Study on Kuranishi spaces of complex structures and vector bundles, showing isomorphisms and counterexamples.
problem Understanding Kuranishi spaces of complex structures and vector bundles.
method Analyzing Kuranishi spaces of pairs (M,E) of compact Kähler manifolds and vector bundles. result Isomorphisms and counterexamples of Kuranishi spaces of pairs (M,E) of nilmanifolds and trivial vector bundles. This is a survey of the author's paper arXiv:1409.6908 and in-progress book. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1503.07631), as the geometric structure on moduli spaces of J-holomorphic curves. We propose a new definition of Kuranishi spac…
A Kuranishi space is a topological space with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on moduli spaces of J-holomorphic curves in symplectic geometry. This paper is a brief introduction to the author's book arXiv:0707.3572. Let Y be an orbifold and R a Q-algebra. We define…
New theory of Kuranishi manifolds derived from homotopy L∞ spaces.
problem Defining and structuring homotopy L∞ spaces. method Developed a new theory of Kuranishi manifolds, proving they form a 2-category. result Kuranishi manifolds form a 2-category with invertible 2-morphisms. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of J-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a s…
Study the structure of Kuranishi spaces for pairs of Kähler manifolds and polystable Higgs bundles.
problem Understanding the structure of Kuranishi spaces for pairs of Kähler manifolds and polystable Higgs bundles.
method Analyzing the Kuranishi space structure under specific conditions on Higgs bundles and Kähler manifolds.
result The Kuranishi space of a pair (X,E,θ) is isomorphic to the direct product of the Kuranishi space of (E,θ) and the Kuranishi space of X under certain conditions. This is an expository article on the theory of Kuranishi structure and is based on a series of pdf files we uploaded for the discussion of the google group named `Kuranishi' (with its administrator H. Hofer). There we replied to several questions concerning Kuranishi structure raised by K. Wehrheim. At this stage we su…
This is the first paper in a series which proposes and develops the polyfold Fredholm structure--Kuranishi structure correspondence, identifying these two abstract perturbative structures which are indispensable for constructing and understanding symplectic invariants in the most general settings. In this paper, I pres…
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.
Eigenvalue estimates for Hodge Laplacian on Fano manifolds lead to geometric insights.
problem Eigenvalue estimates for the Hodge Laplacian on Fano manifolds.
method Application of Bochner-Kodaira formula to study the geometry of Fano manifolds.
result The original Kähler form remains Kähler for deformations, and an explicit Ricci potential is given.
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
Constructs Kuranishi structures on pseudo holomorphic disk moduli spaces.
problem Building a system of Kuranishi structures on moduli spaces of pseudo holomorphic disks.
method Using exponential decay estimate from [FOOO7], completes the construction of a Kuranishi structure for a single moduli space.
result Detailed and self-contained account of Kuranishi structures construction completed.
This paper defines Gromov-Witten invariants for exploded manifolds.
problem Defining Gromov-Witten invariants for a new class of manifolds.
method Construction of a virtual fundamental class for Kuranishi categories.
result Independence and compatibility of the invariants with various operations.
Paper constructs Kuranishi charts for symplectic manifold moduli spaces.
problem Constructing equivariant charts for moduli spaces of pseudo-holomorphic curves.
method Deformation theory of unstable marked curves using Lie groupoid language.
result Detailed construction of G-equivariant Kuranishi charts for moduli spaces. Study of double complexes on Iwasawa manifold yields 3 isomorphism types.
problem Understanding the structure of double complexes on the Iwasawa manifold.
method Used Stelzig and Qi-Khovanov's structure theorem for double complexes.
result Identified and described exactly 3 isomorphism types of double complexes.
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 Kf and product structure of Kh. We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several exa…
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…
Normal forms for equivariant maps in infinite dimensions established.
problem Establishing normal forms for equivariant maps in infinite-dimensional manifolds.
method Inspired by Lyapunov-Schmidt reduction and Kuranishi method, uses Slice Theorem for Fréchet manifolds.
result Abstract moduli spaces of equivariant maps are locally modeled on quotient by a compact group.
This is the first part of the article we promised at the end of [FOOO13, Section 1]. We discuss the foundation of the virtual fundamental chain and cycle technique, especially its version appeared in [FOn] and also in [FOOO4, Section A1, Section 7.5], [FOOO7, Section 12], [Fu2]. In Part 1, we focus on the construction …
The paper proves a theorem about constructing Higgs bundle moduli space.
problem Constructing the moduli space of Higgs bundles on a closed Riemann surface.
method Uses Kuranishi slice method and GIT quotient to prove the moduli space is a complex space locally modeled on a quadratic cone.
result The moduli space of Higgs bundles is a complex space locally modeled on an affine GIT quotient of a quadratic cone.
This part 2 discusses virtual fundamental chain and cycle technique for K-systems.
problem Foundation of virtual fundamental chain and cycle technique for K-systems.
method Consider a system of spaces with Kuranishi structures and their simultaneous perturbations.
result Discuss the virtual fundamental chain and cycle technique for K-systems.
Deforms Seiberg-Witten solutions in 3D Lie group representations.
problem Non-compact moduli spaces of Seiberg-Witten solutions.
method Constructs Kuranishi models and discusses deformations.
result Fueter sections can be deformed to Seiberg-Witten solutions.
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.
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.
Develops formal moduli theory for splitting complex supermanifolds.
problem Tackles the splitting problem of complex supermanifolds.
method Constructs a filtered dg Lie algebra to control splittings and transfers the theory to a minimal filtered L∞-model. result Recover classical obstruction classes as leading terms of Maurer-Cartan representatives and proves the existence of higher obstructions.
Floer homology constructed for monotone Lagrangians in smooth divisor complements.
problem Computing Floer homology for Lagrangians in smooth divisor complements.
method Compactification of moduli spaces of holomorphic discs and strips, introduction of Kuranishi structures.
result RGW compactifications admit Kuranishi structures, enabling construction of Floer homology.
Constructs universal local deformations for curves and differential forms.
problem Local deformations of curves and differential forms under preservation of periods.
method Develops Kuranishi families for pairs of curves and meromorphic 1-forms, focusing on hyperelliptic cases.
result First paper in a series developing a deformation theory for spectral curve data of integrable systems.
Develops a method to compute Morse homology for clean but not necessarily transverse intersections.
problem Computing Morse homology for clean but not necessarily transversely intersecting manifolds.
method Constructs minimal semi-global Kuranishi structures for moduli spaces of Morse trajectories, generalizing obstruction bundle gluing.
result Obtains iterated gluing equals simultaneous gluing, maintaining computability.
Defines a new metric on Fano Kaehler-Ricci solitons.
problem No specific problem stated; focuses on defining a new metric.
method Defines a Weil-Petersson type metric on the space of shrinking Kaehler-Ricci solitons.
result Proves the independence of the Weil-Petersson metric from choices of Kaehler-Ricci soliton metrics and shows its Kaehler property.
Introduces linear K-systems for Hamiltonian Floer theory.
problem Constructing Floer cohomology for Hamiltonian systems on compact manifolds.
method Geometric realization of linear K-systems via pseudo-holomorphic curves and inductive construction of Kuranishi structures.
result Construction of Floer cohomology and isomorphism with singular cohomology.
A model structure is defined on the category of derived differentiable schemes, and it is used to analyse the truncation 2-functor from derived manifolds to d-manifolds. It is proved that the induced 1-functor between the homotopy categories is full and essentially surjective, giving a bijection between the sets of equ…
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.
The study explores polarized deformations of SKT Calabi-Yau manifolds using Aeppli classes.
problem Understanding polarized deformations of SKT Calabi-Yau manifolds.
method Introducing small deformations polarized by Aeppli classes and investigating their properties.
result Existence of primitive elements in Bott-Chern classes and metrics comparison.
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.
In this paper, by using the Kuranishi coordinates on the Teichmüller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces.
Let X be an irreducible symplectic manifold and Def(X) the Kuranishi space. Assume that X admits a Lagrangian fibration. We prove that X can be deformed preserving a Lagrangian fibration. More precisely, there exists a smooth hypersurface H of Def(X) such that the restriction family over H admits a family of Lagrangian…
Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.
problem Existence and dimension of moduli spaces of conically singular instantons.
method Develop Fredholm deformation theory, investigate cokernel of instanton operator.
result Formula for virtual dimension of moduli space of conically singular instantons with structure group P(U(n)).
Termination proof for Cartan's method in constant type problems.
problem Proving termination of Cartan's equivalence method for constant type problems.
method Groupoid approach to Lie pseudo-groups and Cartan-Kuranishi theorem.
result Cartan's method terminates at involution or complete reduction for constant type problems.
We establish a canonical gluing procedure for Seiberg-Witten monopoles on the two pieces of a closed, oriented 4-manifold X which is split along a 3-dimensional closed, oriented submanifold. We only assume that the (unperturbed) character variety is Kuranishi-smooth and the limiting maps are transversal -- then we will…
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…
In this paper, we derive the family switching formula of -n two-sphere fiber bundle embedded in a smooth four-manifold fiber bundle. In the smooth category, it is a partial generalization of Fintushel-Stern's argument for four-manifolds. We also derive an algebraic analogue of the family switching formula, allowing the…
The variation of Hodge structure of a Calabi-Yau 3-fold induces a canonical Kähler metric on its Kuranishi moduli space, known as the Weil-Petersson metric. Similarly, special pseudo Kähler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classific…
The paper studies how certain solitons on Fano manifolds extend to nearby deformations.
problem Conditions for weighted solitons to extend to nearby deformations of Fano manifolds.
method Analyzes the Kuranishi family of Fano manifolds and uses equivariant automorphism groups.
result All members of the Kuranishi family of a Fano manifold with a weighted soliton have weighted solitons if and only if the dimensions of their T-equivariant automorphism groups are equal to that of the original manifold.
Essential dimension of hyperkähler manifolds families is at most 1.
problem Essential dimension of complex manifold families over compact bases.
method Proved essential dimension not greater than 1 for hyperkähler manifolds families over compact simply connected bases.
result Essential dimension of hyperkähler manifolds families is at most 1.
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…
We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds.…
New flow category for contact manifolds from Reeb orbits.
problem No direct problem stated; focuses on new construction.
method Adapting Kuranishi charts to contact setting, associating flow category based on Reeb orbits and pseudo-holomorphic buildings.
result Lifts contact homology and associates flow bimodule to exact symplectic cobordisms.