Related papers: Tamed Symplectic forms and Generalized Geometry

We prove that all complex analytic subvarieties of a generic compact hyperkaehler manifold are even-dimensional. Moreover, these subvarieties are holomorphically symplectic.

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser's theorem for simultaneous isotopies of two families of symplectic forms. We also consider the…

We prove that an integrable system over a symplectic manifold, whose symplectic form is covariantly constant w.r.t. the Gauss-Manin connection, carries a natural hyper-symplectic structure. Moreover, a special Kaehler structure is induced…

We consider the symplectic groupoid of pairs $(B, A)$ with $A$ real unipotent upper-triangular matrix and $B\in GL_n$ being such that $\tilde A=BAB^T$ is also a unipotent upper-triangular matrix. Fock and Chekhov defined a Poisson map of…

Geometric quantization of a Poisson manifold need not imply quantization of its symplectic leaves. We provide the leafwise geometric quantization of a Poisson manifold, seen as a foliated one, whose quantum algebra restricted to each leaf…

We introduce geometric quantization for constant rank presymplectic structures with Riemannian null foliation and compact leaf closure space. We prove a quantization-commutes-with-reduction theorem in this context. Examples related to…

We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a…

We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is…

Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is…

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…

We present a method to construct irreducible symplectic varieties by studying terminalisations of quotient of hyper-K\"ahler manifolds by non-natural group actions. In particular, we construct irreducible symplectic varieties of dimension…

A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper we show that the isomorphism class of the…

Using the idea of a generalized Kaehler structure, which is a pair of commuting generalized complex structures, we construct bihermitian metrics on the projective plane and the product of two projective lines, and show that any such…

In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents.…

We solve the problem of determining the fundamental degrees of freedom underlying a generalized K\"ahler structure of symplectic type. For a usual K\"ahler structure, it is well-known that the geometry is determined by a complex structure,…

We analyze the symplectic and complex structures on the panelled web 4-manifolds. In particular, we give infinite family of examples of almost complex but not symplectic and not complex 4-manifolds in the non-simply connected case.

Previously work of the author with Meier and Starkston showed that every closed symplectic manifold $(X,\omega)$ with a rational symplectic form admits a trisection compatible with the symplectic topology. In this paper, we describe the…

In this paper we introduce the notion of a smooth structure on a stratified space, the notion of a Poisson smooth structure and the notion of a weakly symplectic smooth structure on a stratified symplectic space, refining the concept of a…

In this note we study the behavior of symplectic cohomology groups under symplectic deformations. Moreover, we show that for compact almost-K\"ahler manifolds $(X,J,g,\omega)$ with $J$ $\mathcal{C}^\infty$-pure and full the space of de Rham…

It is a well known fact that every embedded symplectic surface $\Sigma$ in a symplectic 4-manifold $(X^4,\omega)$ can be made $J$-holomorphic for some almost-complex structure $J$ compatible with $\omega$. In this paper we investigate when…