Philippe Monnier
It is known that the computation of the Poisson cohomology is closely related to the classification of singularities of Poisson structures. In this paper, we will first look for the normal forms of germs at (0,0) of Poisson structures on the real (or complex) plane and recall a result given by Arnold. Then, we will compute locally the Poisson cohomology of a particular type of Poisson structure.
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Poisson cohomology of "book" Poisson structures
研究书状李代数的线性泊松结构的泊松上同调，通过计算其上同调来揭示结构特性。
We compute the Poisson cohomology of a class of Poisson manifolds that are symplectic away from a collection $D$ of hypersurfaces. These Poisson structures induce a generalization of symplectic and cosymplectic structures, which we call a k-cosymplectic structure, on the intersection of hypersurfaces in $D$.
On the First Cohomology of Infinitesimal Poisson Algebras
本文研究无穷小泊松代数的第一上同调，用底层泊松子流形的内蕴上同调刻画其结构，建立到第一泊松上同调的自然映射，给出上同调消失的充要条件，并应用于辛叶和部分分裂一阶射流情形。
We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure $π$ is defined on the tubular neighbourhood of the singular locus $Z_ω$ of the 2-form $ω$, it is of maximal rank 4 and it vanishes on a degeneracy set containing $Z_ω$. We compute its smooth Poisson cohomology, which depends on the modular vector field and it is finite dimensional. We conclude with a discussion on the relation between the Poisson structure $π$ and the overtwisted contact structure associated to a near-symplectic 4-manifold…
This is the first of two papers, in which we prove a version of Conn's linearization theorem for the Lie algebra $\mathfrak{sl}_2(\mathbb{C})\simeq \mathfrak{so}(3,1)$. Namely, we show that any Poisson structure whose linear approximation at a zero is isomorphic to the Poisson structure associated to $\mathfrak{sl}_2(\mathbb{C})$ is linearizable. In this first part, we calculate the Poisson cohomology associated to $\mathfrak{sl}_2(\mathbb{C})$, and we construct bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In the second part, we will obt…
A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, Kähler manifolds, and n…
Let M be a paracompact smooth manifold of dimension n; A a Weil algebra and M^A the Weil bundle associated. We define and describe the notion of \widetilded-Poisson cohomology and of \widetilded^A -Poisson cohomology on M^A.
Let M be a Poisson manifold and A a Weil algebra. We describe an isomorphism of cohomolgy algebra and proves that Poisson cohomology with values in A is isomorphic to the tensor product of A with Poisson cohomolgy with real values.
We describe the generalized Kuranishi spaces of solvmanifolds with left-invariant complex structures. By using such description, we study the stability of left-invariantness of deformed generalized complex structures and smoothness of generalized Kuranishi spaces on certain classes of solvmanifolds. We also give explicit finite dimensional cohcain complexes which computes the holomorphic Poisson cohomology of nilmanifolds and solvmanifolds.
We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H, and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra associated to G/H. We also construct a Poisson groupoid over G/H which is symplectic near the identity section. This note serves as preparation for forthcoming papers, in which we will compute explicitly th…
In this paper, we introduce the notions of logarithmic Poisson structure and logarithmic principal Poisson structure; we prove that the latter induces a representation by logarithmic derivation of the module of logarithmic Kahler differentials; therefore, it induces a differential complex from which we derive the notion of logarithmic Poisson cohomology. We prove that Poisson cohomology and logarithmic Poisson cohomology are equal when the Poisson structure is logsymplectic. We give an example of non logsymplectic but logarithmic Poisson structure for which these cohomologies are equal. We als…
We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes will permit us to show that a regular Poisson manifold whose symplectic foliation is of codimension one is unimodular if and only if its symplectic foliation is Riemannian foliation. It permit us also to construct examples of unimodular Poisson manifolds and other which are not unim…
We study the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, first defined by Lichnerowicz using contravariant tensor fields. We show that for a regular Poisson manifold M, the TP-cohomology coincides with the leafwise de Rham (or Cech) cohomology of the symplectic foliation of M. Its computation in various degrees leads to open, non trivial problems. To get a better understanding of these difficulties, we study explicitly many examples coming from nilpotent and 3-dimensional (real) Lie algebras. For the latter, we compare the TP-cohomology and the usual Poisson coh…
We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.
