Global Chern currents and Baum Bott currents defined on arbitrary complex manifolds.
problem Defining Chern classes and Baum Bott residues on complex manifolds without global resolutions.
method Combining Green's techniques with previous constructions to yield representatives of Chern classes and Baum Bott residues, using local resolutions and metrics.
result Transgression formula for the representatives, showing they differ by a current of the form dN. Proves Verdier duality for sheaves on stratified spaces.
problem Verdier duality for constructible sheaves on stratified spaces.
method Uses conically smooth stratified spaces and Lurie's Verdier duality.
result Shows equivalence between constructible sheaves and cosheaves.
The article constructs differential operators for parabolic geometries.
problem Developing a machinery for differential operators in parabolic geometries.
method Starting from a relative tractor bundle, constructs a sequence of differential operators.
result Provides a resolution of a sheaf for many geometries.
The paper develops residue currents for cohesive modules and proves a generalized Poincaré-Lelong formula.
problem Analyzing coherent sheaves on complex manifolds using global analytic methods.
method Developing residue currents for cohesive modules and proving their properties.
result Proves a generalized Poincaré-Lelong formula for cohesive modules.
The paper constructs Levi flat structures using structure sheaves and differential complexes.
problem Global solvability and regularity of Levi flat structures.
method Employing formal integrability and differential complexes, the paper constructs a resolution for the structure sheaf.
result Global exactness and Sobolev regularity of the differential complex for Levi flat structures.
A singular (or Hermann) foliation on a smooth manifold M can be seen as a subsheaf of the sheaf X of vector fields on M. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie brack…
We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with klt singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singu…
Homological mirror symmetry proved for symmetric squares of punctured spheres.
problem Proving homological mirror symmetry for symmetric squares of punctured spheres.
method Constructed quasi-equivalences between wrapped Fukaya categories and derived categories of coherent sheaves, using categorical resolutions and localisation.
result Wrapped Fukaya category of symmetric square quasi-equivalent to coherent sheaves on a singular surface.
The paper proves a theorem and characterizes connections over normal varieties.
problem The study addresses the stability and connections over normal varieties.
method The authors prove a complete version of the Donaldson-Uhlenbeck-Yau theorem and use it to show the polystability of reflexive sheaves.
result An admissible Hermitian-Yang-Mills connection defines a polystable reflexive sheaf and gives a lower bound for discriminants.
Paper constructs solutions for a class of overdetermined systems.
problem Constructing solutions for a class of overdetermined systems.
method Resolution of the solution sheaf, sufficient condition for global exactness, gluing techniques, local solvability of the Treves complex.
result Obtained a sufficient condition for global exactness, leading to gluing techniques for local solutions.
Study co-Higgs sheaves on toric varieties, finding explicit examples.
problem Characterizing and understanding co-Higgs sheaves on toric varieties.
method Characterization and explicit computation of examples.
result Explicit examples of co-Higgs sheaves on toric varieties computed.
The paper studies equivariant sheaves on toric varieties and their quotients.
problem Understanding stability of sheaves on toric GIT quotients.
method Defining equivariant sheaves and showing stability preservation under certain conditions.
result Stability of sheaves on toric GIT quotients is related to combinatorial criteria.
We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. We prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective alge…
Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.
problem Understanding sheaves of Lie-Rinehart algebras and their morphisms.
method Introduced morphisms and comorphisms, proved factorization theorems, and defined higher homotopy groups and groupoids.
result Sheaves of Lie-Rinehart algebras over smooth manifolds induce partitions into orbits of the fundamental groupoid.
Characterizes tangent cones for specific connections on reflexive sheaves.
problem Analyzing tangent cones of admissible Hermitian-Yang-Mills connections over reflexive sheaves.
method Algebro-geometric characterization of analytic tangent cones.
result Complete characterization of tangent cones for admissible Hermitian-Yang-Mills connections over reflexive sheaves.
Constructs coordinate systems from spectral curve sheaves.
problem Creating coordinate systems from spectral curve sheaves.
method Finite-gap integration methods for orthogonal curvilinear coordinates.
result Constructs coordinate systems over reducible spectral curves.
New potentials found for sheaves on Calabi-Yau 4-folds.
problem Understanding sheaves on Calabi-Yau 4-folds.
method Derived Quot-stacks and Lagrangian distributions.
result Globally defined −1-shifted potentials on sheaves. For a real or complex semisimple Lie group G and two nested parabolic subgroups Q⊂P⊂G, we study parabolic geometries of type (G,Q). Associated to the group P, we introduce a class of relative natural bundles and relative tractor bundles and construct some basic invariant differential operators on …
Paper constructs Chern character for coherent sheaves.
problem Chern character for coherent sheaves with values in Bott-Chern cohomology.
method Based on Block's fundamental construction, constructs Chern character.
result Proves Riemann-Roch-Grothendieck formula for coherent sheaves.
