arXiv research

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…

The paper constructs Levi flat structures using structure sheaves and differential complexes.

Proves Verdier duality for sheaves on stratified spaces.

This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing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 …

Lecture notes introduce differential geometry using sheaves and differential operators.

The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.

Let M be a K3 surface or an even-dimensional compact torus. We show that the category of coherent sheaves on M is independent from the choice of the complex structure, if this complex structure is generic.

Desingularizes singular spaces using sheaves and groupoids.

The paper proves a key inequality for a specific type of complex spaces.

In this paper, we study semistable Higgs sheaves over compact Kähler manifolds, we prove that there is an approximate admissible Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolove type inequality holds on a semi-stable reflexive Higgs sheaf.

The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) $G$-module, where $G$-modules are struc…

This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with di…

Smooth structures on diffeological spaces and sheaves, resolving conjectures.

The paper studies coherent sheaves on subvarieties of Hopf manifolds.

In the framework of Abstract Differential Geometry, we show that to a given principal sheaf and a representation of its stuctural sheaf in $A^n$, where A is a sheaf of associative, commutative, unital algebras (over R or C), we associate a vector sheaf. Moreover, under some natural assumptions on the compatibility of t…

In this article, we introduce symbol calculus on a projective scheme. Using holomorphic Poisson structures, we construct deformations of ring structures for structure sheaves on projective spaces.

Study co-Higgs sheaves on toric varieties, finding explicit examples.

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…

Shows flexible sheaves as fibrant objects for Gromov's h-principle.

Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.

Three definitions of graded vector bundles are shown to be equivalent.

The paper extends a conjecture about Euler characteristics of perverse sheaves on certain Kähler manifolds.

Constructs coordinate systems from spectral curve sheaves.

New potentials found for sheaves on Calabi-Yau 4-folds.

Study projective KLT varieties with projectively flat cotangent sheaves.

Kontsevich and Soibelman introduced a notion of orientation data on Calabi-Yau category. It can be viewed as a consistent choice of spin structure on moduli space of objects in the given category. The orientation data plays an important role in Donaldson-Thomas theory. Let X be a projective, simply connected and torsio…

Paper constructs Chern character for coherent sheaves.

Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.

Develops equivariant Chern characters for coherent sheaves with 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 $L_{\infty}$ morphism lifting semiregularity maps.

Defines sheaves and Čech cohomology for diffeological spaces and classifies principal bundles.

Given a smooth projective toric variety $X_Σ$ 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 $Coh(X_Σ)$ into the dg derived category of constructible sheaves on a torus $Sh(T^n, Λ_Σ)$. 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.

Monoidal categorifies genus zero skein algebra using K-theory.

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.

Generalized Nakano positivity for certain singular cases.

For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption o…

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.

Geometrically computes sheaves linking HOMFLY-PT homology to Hilbert schemes.

Study homotopy sheaves on categories and their presheaves, proving descent properties.

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 $\bar{\partial}$-coherent sheaves. We also include here the complete proof of our main Theorem.

A complex contact structure $γ$ is defined by a system of holomorphic local $1$-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle ${\rm Ker}\, γ$ of the tangent bundle and a line bundle $L$. In this paper, we prove that the sheaf of holomorphic $k$-vectors on a compl…

New homotopy theory reveals the structure of stable curves.

Extends six operations to sheaves in any symmetric monoidal category.