The subject for investigation in this note is concerned with holomorphic Poisson structures on nilmanifolds with abelian complex structures. As a basic fact, we establish that on such manifolds, the Dolbeault cohomology with coefficients in holomorphic polyvector fields is isomorphic to the cohomology of invariant forms with coefficients in invariant polyvector fields. We then quickly identify the existence of invariant holomorphic Poisson structures. More important, the spectral sequence of the Poisson bi-complex associated to such holomorphic Poisson structure degenerates at $E_2$. We will a…
Continuing a work of Ph.~Monnier, we determine the Gerstenhaber algebra structure over the Poisson cohomology groups for a large class of Poisson structures with isolated singularities over the plane. It reveals that there exists a GAGA type phenomenon. We give an explicit description via generators and relations for the case of simple singularities.
It is known that there exist singular Poisson structures in $4$-manifolds, whose symplectic foliation is given by singular fibrations over surfaces. In this work, we describe the effect of the monodromy of the fibration on the Poisson cohomology groups.
We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in each of the distinguished holomorphic coordinate charts determined by the open cones of the associated simplicial fan. As an approximation to the smooth cohomology problem in each ${\mathbb C}^n$ chart, we consider the Poisson differential on the complex of polynomial multi-vector fields. For the algebraic problem, we compute $H^0$ and $H^1$ under the assum…
In this note, we compute the Poisson cohomology groups for any Poisson Del Pezzo surface.
We study the variety of Poisson structures and compute Poisson cohomology for two families of Fano threefolds - smooth cubic threefolds and the del Pezzo quintic threefold. Along the way we reobtain by a different method earlier results of Loray, Pereira and Touzet in the special case we are considering.
$θ$-almost twisted Poisson cohomology
本文提出$θ$-几乎扭曲泊松结构，刻画低维情形，构造Lie-Rinehart代数与上同调，并在$\mathbb{R}^5$上给出实例。
We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the $E_2$-term equal to the tensor product of the cohomology of the Lie algebra and the equivariant Poisson cohomology of the manifold. The latter is defined as the equivariant cohomology of the multi-vector fields made into a G-differential complex by means of the momentum mapping.…
Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of Kähler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for an arbitrary Poisson algebra. A geometric version thereof includes the canonical homology and Poisson cohomology of a Poisson manifold introduced by Brylinski, Koszul, and Lichnerowicz, and absorbes the latter in standard homological algebra by expressing them as Tor and Ext group…
It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.
A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bi-complex where one of the two operators is the classical $\overline\partial$-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex m…
A geometric description of the first Poisson cohomology groups is given in the semilocal context, around (possibly singular) symplectic leaves. This result is based on the splitting theorems for infinitesimal automorphisms of coupling Poisson structures which describe the interaction between the tangential and transversal data of the characteristic distributions. As a consequence, we derive some criteria of vanishing of the first Poisson cohomology groups and apply the general splitting formulas to some particular classes of Poisson structures associated with singular symplectic foliations.
We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular points of order $k \geq 1$ for Poisson structures and Lie algebroids. Finally, we apply our results …
We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplectic setting. In particular, we construct scattering-symplectic spheres and scattering symplectic gluings between strong convex symplectic fillings of a contact manifold. By giving an explicit computation of the Poisson cohomology of a scattering symplectic manifold, we also introd…
We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with $D$ independent variables. We find that the second and third cohomology groups are generically non-vanishing in $D>1$. Hence, in contrast with the $D=1$ case, the deformation theory in the multivariable case is non-trivial.
Poisson-like cohomologies associated with some Lie superalgebras
本文将Poisson流形的Poisson上同调群抽象为一般李超代数的类Poisson上同调群，推广了Euler数概念，并发现特殊情况下de Rham上同调群与之匹配。
Given a compact oriented surface, we classify log Poisson bi-vectors whose degeneracy loci are locally modeled by a finite set of lines in the plane intersecting at a point. Further, we compute the Poisson cohomology of such structures and discuss the relationship between our classification and the second Poisson cohomology.
BV Algebras with Applications to Lie Bialgebras and Poisson Cohomology
本文借助微分 Gerstenhaber 代数与 BV 代数重新考察 Poisson 几何的模类和 Lie 双代数的内蕴双导子，并将结果应用于两类上同调。
We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that $BH^p_d(\hat{F}, d_1,d_2)$ is isomorphic to $\mathbb{R}$ for $(p,d)=(0,0)$, to $C^\infty (\mathbb{R})$ for $(p,d)=(1,1)$, $(2,1)$, $(2,3)$, $(3,3)$, and vanishes otherwise.
