Study finite deformations from heterotic superpotential, leading to new complex effective action.
problem Finite deformations of the Hull--Strominger system.
method Expanding the heterotic superpotential around a supersymmetric vacuum, identifying complex coordinates, and using Maurer--Cartan equation.
result Generalizes complex effective action of Kodaira--Spencer and holomorphic Chern--Simons theory, with a supersymmetric locus described by an L3 algebra. 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 degeneration of spectral sequence in complex manifolds under deformations.
problem Behavior of spectral sequence degeneration in complex manifolds under small deformations.
method Deformation theory, pseudo-differential operators, Kodaira-Spencer techniques.
result Degeneration at second step is open under certain conditions but not without them.
Study geometric structures on LVM threefolds, focusing on resonant structures.
problem Understanding deformations of geometric structures on LVM threefolds.
method Using the Ehresmann-Thurston principle and Kuranishi family construction.
result Construction of a family containing all LVM threefolds and complete at every point.
Simplified method for L∞ algebra of Dirac structures.
problem Deformation of Dirac structures.
method Simplified method for L∞ algebra. result Canonical L∞-isomorphism of L∞ algebras. Study curvature of direct image bundles in deformations of maps.
problem Understanding curvature in deformations of maps with fixed targets.
method Analyzing curvature of direct image bundles related to deformation data.
result Proved seminegativity for a vector bundle of relative forms.
The paper develops a deformation theory for Dolbeault cohomology classes.
problem Understanding the variations of Dolbeault cohomology classes.
method Established a deformation theory using the power series method and proved the extension equation.
result Proved the existence and unobstructedness of deformations under certain conditions.
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…
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…
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 theory connects string theory to swampland distance conjecture.
problem Connecting string theory to swampland distance conjecture.
method Deformations of the heterotic superpotential, treating separately for large fluxes or large distances, integrating out fields to obtain a new field theory.
result New holomorphic theory defined, connects to swampland distance conjecture.
This research constructs hypercomplex structures from twistor spaces.
problem Understanding hyperkähler metrics and structures.
method Utilizing twistor spaces and Kodaira-Spencer deformation theory.
result Facilitates construction of hypercomplex structures on parameter spaces.
We shall develop a new deformation theory of geometric structures in terms of closed differential forms. This theory is a generalization of Kodaira -Spencer theory and further we obtain a criterion of unobstructed deformations. We apply this theory to certain geometric structures: Calabi-Yau, HyperKähler, $\G$ and $\Sp…
We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of Kähler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of ho…
Historically tensor calculus emerged in an attempt to formalize Rie- mann's ideas. We show that tensor calculus can be based also on Lie's idea of a transformation group and this approach leads quite naturally to the concept of deformation of a transformation group and the Kodaira- Spencer map.
The goal of the memoir is to develop a new cohomology theory which encompasses De Rham and Dolbeault cohomology as well as Deligne Beilinson cohomology, in the context of general complex analytic manifolds. The special case of the Iwasawa manifold is investigated as a typical example of what occurs in the non Kähler ca…
The abstract presents power series proofs for local stabilities of Kähler and balanced structures.
problem Local stabilities of Kähler and balanced structures on complex manifolds.
method Power series method applied to a natural map of complex differential forms.
result New local stability theorems for balanced structures and p-Kähler structures.
This article gives an exposition of the deformation theory for pairs (X,E), where X is a compact complex manifold and E is a holomorphic vector bundle over X, adapting an analytic viewpoint à la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer--Cartan equa…
Abstract: Almost Kähler Hodge numbers vary with metric choices.
problem Variation of almost Kähler Hodge numbers with metrics.
method Analysis of almost complex Hodge numbers under different almost Kähler metrics.
result The almost Kähler Hodge number h0,1 varies with metric choices. Let M=G/Γ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space C(g) of invariant complex structures on M, the Dolbeault cohomology of M is isomorphic to the one of the differential bigraded algebra ass…
Extends Kodaira Spencer to higher-dimensional Calabi-Yau manifolds.
problem No specific problem stated; extension of functional.
method Extension to Calabi-Yau manifolds of arbitrary dimension.
result Extension of the Kodaira Spencer functional.
We characterise, in the setting of the Kodaira-Spencer deformation theory, the twistor spaces of (co-)CR quaternionic manifolds. As an application, we prove that, locally, the leaf space of any nowhere zero quaternionic vector field on a quaternionic manifold is endowed with a natural co-CR quaternionic structure. Also…
Paper solves a metric-independent problem on almost Kähler 4-manifolds.
problem Find a metric-independent generalization of Bott-Chern and Aeppli numbers.
method Introduced a new approach to generalize Bott-Chern and Aeppli numbers.
result Found a solution valid on almost Kähler 4-manifolds.
Holomorphic supergravity theory simplifies anomaly cancellation in heterotic moduli.
problem Anomaly cancellation in heterotic moduli space.
method Formulated a ten-dimensional version of Kodaira-Spencer gravity, quantized fluctuations, and showed partition function simplification.
result Holomorphic supergravity theory simplifies anomaly cancellation and relates to type I Kodaira-Spencer theory.
There is a natural sequence of CY manifolds that are double covers of the projective g dimensional spaces ramified over 2g+2 hyperplanes. We observe that some of them are obtained as quotient of the action of the semi-direct product of g-1 copies of cyclic Z/2Z groups with the symmetric g group on the product of g-copi…
Solves generalized Kähler Calabi-Yau problem on compact manifolds.
problem Calabi conjecture in generalized Kähler geometry.
method New local deformation result, Bismut Ricci curvature transgression formula, generalized Kähler-Ricci flow.
result Global existence and convergence of flow for initial data in generalized Kähler class of Kähler Calabi-Yau structure.
Survey on SYZ mirror symmetry using Morse theory.
problem Understanding SYZ mirror symmetry with quantum corrections.
method Using Witten-Morse theory and Fukaya's combinatorial structures.
result Explicit relation between geometric and combinatorial structures.
For a given multicusp f=c(θ0,...,θi) (1≤i), we present a direct sum decomposition theorem of the source space of iωˉf, where iωˉf is a higher version of the reduced Kodaira-Spencer-Mather map ωˉf. As a corollary of our direct sum decomposition theorem, we show that for any $i\in \mathb…
In a family of compact, canonically polarized, complex manifolds equipped with Kähler-Einstein metrics the first variation of the lengths of closed geodesics was previously shown in by the authors in [arXiv:0808.3741v2] to be the geodesic integral of the harmonic Kodaira-Spencer form. We compute the second variation. F…
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.
The abstract introduces a map for complex forms and deformations, proving deformation invariance theorems.
problem Deformation properties of complex structures on manifolds.
method Introduces a map from complex forms to infinitesimal deformations, uses this map to generalize an extension formula.
result Proves several deformation invariance theorems for Hodge numbers.
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.
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…