arXiv research

The paper normalizes Poisson saturation of coregular submanifolds.

In this paper, we study deformations of compact holomorphic Poisson submanifolds which extend Kodaira's series of papers on semi-regularity (deformations of compact complex submanifolds of codimension 1), deformations of compact complex submanifolds of arbitrary codimensions, and stability of compact complex submanifol…

Study of Poisson homeomorphisms and rigidity of coisotropic submanifolds.

The first cohomology of Poisson algebras is described and conditions for its vanishing are established.

Local model for Poisson manifolds around submanifolds.

Study Poisson algebras for Hamiltonian systems linearization.

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler stru…

We consider Lagrangian-like submanifolds in certain even-dimensional 'symplectic-like' Poisson manifolds. We show, under suitable transversality hypotheses, that the pair consisting of the ambient Poisson manifold and the submanifold has unobstructed deformations and that the deformations automatically preserve the Lag…

Study submanifolds in Koszul-Vinberg geometry, a blend of Poisson and pseudo-Riemannian structures.

We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisso…

We consider existence and uniqueness of two kinds of coisotropic embeddings and deduce the existence of deformation quantizations of certain Poisson algebras of basic functions. First we show that any submanifold of a Poisson manifold satisfying a certain constant rank condition sits coisotropically inside some larger …

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manife…

We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B-field equivalent to a holomorphic Poisson structure. This is intimately tied with our second main re…

We introduce the notion of twisted generalized complex submanifolds and describe an equivalent characterization in terms of Poisson-Dirac submanifolds. Our characterization recovers a result of Vaisman. An equivalent characterization is also given in terms of spinors. As a consequence, we show that the fixed locus of a…

We detail the construction of a weak Poisson bracket over a submanifold of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson …

Let (M, π ) be a Poisson manifold. A Poisson submanifold $P \in M$ gives rise to an algebroid $AP \rightarrow P$, to which we associate certain chomology groups which control formal deformations of π around P . Assuming that these groups vanish, we prove that π is formally rigid around P , i.e. any other Poisson struct…

The Poisson sigma model is a widely studied two-dimensional topological field theory. This note shows that boundary conditions for the Poisson sigma model are related to coisotropic submanifolds (a result announced in [math.QA/0309180]) and that the corresponding reduced phase space is a (possibly singular) dual pair b…

The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.

This is an expository and introductory note on some results obtained in "Coisotropic embeddings in Poisson manifolds" (ArXiv math/0611480). Some original material is contained in the last two sections, where we consider linear Poisson structures.

We write down the local equations that characterize the submanifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of TM endowed with the tangent Dirac structure. In the Poisson case, these formulas prove again a result of Xu: the submanifold N has a normal …

The paper generalizes hyperkahler metrics near Lagrangian submanifolds.

We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the $L_\infty$-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo-Felder. In the symplectic case, we recover results previously obtained by Oh-Park. Moreover we consider the extended de…

New Poisson structures on algebras linked to derivatives.

Characterizes blowups of Dirac structures on manifolds.

We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a v…

We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express th…

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a…

The notion of a Dirac submanifold of a Poisson manifold was studied by Xu (arXiv:math.SG/0110326). We give an interpretation of Xu's definition in terms of a general notion of tensor fields soldered to a normalized submanifold. Then, this interpretation is used to define Dirac submanifolds of a Jacobi manifold. Several…

Study star products on Poisson manifolds compatible with reduction.

Study of symplectic and Poisson reduction, proposing Poisson implosion.

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 this paper, we attach an $L_\infty$-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), Lê-Oh (locally conformal symplectic case). As a new special case, we attach an $L_\infty$-alg…

This thesis studies normal forms for Poisson structures around symplectic leaves using several techniques: geometric, formal and analytic ones. One of the main results (Theorem 2) is a normal form theorem in Poisson geometry, which is the Poisson-geometric version of the Local Reeb Stability (from foliation theory) and…

Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distributio…

We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrang…

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold $(M,ω)$, we construct a weak symplectic structure on each leaf ${\textbf I}_{w}$ of a foli…

The paper studies deformations of Lagrangian submanifolds using algebraic tools.

A description of time-dependent Mechanics in terms of Lagrangian submanifolds of Dirac manifolds (in particular, presymplectic and Poisson manifolds) is presented. Two new Tulczyjew triples are discussed. The first one is adapted to the restricted Hamiltonian formalism and the second one is adapted to the extended Hami…

We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold $S$ controls the coisotropic deformation problem of $S$ under both Hamiltoni…

Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, π)$, we prove that there exists a holomorphic symplectic struct…

Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define a…

Taking configuration space as a Lie group, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew's symplecti…

Quantization of (-1)-shifted derived Poisson manifolds via BV-infinity operators.

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure $Π$. We determine the irreducible components of $\Lagr$, and we show that each irreducible component is a smo…

Novel coarse extrinsic curvature for Riemannian submanifolds.

New Poisson structures on hypersurface algebroids discovered.

We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)Kähler geometry.