We try to generalize the Poisson cohomology of a 2-dimensional Poisson manifold to the n-vectors on a n-dimensional manifold. We define several cohomologies and we compute locally some of them, in the case of germs at 0 of n-vectors on a real or complex vector field of dimension n.
In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we present an extension of the van Est isomorphism to groupoids. This immediately implies a version of Haefliger's conjecture for differentiable cohomology. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu). As a second application we extend van Est's argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately gives (a slight improvement …
We characterize the harmonic forms on a flag manifold $K/T$ defined by Kostant in 1963 in terms of a Poisson structure. Namely, they are ``Poisson harmonic" with respect to the so-called Bruhat Poisson structure on $K/T$. This enables us to give Poisson geometrical proofs of many of the special properties of these harmonic forms. In particular, we construct explicit representatives for the Schubert basis of the $S^1$-equivariant cohomology of $K/T$, where the $S^1$-action is defined by $ρ$. Using a simple argument in equivariant cohomology, we recover the connection between the Kostant harmoni…
On every split supermanifold equipped with the Rothstein even super-Poisson bracket we construct a deformation quantization by means of a Fedosov-type procedure. In other words, the supercommutative algebra of all smooth sections of the dual Grassmann algebra bundle of an arbitrarily given vector bundle E (equipped with a fibre metric) over a symplectic manifold M will be deformed by a series of bidifferential operators having first order supercommutator proportional to the Rothstein superbracket. Moreover, we discuss two constructions related to the above result, namely the quantized BRST-coh…
Operations on the de Rham cohomology of Poisson and Jacobi manifolds
本研究证明泊松与雅可比流形德拉姆上同调的（同伦）超交换代数结构同伦平凡，仅保留通用的（同伦）交换代数结构，并将类似结论推广至广义泊松超流形的交换Batalin-Vilkovisky-∞代数结构。
Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.
We study higher-order conservation laws of the non-linearizable elliptic Poisson equation $ \frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u) $ as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. The S^1--symmetry of the PDE leads to a normal form for the undifferentiated conservation law as well. We show that for higher-order conservation laws to exist, it is necess…
Let X(Σ) be a smooth projective toric variety for a complex torus T_\C. In this paper, a real T_\C-invariant Poisson structure Π_Σis constructed on the complex manifold X(Σ), the symplectic leaves of which are the T_\C-orbits in X(Σ). It is shown that each leaf admits a Hamiltonian action by a sub-torus of the compact torus T\subset T_\C. However, the global action of T_\C on (X(Σ),Π_Σ) is Poisson but not Hamiltonian. The main result of the paper is a lower bound for the first Poisson cohomology of these structures. For the simplest case, X(Σ)=\CP^1, the Poisson cohomology is computed using a …
Log Floer cohomology for oriented log symplectic surfaces
本文针对在低维子流形上退化的对数辛 Poisson 结构，在可定向曲面上定义对数 Floer 上同调，证明其同痕不变性及单拉格朗日情形下与对数德拉姆上同调的同构。
We endow the de Rham cohomology of any Poisson or Jacobi manifold with a natural homotopy Frobenius manifold structure. This result relies on a minimal model theorem for multicomplexes and a new kind of a Hodge degeneration condition.
The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the cohomology of the PVAs associated with two-dimensional, two-components Poisson brackets of hydrodynamic type at the third differential degree. This allows us to obtain their corresponding Poisson-Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to …
We exhibit a Poisson module restoring a twisted Poincare duality between Poisson homology and cohomology for the polynomial algebra R=C[X_1,...,X_n] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier.
