Related papers: The perverse filtration and the Lefschetz Hyperpla…
In this paper we define a new cohomology theory for a $B$-algebra $A$. We use this cohomology to study deformations of algebras $A[[t]]$, that have a $B$-algebra structure.
We show a Lefschetz theorem for irreducible overconvergent $F$-isocrystals on smooth varieties defined over a finite field. We derive several consequences from it.
In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application…
We prove an equivalence between filtrations of primitive bialgebras and filtrations of factorizable perverse sheaves, generalizing the results obtained by Kapranov-Schechtman. Under this equivalence, we find that the word length filtration…
We give a definition and study the basic properties of the irregular Hodge filtration on the exponentially twisted de Rham cohomology of a smooth quasi-projective complex variety.
We introduce a way of describing cohomology of the symmetric groups with coefficients in Specht modules over Z or F_p. We study i-th-degree cohomology for i in {0,1,2}. The focus lies on the isomorphism type of second-degree cohomology of…
We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of…
We introduce an idea of constructing Lefschetz fibrations of Weinstein manifolds from Weinstein handle decompositions on them. We prove theorems that formulate the idea for the cases of cotangent bundles and some plumbings. As a corollary,…
We discuss the relationship between kernels, images and cokernels of morphisms between perverse sheaves and induced maps on stalk cohomology.
By the Lefschetz hyperplane theorem, if X is a smooth quasi-projective variety and C a general curve section of X then the fundamental group of C surjects onto the fundamental group of X. Here we consider when this conclusion holds for a…
In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with…
We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky-MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity…
This is a mostly expository paper, intended to explain a very natural relationship between two a priori distinct notions appearing in the literature: Generic Vanishing in the context of vanishing theorems and birational geometry, and…
Using Saito's theory of mixed Hodge modules, we study a generalization of Hellus-Schenzel's "cohomologically complete intersection" property. This property is equivalent to perversity of the shifted constant sheaf. We relate the generalized…
For a stratified pseudomanifold $X$, we have the de Rham Theorem $ \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} - \per{p}}{*}{X}, $ for a perversity $\per{p}$ verifying $\per{0} \leq \per{p} \leq \per{t}$, where $\per{t}$ denotes the top…
In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we…
In this article we establish an explicit link between the classical theory of deformations \`a la Gerstenhaber -- and a fortiori with the Hochschild cohomology-- and (weak) PBW-deformations of homogeneous algebras. Our point of view is of…
Motivated by definitions in mixed Hodge theory, we define the weight filtration and the monodromy weight filtration on the combinatorial intersection cohomology of a fan. These filtrations give a natural definition of the multivariable…
We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians of curves over an algebraically closed non-Archimedean field. Let $J$ be the Jacobian of a curve $X$, and let $W_d \subset J$ be the locus of…
We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral…