Study complex deformations of compact complex surfaces in Calabi-Yau four-folds.
problem Explaining why complex and Cayley deformations of a compact complex surface are the same.
method Study complex deformations of compact complex submanifolds of Calabi-Yau manifolds.
result Prove that the moduli space of complex deformations of any compact complex embedded submanifold of a Calabi-Yau manifold is a smooth manifold.
The paper studies deformations of cohesive modules on complex manifolds.
problem Deformation theory of cohesive modules on compact complex manifolds.
method Development of Kuranishi maps and obstructions for deformations of cohesive modules.
result Generalization of deformation theory for holomorphic vector bundles and coherent sheaves.
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 on symplectic structures and their deformations.
problem Preservation of complex symplectic structures under deformations.
method Analyzes various cohomologies and conditions for deformations.
result Obtains topological obstructions for compact complex symplectic manifolds.
The paper discusses q-deformations of the Aomoto complex.
problem Deformation of cochain complexes associated with hyperplane arrangements.
method Replaces entries of coboundary maps with q-analogues and analyzes the resulting structures. result The q-deformation can be a cochain complex under certain conditions and yields local system cohomology groups. Study complex structure deformations on Lie algebras and Dolbeault cohomology.
problem Deformations of complex structures on Lie algebras and their associated Dolbeault cohomology.
method Construct a complete deformation of complex structures similar to the Kuranishi family, showing extension isomorphism validity.
result Analytic open subset of deformations where Dolbeault cohomology can be computed by left invariant tensor fields.
Kuranishi's proof of complex deformation theory revisited
problem Existence of complex deformations on compact complex manifolds
method Hamilton-Nash-Moser implicit function theorem
result Revisits classical proof with modern tools
In this paper, we use Pacard-Xu's methods to discuss the complex deformation of constant scalar curvature metrics in the case of fixed and varying complex structures. Moreover, we also discuss the complex deformation of Kähler Ricci solitons.
Study canonical deformations of complex forms and their cohomology properties.
problem Understanding canonical deformations and cohomology of complex manifolds.
method Analyzes canonical Aeppli deformations and their relations to deformed cohomology.
result Proves the jumping formula for deformed Aeppli cohomology and constant dimension conditions.
Deform quantization recovers scalar curvature in complex structures.
problem Recovering scalar curvature in complex structures.
method Formal moment map construction on almost complex structures.
result Formal moment map deforms scalar curvature moment map in integrable cases.
This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into PU(n,1) and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sulliva…
Study on deformations of singular submanifolds in complex geometry.
problem Deformation theory of conically singular Cayley submanifolds.
method Proved expected dimension of moduli space and compared complex and Cayley deformations.
result Moduli space is smooth for 2D complex submanifolds of Calabi-Yau 4-folds.
The paper studies deformations of astheno-Kähler metrics on complex manifolds.
problem Stability of astheno-Kähler metrics under complex structure deformations.
method Proves necessary cohomological conditions for astheno-Kähler metrics along deformations.
result Provides obstructions to the existence of astheno-Kähler metrics on specific nilmanifolds.
The study counts cusps in deformed complex polynomials near the origin.
problem Understanding singularities in deformed complex polynomials.
method Analyzes deformations of complex polynomials to real maps with fold and cusp singularities.
result Calculates the number of cusps in a small neighborhood of the origin.
Extends extension formulas for Hodge numbers on complex manifolds.
problem Deformation invariance of Hodge numbers on complex manifolds.
method Introduces a canonical isomorphism between complex differential forms on a manifold and its infinitesimal deformations, generalizing an extension formula.
result Proves several deformation invariance theorems for Hodge numbers.
This paper studies deformations of hyperbolic surfaces with special structures.
problem Infinitesimal deformations of hyperbolic surfaces with boundary and ideal vertices.
method Description of the admissible cone of deformations in terms of the arc complex.
result Realization of the admissible cone and its faces as arc complexes for specific surface families.
We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are uno…
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.
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. Analyzes complex structure deformations using cohomology contraction methods.
problem Deforming complex structures and identifying obstructions.
method Refined power series method for (p,q)-forms and complex structures, using Frölicher spectral sequence. result All obstruction classes lie in the kernel of contraction maps under natural vanishing conditions.
Stability of SKT metrics under deformations on complex manifolds.
problem Stability of strong Kähler with torsion metrics under small deformations.
method Finding necessary conditions for stability of SKT metrics along a family of complex manifolds.
result Necessary conditions for the stability of SKT metrics on a smooth curve of Hermitian metrics.
Study Lie algebras with complex structures, focusing on degenerations and deformations.
problem Understanding the space of Lie algebras with complex structures and their transformations.
method Identifying invariants that remain consistent under degenerations and applying to four-dimensional case.
result Found invariants that help in understanding the behavior of Lie algebras under complex structures.
Study of deformed Bott-Chern cohomology on complex manifolds.
problem Deformation theory and cohomology of complex manifolds.
method Introduce a double complex structure and study its Bott-Chern cohomology.
result Established a deformation theory for Bott-Chern cohomology and computed deformed cohomology for specific manifolds.