This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse t-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic …
Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.
problem Understanding the Kobayashi-Hitchin correspondence for specific sheaves.
method Using Hermitian-Yang-Mills flow on Kähler manifolds with simple normal crossing divisors.
result Established the correspondence for saturated reflexive parabolic sheaves.
Augmentations and sheaves linked for Legendrian graphs.
problem Understanding categorical Legendrian isotopy invariants.
method Equivalence between augmentation category and DG category of sheaves.
result Proved 'augmentations are sheaves' for Legendrian graphs.
Partial AHS-structures extend G-structures and Cartan geometries to manifolds with involutive distributions.
problem Extending G-structures and Cartan geometries to manifolds with involutive distributions.
method Developing a canonical Cartan geometry for partial AHS-structures and constructing BGG sequences.
result Partial AHS-structures have analogs of BGG sequences, providing fine resolutions of sheaves.
Develops equivariant Chern characters for coherent sheaves with group actions.
problem Computing Chern characters for coherent sheaves on manifolds with group actions.
method Introduces equivariant Chern characters and proves Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.
result Establishes a Riemann-Roch-Grothendieck theorem for coherent sheaves with finite group actions.
We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.
The paper connects connections on sheaves to an L∞ morphism lifting semiregularity maps.
problem Understanding connections on sheaves and their relationship to semiregularity maps.
method Proves a canonical association of a connection of type (1,0) on a sheaf to an L∞ morphism. result Establishes a connection between connections on sheaves and an L∞ morphism lifting semiregularity maps. Defines sheaves and Čech cohomology for diffeological spaces and classifies principal bundles.
problem Defining sheaves and Čech cohomology for diffeological spaces.
method Defines sheaves for diffeological spaces and constructs Čech cohomology. Uses Čech cohomology to classify principal bundles.
result First degree Čech cohomology classes classify diffeological principal G-bundles. Given a smooth projective toric variety XΣ of complex dimension n, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves Coh(XΣ) into the dg derived category of constructible sheaves on a torus Sh(Tn,ΛΣ). Recently, K…
We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.
Develops a new method to study algebraic tangent cones of sheaves using valuations.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
Let M be a hyperkaehler manifold, and F a torsion-free and reflexive coherent sheaf on M. Assume that F (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then F is stable and its singulariti…
Unified framework for Morita invariant cohomology of Lie groupoids.
problem Proving Morita invariance of cohomology theories for Lie groupoids.
method Viewing cohomology as sheaves of modules on the nerve of the groupoid and establishing criteria for Morita invariance.
result Established criteria for Morita invariant cohomology theories.
Generalized Nakano positivity for certain singular cases.
problem Nakano positivity of direct image sheaves for singular cases.
method Generalization of Berndtsson's result to singular cases.
result Nakano positivity for direct image sheaves in special singular cases.
We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit Kähler-Einstein metrics. We also show t…
Sheaves on graphs link to noncommutative geometry.
problem Exploring noncommutative geometry concepts on graphs.
method Sheaf theory and simplicial sets.
result Enhanced understanding of discrete noncommutative geometry.
Geometrically computes sheaves linking HOMFLY-PT homology to Hilbert schemes.
problem Linking HOMFLY-PT homology to geometric structures on Hilbert schemes.
method Geometric sheaf theory, Hochschild homology formality, Hilbert schemes of points.
result Established formalism connecting HOMFLY-PT homology to coherent sheaves on Hilbert schemes.
Study homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
This is a large audience version of our previous work (see math.AG/0301146) in which we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of ∂ˉ-coherent sheaves. We also include here the complete proof of our main Theorem.
Extends six operations to sheaves in any symmetric monoidal category.
problem Extending six operations to a broader class of sheaves.
method Develops formalism for sheaves in any closed symmetric monoidal ∞-category, proving properties of locally contractible geometric morphisms and relating pullbacks and colimits.
result Establishes the six functor formalism for a wider range of sheaves, including those with values in any closed symmetric monoidal ∞-category.
In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs th…
In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, Sesum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kahler-Einstein metrics.
Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.
We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geom…
The flow converges without Kähler-Einstein and develops ideal sheaves.
problem Analyzing convergence of inverse Monge-Ampere flow without Kähler-Einstein metrics.
method Generalizing the flow and providing conditions for convergence and ideal sheaves development.
result The flow converges without Kähler-Einstein metrics and develops Nadel multiplier ideal sheaves.
New L∞ liftings derived from Chern-Simons classes for coherent sheaves.
problem Liftings of semiregularity maps for coherent sheaves on complex manifolds.
method Introducing Chern-Simons classes for curved DG-pairs and proving canonical liftings.
result Canonical L∞ liftings of Buchweitz-Flenner semiregularity maps. Lecture notes introduce differential geometry using sheaves and differential operators.
problem Exploring differential geometry concepts.
method Using sheaves, differential operators, and horizontal subbundles.
result Presented an approach to fundamental differential geometry structures.
We study the basic properties of Higgs sheaves over compact Kähler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of a…
We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that general…