arXiv research

Holomorphic Jacobi structures enrich the theory of Poisson manifolds.

Holomorphic Poisson structures on nilmanifolds have degenerate spectral sequences.

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 …

Global Poisson structures on $S^4$ studied using twistor methods.

Holomorphic Poisson cohomology on nilmanifolds identified and characterized.

The paper explores deformations of compact holomorphic Poisson submanifolds.

The paper solves a long-standing problem by providing a symplectic realization for holomorphic Poisson manifolds.

Formula derived for holomorphic Poisson blow-ups.

Authors find a Hodge-type decomposition for holomorphic Poisson cohomology on nilmanifolds.

It is known that holomorphic Poisson structures are closely related to theories of generalized Kähler geometry and bi-Hermitian structures. In this article, we introduce quantization of holomorphic Poisson structures which are closely related to generalized Kähler structures /bi-Hermitian structures. By resulting nonco…

The paper generalizes hyperkahler metrics near Lagrangian submanifolds.

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…

Study neighbourhoods of submanifolds in generalized complex geometry.

In this paper, we define a concept of a family of compact holomorphic Poisson manifolds on the basis of Kodaira-Spencer's deformation theory and deduce the integrability condition. We prove an analogue of their `Theorem of existence for complex analytic structures' under some analytic assumption, and establish an analo…

We first extend the notion of connection in the context of Courant algebroids to obtain a new characterization of generalized Kaehler geometry. We then establish a new notion of isomorphism between holomorphic Poisson manifolds, which is non-holomorphic in nature. Finally we show an equivalence between certain configur…

New method to deform generalized Kähler structures.

The study introduces a new equivalence for Poisson modules on complex projective varieties.

The Serre construction of rank two holomorphic bundles with a section is adapted to construct generalized holomorphic bundles on a generalized complex 4-manifold from the data of a set of points on an elliptic curve. The motivation is the special case of rank two Poisson modules on a complex surface with a holomorphic …

Investigates conditions for spectral sequence degeneracy in holomorphic Poisson structures.

Introduces compatibility between Dirac structures and Nijenhuis tensors.

The paper proposes a noncommutative deformation of toric varieties.

Study deformations of holomorphic Poisson maps, extending Horikawa's work.

The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.

Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.

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 explic…

We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors.…

The paper explores complex Poisson structures on smooth functions in complex manifolds.

Study of mixed product Poisson structures on Lie group and bialgebra manifolds.

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 a…

We introduce the notion of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures. It combines symplectic groupoids, holomorphic Lie groupoids and holomorphic Poisson groupoids into a unified framework. Their infinitesimal, Glanon Lie algebroids are studied. We prove that …

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.

We characterize HKT structure in terms of nondegenrate complex Poisson bivector on hypercomplex manifold. We extend the characterization to the twistor space. After considering the flat case in detail, we show that the twistor space of hyperkaehler manifold admits a holomorphic Poisson structure. We briefly mention the…

Playing off against each other the real and complex structures, we elucidate the local structure of certain representation spaces in the world of Poisson geometry. Particular cases of these spaces arise as moduli spaces of semistable holomorphic vector bundles on Riemann surfaces.

We introduce K-deformations of generalized complex structures on a compact Kahler manifold $M=(X, J)$ with an effective anti-canonical divisor and show that obstructions to K-deformations of generalized complex structures on $M$ always vanish. Applying the stability theorem of generalized Kahler structures, together wi…

The paper examines Poisson deformations with examples.

Solves the generalized Kähler structure problem for symplectic type.

In this note, we study the Koszul-Brylinski homology of holomorphic Poisson manifolds. We show that it is isomorphic to the cohomology of a certain smooth complex Lie algebroid with values in the Evens-Lu-Weinstein duality module. As a consequence, we prove that the Evens-Lu-Weinstein pairing on Koszul-Brylinski homolo…

In this thesis, we study deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes in the framework of Kodaira-Spencer's analytic deformation theory and Grothendieck's algebraic deformation theory.

In this article, we introduce symbol calculus on a projective scheme. Using holomorphic Poisson structures, we construct deformations of ring structures for structure sheaves on projective spaces.

Study of quasi-classical generalized F and CRF structures.

In this paper, we construct tools from the holomorphic twistor spaces that we introduced in \cite{Gindi1} to derive results about the complex geometries of their base manifolds. In particular, we develop a new approach to studying generalized Kahler manifolds that leads to insights into their real and holomorphic Poiss…

The paper classifies Poisson structures on toric manifolds and computes their cohomology.

Develops new Poisson structures for moduli spaces.

Local Poisson groupoids over mixed product Poisson structures defined and applied.

We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to obtain $A_{\infty}$ algebra structures and some canonically defined deformations of s…

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 structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual c…

Study blow-ups in generalized complex geometry using holomorphic ideals.

Establishes functoriality of Baum-Bott residues under specific conditions.