In this paper, we study deformations of holomorphic Poisson maps which extend Horikawa's series of papers on deformations of holomorphic maps in the context of holomorphic Poisson deformations. In appendices, we present deformations of Poisson morphisms in the language of functors of Artin rings which is the algebraic …
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48 results for “holomorphic deformations”
Study on deformations of holomorphic Cartan geometries, focusing on flat cases.
problem Deformation of holomorphic Cartan geometries on complex manifolds.
method Computing infinitesimal deformations and analyzing the forgetful map.
result The forgetful map from infinitesimal deformations of a flat holomorphic Cartan geometry to the underlying flat principal bundle is an isomorphism.
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…
Introducing the deformation theory of holomorphic Cartan geometries, we compute infinitesimal automorphisms and infinitesimal deformations. We also prove the existence of a semi-universal deformation of a holomorphic Cartan geometry.
Study on deforming complex manifolds and Higgs bundles.
problem Deforming holomorphic-Higgs pairs on complex manifolds.
method Introduced a DGLA and derived the Maurer-Cartan equation to govern the deformation.
result Proved the local completeness of the Kuranishi family of the deformed holomorphic-Higgs pair.
Study deformations of Calabi-Yau foliations using Kuranishi spaces.
problem Deforming Calabi-Yau foliations and understanding their properties.
method Analysis of three types of deformations (unfoldings, holomorphic, transversally holomorphic) using Kuranishi spaces.
result Smoothness of and product structure of .
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.
New symplectic forms derived from Lagrangian fibrations on symplectic manifolds.
problem Deriving new symplectic forms from existing ones.
method Proving existence of degenerate twistorial deformations.
result Existence of degenerate twistorial deformations preserving complex structures.
Degenerate twistor deformations of Kähler manifolds are also Kähler.
problem Understanding the Kähler structure of degenerate twistor deformations.
method Using positive currents, Hahn–Banach theorem, and Huybrechts's theorem.
result Degenerate twistor deformations of compact holomorphically symplectic Kähler manifolds are Kähler.
We show how wall-crossing formulas in coupled 2d-4d systems, introduced by Gaiotto, Moore and Neitzke, can be interpreted geometrically in terms of the deformation theory of holomorphic pairs, given by a complex manifold together with a holomorphic vector bundle. The main part of the paper studies the relation between …
Introduces new holomorphic contact structures and proves unobstructedness theorems.
problem Generalizing classical holomorphic contact and symplectic structures.
method Introducing new classes of holomorphic -contact and -symplectic manifolds, observing their properties, and proving structure and unobstructedness theorems.
result Generalizes classical results on small deformations of complex structures.
Study on special metrics and deformations of solvmanifolds.
problem Existence and properties of Kähler metrics on solvmanifolds.
method Investigation of strong Kähler with torsion metrics and balanced metrics on deformations of specific solvmanifolds.
result Non-existence of certain metrics on specific solvmanifolds.
On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Mapsmath.DS
This paper shows similarities in deformation spaces of Kleinian groups and anti-holomorphic maps.
problem Comparing deformation spaces of Kleinian reflection groups and anti-holomorphic rational maps.
method Established an analogue of Thurston's compactness theorem for critically fixed anti-rational maps and characterized deformation space interactions.
result Deformation spaces of Kleinian reflection groups and anti-holomorphic rational maps share striking similarities.
A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact manifold with admits only finitely…
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 properties of holomorphic -contact manifolds, including non-Kähler hyperbolicity and deformations.
problem Characterize the geometric and algebraic properties of holomorphic -contact manifolds.
method Explores non-Kähler hyperbolicity, differential calculus, and -contact deformations, proving unobstructedness theorems.
result Proves a Bogomolov-Tian-Todorov-type unobstructedness theorem for -contact deformations up to order two.
On a noncommutative deformation of holomorphic line bundles on complex tori and the SYZ transformmath.DG
Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.
problem Noncommutative deformations of holomorphic line bundles on complex tori.
method Real nonformal deformation quantization and SYZ construction.
result Extended construction of noncommutative deformations of holomorphic line bundles.
The paper studies deformations of Lagrangian fibrations on symplectic manifolds.
problem Understanding deformations of Lagrangian fibrations on holomorphic symplectic manifolds.
method Analyzes degenerate twistor deformations and meromorphic sections.
result Compact hyperkahler manifolds with primitive fibers admit meromorphic sections.
In this paper, we study obstructed and unobstructed (holomorphic) Poisson deformations with classical examples in deformation theory.
The paper proves conditions for solutions of -equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.
problem Conditions for existence of solutions of -equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.
method Adiabatic limit technique
result Conditions for existence of solutions of -equation and deformed Hermitian-Yang-Mills equation on holomorphic submersions.
The article studies critical points of a new energy functional in higher dimensions.
problem Investigating critical points of a new energy functional in higher dimensions.
method Holomorphic deformations, closed and open properties, differential of the functional.
result Properties of critical points under holomorphic deformations are closed and open.
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.
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…
Deformed holomorphic Chern-Simons theory yields new instantons.
problem Deforming classical holomorphic Chern-Simons theory on Calabi-Yau manifolds.
method Deformation of complex structure by a parameter \( h \) leading to new instanton solutions.
result Existence of instanton solutions invariant under re-scalings of \( h \) and their connection to \( G_2 \)-instantons.
New findings on complex manifold properties under deformations.
problem Properties of Dolbeault and Bott-Chern formalities are not preserved under holomorphic deformations.
method Construction of a complex manifold to demonstrate non-preservation of properties.
result Existence of a manifold satisfying -lemma but with non-vanishing Aeppli-Bott-Chern-Massey product.
Paper discusses conditions for deforming coupled Kähler-Einstein metrics.
problem Conditions for deforming coupled Kähler-Einstein metrics.
method Analyzes deformation of coupled Kähler-Einstein metrics on Fano manifolds.
result Necessary and sufficient condition for deformation of coupled Kähler-Einstein metrics.
New deformations of lattice cohomology help calculate knot invariants.
problem Calculating knot invariants using lattice cohomology.
method Using holomorphic triangles counting and lattice cohomology.
result Combinatorial formulae for the upsilon invariant are derived.
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…
The paper explores Kodaira dimension on almost complex manifolds.
problem Understanding Kodaira dimension on non-integrable almost complex manifolds.
method Generalization of Kodaira dimension to almost complex manifolds and study of its behavior under deformations.
result Kodaira dimension is invariant under holomorphic deformations for smooth projective manifolds but not for non-projective manifolds.
We give a new characterization of generalized Kähler structures in terms of their corresponding complex Dirac structures. We then give an alternative proof of Hitchin's partial unobstructedness for holomorphic Poisson structures. Our main application is to show that there is a corresponding unobstructedness result for …
This research classifies singular foliations and finds a universal deformation.
problem Classifying singular foliations on .
method Topological universal deformation through fixed invariants.
result Every equisingular deformation uniquely factors through the topological universal deformation.
We introduce K-deformations of generalized complex structures on a compact Kahler manifold with an effective anti-canonical divisor and show that obstructions to K-deformations of generalized complex structures on always vanish. Applying the stability theorem of generalized Kahler structures, together wi…
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 determine obstructedness or unobstructedness of (holomorphic) Poisson deformations of ruled surfaces over an elliptic curve.
We show that the property of existence of solution to the Strominger system in dimension six is neither open nor closed under holomorphic deformations of the complex structure. These results are obtained both in the case of positive slope parameter as well as in the case of negative slope parameter in the anomaly cance…
We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with transversely Riemannian foliations. We use thi…
Investigates -equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of -equation on holomorphic vector bundles.
method Introduces and studies -equation, provides algebraic and numerical criteria.
result Provides an algebraic condition (asymptotic -stability) and a numerical criterion for vortex bundles.
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…
Recently, Cappell and Miller extended the classical construction of the analytic torsion for de Rham complexes to coupling with an arbitrary flat bundle and the holomorphic torsion for -complexes to coupling with an arbitrary holomorphic bundle with compatible connection of type . Cappell and Mil…
We present a geometrical framework which incorporates higher derivative corrections to the action of N = 2 vector multiplets in terms of an enlarged scalar manifold which includes a complex deformation parameter. This enlarged space carries a deformed version of special Kahler geometry which we characterise. The holomo…
Study deformations of compact Kähler hyperbolic manifolds.
problem Investigate the deformation openness of Kähler hyperbolicity.
method Propose modified versions of Kähler hyperbolicity as a tool.
result Provide a first step towards understanding deformations.
The paper proposes a noncommutative deformation of toric varieties.
problem Deforming complex structures into generalized complex structures.
method Using holomorphic Poisson tensors and generalized complex branes.
result Noncommutative deformation of homogeneous coordinate rings.
We prove unobstructed deformations for compact Kaehlerian even-dimensional Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities. Examples include Hilbert schemes of del Pezzo surfaces.
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and -stability.
result Equivalence between dHYM solutions and -stability for vortex type bundles.
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 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is simila…
Let X be a Stein manifold, A a closed complex subvariety of X, and f a continuous map from X to a complex manifold Y whose restriction to A is holomorphic. After a homotopic deformation of the Stein structure outside a neighborhood of A in X (and of its smooth structure when X is a Stein surface)we find a holomorphic m…
The paper extends local h-principles to complex structures on Stein manifolds.
problem Existence of local h-principles for complex structures on Stein manifolds.
method Introducing realifications of partial holomorphic relations and proving h-principles for them.
result Local h-principles can be extended to complex structures on Stein manifolds.