Related papers: Intersection Cohomology of reductive varieties
We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this…
We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the…
Let $X^{2n}\subseteq \mathbb{P} ^N$ be a smooth projective variety. Consider the intersection cohomology complex of the local system $R^{2n-1}\pi{_*}\mathbb{Q}$, where $\pi$ denotes the projection from the universal hyperplane family of…
In this survey article we describe the geometry of toric hyperk\"ahler varieties, which are hyperk\"ahler quotients of the quaternionic vector spaces by tori. In particular, we discuss the Betti numbers, the cohomology ring, and variation…
We study the topology of toric maps. We show that if $f\colon X\to Y$ is a proper toric morphism, with $X$ simplicial, then the cohomology of every fiber of $f$ is pure and of Hodge-Tate type. When the map is a fibration, we give an…
We introduce a recursive decomposition algorithm for the Betti diagram of a complete intersection using the diagram of a complete intersection defined by a subset of the original generators. This alternative algorithm is the main tool that…
In this paper we shall give formulas for the pairings of intersection cohomology classes of complementary dimensions in the intersection cohomology of geometric invariant theoretic quotients for which semistability is not necessarily the…
Topologically, compact toric varieties can be constructed as identification spaces: they are quotients of the product of a compact torus and the order complex of the fan. We give a detailed proof of this fact, extend it to the non-compact…
In this review, novel non-standard techniques for the computation of cohomology classes on toric varieties are summarized. After an introduction of the basic definitions and properties of toric geometry, we discuss a specific computational…
We explain a method for calculating the cohomology of line bundles on a toric variety in terms of the cohomology of certain constructible sheaves on the polytope. We show its effective use by means of some examples.
The motivation of this work is to construct an analog of compactified moduli of abelian varieties and toric pairs in the case of non-commutative algebraic group G. We introduce a class of "stable reductive varieties" which contain connected…
Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of…
Given an arrangement of subtori of arbitrary codimension in a torus, we compute the cohomology groups of the complement. Then, using the Leray spectral sequence, we describe the multiplicative structure on the graded cohomology. We also…
We construct a concrete isomorphism from the permutohedral variety to the regular semisimple Hessenberg variety associated to the Hessenberg function $h_+(i)=i+1$, $1\le i\le n-1$. In the process of defining the isomorphism, we introduce a…
We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential $n$-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology…
Extracting useful information from large data sets can be a daunting task. Topological methods for analyzing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying…
Hypertoric varieties are hyperk\"ahler analogues of toric varieties, and are constructed as abelian hyperk\"ahler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold…
We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses.…
In this text, We compute the equivariant cohomology of Bott-Samelson varieties. Thanks to this computation, we give a new demonstration for the formulas proved by Sarah Billey for the equivariant cohomology of Schubert varieties.
In this paper, we prove the integrality conjecture for quotient stacks arising from weakly symmetric representations of reductive groups. Our main result is a decomposition of the cohomology of the stack into finite-dimensional components…