=0$. ∎

As we have seen above, the geometric study of the symplectic foliation $\mathscr{F}_{\nabla}$ reduces, up to a quasi-étale Poisson cover, to the foliation $\mathscr{F}_{\nabla_{0}}$ on $\mathbb{P}(E_{0},\nabla_{0})\to(Y,\sigma_{0})$.

Corollary 5.12**.**

Let $\mathscr{F}_{\nabla_{0}}$ be the symplectic foliation induced on $\pi:\mathbb{P}(E_{0})\to(Y,\sigma_{0})$. Then at least one of the following holds.

• (a)

$\mathscr{F}_{\nabla_{0}}$* is a dimension 2 foliation which is a pull-back of a foliation by curves on $(Y,0)$.*

• (b)

$\mathscr{F}_{\nabla_{0}}$* is a Riccati foliation of codimension one on $\mathbb{P}(E_{0})$, if $(Y,\sigma_{0})$ is generically symplectic.*

• (c)

$\mathscr{F}_{\nabla_{0}}$* is a Riccati foliation of codimension one on $\mathbb{P}(E_{0})$ which is given by a morphism \mbox{{\mathscr{A}}}\to d_{\text{\rm refl}}\pi(\pi^{*}(T\mathscr{F}_{\sigma_{0}}^{*}))\subset\Omega_{\mathbb{P}(E_{0})}^{[1]}, where ${\mathscr{A}}$ is a line bundle and $d_{\text{\rm refl}}\pi:\pi^{*}\Omega_{Y}^{[1]}\to\Omega_{\mathbb{P}(E_{0})}^{[1]}$ is the pull-back morphism of reflexive forms.*

• (d)

There exist a rational Poisson vector field $v$ generically transversal to $\mathscr{F}_{\sigma_{0}}$ such that $\mathscr{F}_{\nabla_{0}}$ has dimension $2k+2$ and it is the pull-back of the foliation induced by $v$ and $\mathscr{F}_{\sigma_{0}}$. In particular, if $\mathrm{dim}\,(Y)=2k+1$, then $\mathbb{P}(E_{0})$ is generically symplectic and there exist a rational Poisson map $\zeta:Y\dashrightarrow B$ generically transversal to $\mathscr{F}_{\sigma_{0}}$, where $B$ is a generically symplectic variety with $\mathrm{dim}\,(B)=2k$ and the induced map $\mathbb{P}(E_{0})\dashrightarrow B$ is Poisson.

Proof.

If $(Y,\sigma_{0})$ is generically symplectic, then $\nabla_{0}$ corresponds to a meromorphic flat connection. Hence, $\mathscr{F}_{\nabla_{0}}$ is a Riccati foliation of codimension one. If $\nabla_{0}$ corresponds to a partial flat meromorphic connection

[TABLE]

then $\mathscr{F}_{\nabla_{0}}$ is a Riccati foliation of codimension one, which is given by the meromorphic 1-form

[TABLE]

where $\omega_{i}$’s are meromorphic sections of $\pi^{*}(T\mathscr{F}_{\sigma_{0}}^{*})\subset\pi^{*}\Omega_{Y}^{[1]}$. Therefore, $\mathscr{F}_{\nabla_{0}}$ is induced by a morphism \mbox{{\mathscr{A}}}\to d_{\text{\rm refl}}\pi(\pi^{*}(T\mathscr{F}_{\sigma_{0}}^{*}))\subset\Omega_{\mathbb{P}(E_{0})}^{[1]}, where ${\mathscr{A}}$ is a line bundle and

[TABLE]

is the pull-back morphism which there exist by [28, Theorem 1.4].

Now, recall from the proof of Proposition 5.2 that the foliation $\mathscr{F}_{\nabla_{0}}$ is induced by the bivector

[TABLE]

If $\nabla_{0}$ corresponds to a co-Higgs field on $(Y,0)$, then from Proposition 5.7 we have that $\mathscr{F}_{\nabla_{0}}$ is a dimension 2 foliation which is a pull-back of a foliations by curves. This shows the part (c). Finally, for the case (d) we have the bivector

[TABLE]

where $v(z)=\ell v$, with $\ell$ being the non-zero rational function $f_{0}+2f_{1}z+f_{2}z^{2}$. Thus, $\mathscr{F}_{\nabla_{0}}$ is induced by

[TABLE]

since $\sigma_{0}^{k+1}=0$, $\sigma_{0}^{k}\neq 0$ and $v$ is transversal to $\sigma_{0}^{k}$. This show us that $\mathscr{F}_{\nabla_{0}}$ is induced by the pull-back of the foliation generated by $v$ and $\sigma_{0}^{k}$. Now, if $\mathrm{dim}\,(Y)=2k+1$, then the symplectic foliation has codimension one and the rational vector fields $v$ induces a foliation whose leaves are tangent to the fibers of $\zeta:Y\dashrightarrow B$. The condition $\delta(v)=L_{v}\sigma_{0}=0$, says us that there exist a non-trivial Poisson bivector $\sigma_{B}$ such that $\zeta_{*}\sigma_{0}=\sigma_{B}$. ∎