Let $M^{2n}$ be a Poisson manifold with Poisson bivector field $Π$. We say that $M$ is b-Poisson if the map $Π^n:M\toΛ^{2n}(TM)$ intersects the zero section transversally on a codimension one submanifold $Z\subset M$. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of $(M,Π)$ in the neighbourhood of $Z$ and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomo…
We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex which computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle $A\to X$ is shown to be equivalent to a matched pair of complex Lie algebroids $(T^{0,1}X,A^{1,0})$, in the sense of Lu. The holomorphic Lie algebroid cohomology of $A$ is isomorphic to the cohomology of the elliptic Lie alge…
Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Mackenzie; in particular, we show that Q-groupoids are intermediary objects between Mackenzie's LA-groupoids and double complexes, which include as a special case the simplicial model of equivariant cohomology. There is also a double complex associated to a Q-algebroid, which in the above special case is the BRST model of equivariant cohomology. Other special cases include models for the…
We define Lie and Courant algebroids on Fréchet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fréchet manifold and show that it inherits a Fréchet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$-cotangent bundle Lie algebroid of a weakly symplectic Fréchet manifold $M$ is the Lichnerowicz-Poisson cohomology of $M$.
We give numerous examples of almost Lie algebroids arising as Dirac structures in pre-Courant algebroids, e.g. from twisted Poisson structures, as well as from twisted actions of a Lie algebra. We moreover define a cohomology for them, motivated by a Q-structure, that is trivial for (bundles of) Lie algebras but characterizes the underlying integrable distribution the image of the anchor map.
The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory. In particular, we extend Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology to the realm of extended Poisson manifolds; we establish a sufficient criterion for these to be finite…
Poisson Structures on Trivial Extension Algebras
本文研究平凡扩张代数上的泊松结构，推广了泊松模诱导的已知结构，建立了其与反变导数的一一对应，并以李代数胚表述，同时探讨了第一泊松上同调性质及实例。
We show how to carry out the gauging of the Poisson sigma model in an AKSZ inspired formulation by coupling it to the a generalization of the Weil model worked out in ref. arXiv:0706.1289 [hep-th]. We call the resulting gauged field theory, Poisson--Weil sigma model. We study the BV cohomology of the model and show its relation to Hamiltonian basic and equivariant Poisson cohomology. As an application, we carry out the gauge fixing of the pure Weil model and of the Poisson--Weil model. In the first case, we obtain the 2--dimensional version of Donaldson--Witten topological gauge theory, descri…
On an orientable manifold M, we consider a regular even dimensional foliation F which is globally defined by a set of k-independent 1-forms. We give necessary and sufficient conditions for the existence of a regular Poisson structure on M whose Characteristic foliation is precisely F. Moreover, introducing a special class of the foliated 1-cohomology we describe obstructions for the existence of unimodular Poisson structures with a given characteristic foliation. In the same lines, we also give conditions for the existence of transversally constant Poisson structures.
In this paper, we compute the Gerstenhaber bracket on the Hoch-schild cohomology of $C^\infty(M)\rtimes G$ for a finite group $G$ acting on a compact manifold $M$. Using this computation, we obtain geometric descriptions for all noncommutative Poisson structures on $C^\infty(M)\rtimes G$ when $M$ is a symplectic manifold. We also discuss examples of deformation quantizations of these noncommutative Poisson structures.
In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold $M$ to its Weil bundle $T^A M$ defined by a Frobenius Weil algebra $A$. For a Poisson manifold $(M,w)$, we show that the complete lift $w^C$ and the vertical lift $w^V$ of the Poisson tensor $w$ are Poisson tensors on $T^A M$ and establish their properties. We prove that the complete and the vertical lifts induce homomorphisms of the Poisson cohomology spaces. We compute the modular classes of the lifts of Poisson structures.
We relate Kostant's theorem on the cohomology of a flag manifold $G/B$ with the geometry of the Bruhat-Poisson structure. We express Kostant's harmonic forms in terms of the moment maps (for the torus action) and the Liouville volume forms for the symplectic structures on the Schubert cells induced by the Bruhat-Poisson structure. We do this by writing everything down explicitly in some coordinates on each cell.
We shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a $C^\infty$ logarithmic symplectic structure has unobstructed deformations which are parametrized by an open set of the second de Rham cohomology group of the complement of type changing loci if the type changing loci are smooth. Complex surfaces with smooth effective anti-canonical divisors admit unobstructed deformations of generalize…
The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we found an explicit connection between the Koszul-Brylinski operator of a Poisson manifold and its modular class. As a consequence, we prove that Poisson homology is isomorphic to Poisson cohomology for unimodular Poisson structures.
The characterization of the Nambu-Poisson n-tensors as a subfamily of the Generalized-Poisson ones recently introduced (and here extended to the odd order case) is discussed. The homology and cohomology complexes of both structures are compared, and some physical considerations are made.
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