arXiv research

Study of perverse sheaves on instanton moduli spaces.

Diagrammatically describes algebra equivalent to perverse sheaves on isotropic Grassmannians.

Generalizes intersection homology with broader applicability and new duality.

This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse $t$-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing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

New algebraic construction of knot contact homology using perverse sheaves.

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

Let X be a compact complex manifold, $D_c^b(X)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk(T^*X)$ be the Fukaya category of $T^*X$. A Lagrangian brane in $Fuk(T^*X)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^*X_{\mathbb{C}}$, the holomorphic cotange…

New knot invariant defined using sheaf theory and SL(2,C) connections.

For a fixed parabolic subalgebra p of gl(n,C) we prove that the centre of the principal block O(p) of the parabolic category O is naturally isomorphic to the cohomology ring of the corresponding Springer fibre. We give a diagrammatic description of O(p) for maximal parabolic p and give an explicit isomorphism to Braden…

We develop a generalization to non-Witt spaces of the intersection homology theory of Goresky-MacPherson. The second author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we extend both of…

Let $U$ be a smooth $\mathbb C$-scheme, $f:U\to\mathbb A^1$ a regular function, and $X=$Crit$(f)$ the critical locus, as a $\mathbb C$-subscheme of $U$. Then one can define the "perverse sheaf of vanishing cycles" $PV_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $Φ:U\to U$ is an iso…

This is partly a survey and partly a research article. Some known results and open problems about Kaehler groups (fundamental groups of compact Kaehler manifolds) are discussed. A new notion of Kaehler homomorphism is introduced. This is a homomorphism induced by a holomorphic map between these types of manifolds. Some…

We give a construction of the Floer homology of the pair of {\it non-compact} Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy which moves the infinity but leaves the intersection set of the pair compact. This construction uses the concept of Lagrangian cobordism and ce…

This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.

Unified approach to stability conditions on surfaces with quadratic differentials.

L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of a locally symmetric space. We define the micro-support of an L-module; it is a set of irreducible modules for the Levi quotients of the parabolic Q-subgroups associated to the strata. We prove a vanishing th…

Study intersection cohomology and Lagrangian fibrations in symplectic varieties.

Study intersection homotopy groups in coarsenings of CS sets.

Given a compact stratified pseudomanifold with a Thom-Mather stratification and a class of riemannian metrics over its regular part, we study the relationships between the $L^{2}$ de Rham and Hodge cohomology and the intersection cohomology of $X$ associated to some perversities. More precisely, to a kind of metric whi…

A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifol…

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …

Identifies filtration in Lagrangian fibrations to monodromy weight filtration in degenerations.

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also cal…

The study refines contingency matrices for complex stratification and braid group cohomology.

On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted $L^2$ cohomology for iterated fibred cusp metrics on the smooth stratum. In particular given a Witt space, we identify the $L^2$ cohomology of iterated fibred cusp metrics with the middle perversity i…

Two short proofs show intersection homology's stability.

New connection found between Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds.

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted $L^2$ harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted $L^2$ harmonic fo…

New proof shows coherent sheaves quasi-equivalent to constructible sheaves on torus.

A new method prevents insurance derivatives from incentivizing risky behavior.

Classifies simple sheaves for knot conormals.

Abstract: Review and properties of Gieseker stability for Higgs sheaves.

We introduce a singular chain intersection homology theory which generalizes that of King and which agrees with the Deligne sheaf intersection homology of Goresky and MacPherson on any topological stratified pseudomanifold, compact or not, with constant or local coefficients, and with traditional perversities or superp…

A sheaf-theoretic model connects SL(2,C) Floer homology to 3-manifold invariants.

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

The paper studies equivariant sheaves on toric varieties and their quotients.

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

Characterizes tangent cones for specific connections on reflexive sheaves.

Study tangent cones of reflexive sheaves, proving existence and uniqueness.

Constructs coordinate systems from spectral curve sheaves.

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

Study of holomorphic distributions on projective 3-space, focusing on stable tangent sheaves.

Paper constructs Chern character for coherent sheaves.

Augmentations and sheaves linked for Legendrian graphs.

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

Develops equivariant Chern characters for coherent sheaves with group actions.

The paper proves a key inequality for Higgs sheaves on complex manifolds.

The paper connects connections on sheaves to an $L_{\infty}$ morphism lifting semiregularity maps.