The paper explores deformations of compact holomorphic Poisson submanifolds.
problem Deformations of compact holomorphic Poisson submanifolds.
method Study of deformations through Kodaira's series and algebraic methods.
result Identification of first-order deformations and obstructions.
Study blow-ups in generalized complex geometry using holomorphic ideals.
problem Blow-ups in generalized complex geometry.
method Introduce holomorphic ideal to define blow-ups in smooth manifolds. Identify suitable submanifolds and provide conditions for blow-ups.
result Necessary and sufficient conditions for generalized Poisson submanifolds to carry a canonical holomorphic ideal and for blow-ups to be generalized complex.
Study neighbourhoods of submanifolds in generalized complex geometry.
problem Understanding the structure and deformations of submanifolds in generalized complex geometry.
method Analytical tools including Hodge decompositions and Nash-Moser algorithm.
result Explicit conditions for B-field equivalence of holomorphic Poisson structures.
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…
The paper solves a long-standing problem by providing a symplectic realization for holomorphic Poisson manifolds.
problem Symplectic realization of holomorphic Poisson manifolds.
method Explicit construction of a holomorphic symplectic structure in a neighborhood of the zero section of T∗X. result There exists a holomorphic symplectic structure in a neighborhood of the zero section of T∗X such that the projection map is a symplectic realization of the given Poisson manifold. The paper generalizes hyperkahler metrics near Lagrangian submanifolds.
problem Constructing hyperkahler structures near complex Lagrangian submanifolds.
method Generalization of Feix-Kaledin theorem and deformations of holomorphic symplectic structures.
result Hyperkahler structures can be constructed on symplectic realizations of holomorphic Poisson manifolds.
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…
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…
The paper normalizes Poisson saturation of coregular submanifolds.
problem Normalizing the Poisson saturation of coregular submanifolds.
method Normal form construction and Poisson geometry analysis.
result Local Poisson saturation of coregular submanifolds is an embedded Poisson submanifold with a normal form.
Study deformations of holomorphic Poisson maps, extending Horikawa's work.
problem Deforming holomorphic Poisson maps.
method Algebraic approach using functors of Artin rings.
result Identification of first-order deformations and obstructions.
Formula derived for holomorphic Poisson blow-ups.
problem Invariance of Koszul-Brylinski homology under Poisson blow-ups.
method Blow-up formula derivation for holomorphic Koszul-Brylinski homologies.
result Invariance of E1-degeneracy of Dolbeault-Koszul-Brylinski spectral sequence. Holomorphic Jacobi structures enrich the theory of Poisson manifolds.
problem Holomorphic Poisson structures are limited; holomorphic Jacobi structures offer more.
method Developed holomorphic Jacobi structures and their relationship with other structures.
result Holomorphic Jacobi structures provide a broader framework than holomorphic Poisson structures.
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 …
Solves the generalized Kähler structure problem for symplectic type.
problem Determine the fundamental degrees of freedom in generalized Kähler structures.
method Uses symplectic Morita equivalence and holomorphic Poisson manifolds.
result Generalized Kähler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence, and a generalized Kähler potential.
Holomorphic Poisson structures on nilmanifolds have degenerate spectral sequences.
problem Holomorphic Poisson structures on nilmanifolds with abelian complex structures.
method Established isomorphism between Dolbeault cohomology and invariant polyvector fields cohomology, identified invariant structures, and analyzed spectral sequences.
result Spectral sequence of the Poisson bi-complex degenerates at E2.
Study of Poisson homeomorphisms and rigidity of coisotropic submanifolds.
problem Rigidity and non-rigidity phenomena in Poisson geometry.
method Study of Poisson homeomorphisms, use of clean intersection points, and analysis of characteristic partitions.
result Poisson homeomorphisms preserve symplectic foliations and coisotropic submanifolds are flexible.
The first cohomology of Poisson algebras is described and conditions for its vanishing are established.
problem Understanding the first cohomology of Poisson algebras.
method Description and mapping of first cohomology to intrinsic cohomologies of Poisson submanifolds, formulation of vanishing conditions.
result Necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras are derived.
Holomorphic Poisson cohomology on nilmanifolds identified and characterized.
problem Characterizing the cohomology of holomorphic Poisson structures on nilmanifolds.
method Construction of non-trivial holomorphic Poisson structures and identification of conditions for cohomology isomorphisms.
result Conditions for the cohomology of non-trivial holomorphic Poisson structures to be isomorphic to trivial ones.
Local model for Poisson manifolds around submanifolds.
problem Linearization of Poisson manifolds around submanifolds.
method Constructing a first order local model and giving conditions for it to be a normal form.
result Includes known linearization theorems for fixed points and symplectic leaves.
Global Poisson structures on S4 studied using twistor methods.
problem Global Poisson structures on S4. method Holomorphic Poisson structures on CP3, twistor method. result Examples of Poisson structures on S4 associated with codimension one holomorphic foliations of degree 2 on CP3. Study Poisson algebras for Hamiltonian systems linearization.
problem Linearize dynamics along Poisson submanifolds.
method Use contravariant derivative to characterize Poisson algebras.
result Infinitesimal Poisson algebras provide a framework for Hamiltonization.
