arXiv research

Unified approach to stability conditions on surfaces with quadratic differentials.

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…

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

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…

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…

We provide a generalization of the Deligne sheaf construction of intersection homology theory, and a corresponding generalization of Poincaré duality on pseudomanifolds, such that the Goresky-MacPherson, Goresky-Siegel, and Cappell-Shaneson duality theorems all arise as special cases. Unlike classical intersection homo…

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…

I will explain my joint paper `Instantons moduli spaces and W-algebras' with A.Braverman, M.Finkelberg, arXiv:1406.2381. I will concentrate on the geometric part, that is a study of perverse sheaves on instanton moduli spaces. I place a particular emphasize on the hyperbolic restriction functor and stable envelop, whic…

When an insurance note is also a derivative a serious problem arises because a derivative must be fulfilled immediately. This feature of derivatives prevents claims processing procedures that screen out ineligible claims. This, in turn, creates a perverse incentive for insured holders of notes to commit acts that resul…

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…

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 …

For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect…

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…

We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total space. This builds a new connection between the topology of Lagrangian fibrations a…

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

In this paper, which is mostly a research announcement, we give a new algebraic construction of knot contact homology in the sense of L. Ng [Ng05a]. For a link $L$ in ${\mathbb R}^3$, we define a differential graded (DG) $k$-category $\tilde{\mathscr A}$ with finitely many objects, whose quasi-equivalence class is …

We prove a conjecture raised by M. Goresky and W. Pardon, concerning the range of validity of the perverse degree of Steenrod squares in intersection cohomology. This answer turns out of importance for the definition of characteristic classes in the framework of intersection cohomology. For this purpose, we present a c…

Let $X$ be any subanalytic compact pseudomanifold. We show a De Rham theorem for $L^\infty$ forms. We prove that the cohomology of $L^\infty$ forms is isomorphic to intersection cohomology in the maximal perversity.

New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.

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…

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…

The paper proves a conjecture linking Higgs bundles and Lie algebra actions.

Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an orie…

We indicate two short proofs of the Goresky-MacPherson topological invariance of intersection homology. One proof is very short but requires the Goresky-MacPherson support and cosupport axioms; the other is slightly longer but does not require these axioms and so is adaptable to more general perversities.

Witt spaces are pseudomanifolds for which the middle-perversity intersection homology with rational coefficients is self-dual. We give a new construction of the symmetric signature for Witt spaces which is similar in spirit to the construction given by Miscenko for manifolds. Our construction has all of the expected pr…

We conjecture an expression for the dimensions of the Khovanov-Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture relating the HO…

Let $M$ be a stratum of a compact stratified space $A$. It is equipped with a general adapted metric $g$, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, $g$ has a general type, which is an extension of the type of an adapted metric. A restriction on this …

The paper constructs instanton complexes on stratified pseudomanifolds.

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 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…

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…

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…

Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, in which case intersection cohomology may depend on the choice of the stratification by which it is defined. Topological invariance also…

We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping $G : \C^{n} \to \C^{n - 1}$ where $n \geq 2$. We prove that in the case $G : \C^{3} \to \C^{2}$, if $G$ is a local submersion but is not a fibration, then the homology and the intersection homology with total perversity (with compact s…

Consider the middle perversity intersection cohomology groups of various compactifications of a Hermitian locally symmetric space. Rapoport and independently Goresky and MacPherson have conjectured that these groups coincide for the reductive Borel-Serre compactification and the Baily-Borel-Satake compactification. Thi…

This article is a survey of recent work of the author, together with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza, on the Hodge theory of stratified spaces. We discuss how to resolve a Thom-Mather stratified space to a manifold with corners with an iterated fibration structure and the generalization of …

Cohen, Goresky and Ji showed that there is a Kunneth theorem relating the intersection homology groups $I^{\bar p}H_*(X\times Y)$ to $I^{\bar p}H_*(X)$ and $I^{\bar p}H_*(Y)$, provided that the perversity $\bar p$ satisfies rather strict conditions. We consider biperversities and prove that there is a Künneth theorem r…

Spectral risk measures are attractive risk measures as they allow the user to obtain risk measures that reflect their risk-aversion functions. To date there has been very little guidance on the choice of risk-aversion functions underlying spectral risk measures. This paper addresses this issue by examining two popular …

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invaria…

Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove…

Constructs Morse homology for complex algebraic varieties.

We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the stand…

In this paper we explain how non-abelian Hodge theory allows one to compute the $L^2$ cohomology or middle perversity higher direct images of harmonic bundles and twistor D-modules in a purely algebraic manner. Our main result is a new algebraic description for the fiberwise $L^2$ cohomology of a tame harmonic bundle o…

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

I study the limit of a large random economy, where a set of consumers invests in financial instruments engineered by banks, in order to optimize their future consumption. This exercise shows that, even in the ideal case of perfect competition, where full information is available to all market participants, the equilibr…