Related papers: Blown-up intersection cochains and Deligne's sheav…
We generalize a construction of Barthel-Brasselet-Fieseler-Gabber-Kaup in the setting of complex varieties to the setting of finite type, complex algebraic stacks. Given two such stacks $\mathcal{X},\mathcal{Y}$ with affine stabilizers, and…
In earlier papers arXiv:0802.3120, arXiv:0806.0463 of this series we constructed a sequence of intermediate moduli spaces $\bM^m(c)$ connecting a moduli space $M(c)$ of stable torsion free sheaves on a nonsingular complex projective surface…
We present a theoretical approach which maps polymer blends onto mixtures of soft-colloidal particles. The analytical mesoscale pair distribution functions reproduce well data from united atom molecular dynamics simulations of polyolefin…
With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory and polarization, Deligne's Mixed Hodge Structure on cohomology of complex algebraic varieties is described.
The purpose of these lectures is to introduce the notion of a Stokes-perverse sheaf as a receptacle for the Riemann-Hilbert correspondence for holonomic D-modules. They develop the original idea of P. Deligne in dimension one, and make it…
A notion of Poincar\'e series was introduced by A.Campillo et al. It was developed by Delgado and Gusein-Zade for a multi-index filtration corresponding to the sequence of blow-ups. The present paper suggests the way to generalize the…
We discuss a connection between coherent duality and Verdier duality via a Gersten-type complex of sheaves on real schemes, and show that this construction gives a dualizing object in the derived category, which is compatible with the…
We study the Harvey-Lawson spark characters of level p on complex manifolds. Presenting Deligne cohomology classes by sparks of level $p$, we give an explicit analytic product formula for Deligne cohomology. We also define refined Chern…
Banagl's method of intersection spaces allows to modify certain types of stratified pseudomanifolds near the singular set in such a way that the rational Betti numbers of the modified spaces satisfy generalized Poincar\'{e} duality in…
We define a Real version of smooth Deligne cohomology for manifolds with involution which interpolates between equivariant sheaf cohomology and smooth imaginary-valued forms. Our main result is a classification of Real line bundles with…
For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical…
We show that the cohomology table of any coherent sheaf on projective space is a convergent--but possibly infinite--sum of positive real multiples of the cohomology tables of what we call supernatural sheaves.
This paper is the first in a series. The main goal of the series is to present a geometric construction of certain remarkable tensor categories arising from quantum groups coresponding to the value of deformation parameter $q$ equal to a…
For different cohomology theories (including the Hochschild homology, Hodge cohomology, Chow groups, and Grothendieck groups of coherent sheaves), we identify the cohomology of moduli space of rank 1 perverse coherent sheaves on the blow-up…
In this paper we study the intersection theory on surfaces with abelian quotient singularities and we derive properties of quotients of weighted projective planes. We also use this theory to study weighted blow-ups in order to construct…
In the present review we provide an extensive analysis of the intertwinement between Feynman integrals and cohomology theories in the light of the recent developments. Feynman integrals enter in several perturbative methods for solving non…
We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of $E_{1}$-degeneracy of the Hodge-de Rham spectral…
We elucidate the vector space (twisted relative cohomology) that is Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces - an algebraic invariant…
Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic. Following Drinfeld, one defines a compactification $\widetilde{\on{Bun}}_P$ of the moduli stack of $P$-bundles on $X$. The present paper is concerned…
We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso, to perverse sheaves on symmetric products of the complex line (i.e., on the spaces of monic polynomials stratified by multiplicities of roots). More precisely, we…