Study strip deformations of hyperbolic polygons with decorated vertices.
problem Understanding deformations of hyperbolic polygons with decorated vertices.
method Analyzing strip deformations of ideal hyperbolic polygons with horoballs.
result Arc complexes parameterize uniformly lengthening deformations.
Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. Further it is shown that non-split structures appear in the first case as deformations of a split reduction and in the …
Introduces Witten deformation and its applications in topology.
problem Analyzing and applying Witten deformation in topology.
method Deformation of Dirac operators and analytic proofs.
result Analytic proofs of Poincaré-Hopf index theorem, Real Morse inequalities, Thom-Smale complex quasi-isomorphism, and Atiyah vanishing theorem.
Extending the work of G. Székelyhidi and T. Brönnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex …
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.
Smooth deformation of Moishezon manifolds preserves their Moishezon property.
problem Preserving Moishezon property under smooth deformation.
method Smooth deformation over a unit disk in C.
result Deformation limit of Moishezon manifolds is Moishezon.
Curved Rickard complexes extend link homologies to arbitrary representations.
problem Link homologies for arbitrary representations.
method Definition and study of curved Rickard complexes.
result Deformations of link homologies generalize previous work.
This paper studies a deformation retraction of Teichmüller space and its analogy with well-rounded retractions.
problem Understanding the well-rounded deformation retraction of Teichmüller space.
method Examining the mapping class group-equivariant deformation retraction of Teichmüller space onto a CW complex and comparing it to well-rounded retractions of other spaces.
result The well-rounded deformation retraction of Teichmüller space is analogous to well-rounded retractions of other spaces.
Compact complex manifolds with trivial canonical bundle and ∂∂ˉ-Lemma have smooth deformations and a surjective Albanese map.
problem Understanding the structure and deformations of compact complex manifolds with specific properties.
method Analyzing the Kuranishi space, showing smooth deformations, and studying the Albanese map.
result Compact complex manifolds with trivial canonical bundle and ∂∂ˉ-Lemma have smooth deformations and a surjective Albanese map. Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.
problem Deformation theory of Lie groupoids and algebroids.
method Defining a morphism between deformation complexes and Hochschild complexes, applying to adiabatic groupoids.
result Induced van Est map from geometric to algebraic deformation cohomology.
We consider the stability of Sasaki-extremal metrics under deformations of the complex structure on the Reeb foliation. Given such a deformation preserving the action of a compact subgroup of the automorphism group of a Sasaki-extremal structure, a sufficient condition is given involving the nondegeneracy of the relati…
This thesis studies deformations of VB-algebroids and VB-groupoids in Lie algebroid and groupoid categories.
problem Deformations of VB-algebroids and VB-groupoids in Lie algebroid and groupoid categories.
method Attach cochain complexes to VB-algebroids and VB-groupoids, equip them with DGLA structures, discuss their properties and relationships with deformation complexes of total and base spaces.
result Linear van Est theorem and Morita invariance theorem for VB-groupoids.
In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.
The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an L∞ algebra instead. We develop a simplified method for describing this L∞ algebra a…
Study on complex Grassmannians' rigidity using Einstein deformations.
problem Characterizing integrable infinitesimal Einstein deformations of complex Grassmannians.
method Analyzing the integrability to second order of infinitesimal deformations using Koiso's obstruction polynomial.
result Characterized integrable deformations as an explicit variety in su(n), showing g is isolated for odd n. Formula connects complex forms on deforming spaces.
problem Establishing a formula for complex forms on deforming spaces.
method Holomorphic family of pairs and correspondence formula.
result Formula connects Dolbeault complexes of deforming spaces.
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…
Study on Einstein deformations of negative Kähler Einstein metrics.
problem Understanding Einstein deformations of Kähler Einstein metrics.
method Relate second order Einstein deformation theory to complex geometry, gauge normalise, and use Taylor expansion.
result Taylor expansion to order two of an Einstein deformation is determined by h12 and the divergence of the Kodaira-Spencer bracket. 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.
In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which a…
A new Witten deformation modifies Dolbeault complex properties.
problem Modifying Dolbeault complex properties.
method Introducing a Witten-Novikov type perturbation ∂ˉωˉ of the Dolbeault complex. result Heat invariants of lower order are zero.
We investigate the formal deformation theory of (rank 1) branes on generalized complex (GC) manifolds. This generalizes, for example, the deformation theory of a complex submanifold in a fixed complex manifold. For each GC brane B on a GC manifold (X,J), we construct a formal (pointed) groupoid $…
Study on bending deformations in hyperbolic manifolds, generalizing Johnson and Millson's work.
problem Understanding infinitesimal deformations in branched bending complexes.
method Defining branched bending deformations, giving lower bounds, and constructing examples.
result Lower bounds on the dimension of deformation spaces and examples of specific deformations.
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 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.