Authors find a Hodge-type decomposition for holomorphic Poisson cohomology on nilmanifolds.
problem Investigating conditions for spectral sequence degeneration in holomorphic Poisson cohomology.
method Analyzing spectral sequences associated with bi-complexes on nilmanifolds.
result A Hodge-type decomposition of holomorphic Poisson cohomology is established for a specific class of structures.
The paper examines Poisson deformations with examples.
problem Understanding Poisson deformations with classical examples.
method Study of obstructed and unobstructed Poisson deformations.
result Examples in deformation theory.
The study introduces a new equivalence for Poisson modules on complex projective varieties.
problem Understanding the structure of Poisson modules on complex projective varieties with mild singularities.
method Introducing a weak concept of Morita equivalence in the birational context for Poisson modules.
result Poisson modules are classified into three types based on their properties.
Study of quasi-classical generalized F and CRF structures.
problem Integrability conditions and properties of tensor fields.
method Analysis of tensor fields (A, π) and their relations.
result Established integrability conditions and properties of CRF structures.
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.
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.
problem Understanding submanifolds in Koszul-Vinberg geometry.
method Analyzing submanifolds within the framework of Koszul-Vinberg manifolds, considering developments in Poisson submanifolds.
result Developed methods to analyze submanifolds in this geometric setting.
Study of holomorphic submanifolds in hypercomplex manifolds with specific metrics.
problem Characterizing holomorphic submanifolds in hypercomplex manifolds with Hermitian and Norden metrics.
method Investigation of necessary and sufficient conditions for total umbilicity and geodesicity.
result Conditions for holomorphic submanifolds to be totally umbilical or totally geodesic are derived.
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 …
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…
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…
Minimal real Kähler submanifolds in codimension 6 are holomorphic.
problem Characterizing real Kähler submanifolds in high codimension.
method Using isometric rigidity and properties of the second fundamental form.
result Minimal real Kähler submanifolds in codimension 6 are holomorphic.
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…
Study optimal holomorphic extensions for jets along submanifolds as tensor powers increase.
problem Optimal holomorphic extensions of jets along submanifolds for high tensor powers.
method Careful study of Schwartz kernels and Bergman projectors for asymptotic analysis.
result Explicit asymptotic formula for the extension operator as tensor power tends to infinity.
The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.
problem Understanding deformations of holomorphic Poisson manifolds under certain cohomological assumptions.
method Investigates properties of Koszul-Brylinski homology and Dolbeault cohomology, proving formality of a DGLA.
result The DGLA is shown to be formal, and Maurer-Cartan elements induce complex structure deformations.
Study CR-statistical submanifolds in holomorphic statistical spaces.
problem Characterize CR-statistical submanifolds and their properties.
method Optimization technique to relate Ricci curvature and mean curvature.
result Established relationship between Ricci curvature and mean curvature.
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 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 …
The paper shows conditions under which a Kaehler submanifold is minimal or holomorphic.
problem Characterizing conditions for Kaehler submanifolds to be minimal or holomorphic.
method Analyzing the second fundamental form of the submanifold and proving conditions on its rank.
result For codimension p≤11, a Kaehler submanifold is holomorphic with respect to some complex structure in the ambient space. The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.
problem Analyzing Poisson-flat connections with logarithmic poles.
method Defining an Euler-Poisson principal part and residue theory, establishing a Poisson Poincaré-Dulac theorem.
result Any logarithmic Poisson-flat connection is holomorphically gauge equivalent to a pure Euler-Poisson normal form.
Investigates conditions for spectral sequence degeneracy in holomorphic Poisson structures.
problem Conditions for spectral sequence degeneracy in holomorphic Poisson structures.
method Uses Lie bi-algebroids, generalized complex structures, and hypercohomology of bi-complexes.
result Investigates conditions for spectral sequence degeneracy on the first page.
Study holomorphic last multipliers on complex manifolds.
problem Equivalence between holomorphic and real ODE systems.
method Analyzing last multipliers in complex manifold context.
result Relate holomorphic last multipliers to real last multipliers.
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 prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. As a consequence we give another proof of the following theorem: a holomorphic…
Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.
problem Investigating holomorphic Koszul-Brylinski homology on Poisson manifolds.
method Using Dolbeault cohomology to derive properties of holomorphic Koszul-Brylinski homology.
result Obtained the Leray-Hirsch theorem, Mayer-Vietoris sequence, and Künneth theorem for holomorphic Koszul-Brylinski homology.
An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deform…
Constructs a weak Poisson bracket and lifts it to differential forms.
problem Creating a weak Poisson bracket over submanifolds and foliations.
method Encoding weak Poisson structure into homotopy Poisson structure and lifting to differential forms.
result Lifts a weak Poisson bracket to the algebra of forms with a direct physical interpretation.