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Local intersection cohomology of varieties of complexes

Xin Fang, Markus Reineke

TL;DR

This work computes the local intersection cohomology of the irreducible components of Buchsbaum–Eisenbud varieties of complexes by leveraging Lusztig’s geometric realization of canonical bases through Hall algebras. The authors explicitly construct canonical-basis elements corresponding to the type A quiver realizations of complexes, and they derive a precise, triangular expansion that expresses these elements in terms of PBW-type bases. Using this expansion, they obtain an explicit formula for the Poincaré polynomials of the stalks of local intersection cohomology, presented as sums of $q$-binomial terms over combinatorial data tied to the sparsity set $\Omega$. The results provide complete, computable local IC information for these singular varieties and illustrate how quantum-group techniques can encode detailed topological invariants of algebraic varieties.

Abstract

We compute the local intersection cohomology of the irreducible components of varieties of complexes, by using Lusztig's geometric approach to quantum groups and explicit constructions of elements of Lusztig's canonical bases.

Local intersection cohomology of varieties of complexes

TL;DR

This work computes the local intersection cohomology of the irreducible components of Buchsbaum–Eisenbud varieties of complexes by leveraging Lusztig’s geometric realization of canonical bases through Hall algebras. The authors explicitly construct canonical-basis elements corresponding to the type A quiver realizations of complexes, and they derive a precise, triangular expansion that expresses these elements in terms of PBW-type bases. Using this expansion, they obtain an explicit formula for the Poincaré polynomials of the stalks of local intersection cohomology, presented as sums of -binomial terms over combinatorial data tied to the sparsity set . The results provide complete, computable local IC information for these singular varieties and illustrate how quantum-group techniques can encode detailed topological invariants of algebraic varieties.

Abstract

We compute the local intersection cohomology of the irreducible components of varieties of complexes, by using Lusztig's geometric approach to quantum groups and explicit constructions of elements of Lusztig's canonical bases.

Paper Structure

This paper contains 7 sections, 4 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Varieties of complexes
  3. Quantized enveloping algebras
  4. Quiver approach to canonical bases
  5. Type $A$ quiver and complexes
  6. Construction of elements of canonical bases
  7. Application to local intersection cohomology

Key Result

Lemma 2.1

The irreducible components of ${\rm Com}({\bf d})$ are the orbit closures $\overline{\mathcal{O}({\bf r},{\bf h})}$ for tuples $({\bf r},{\bf h})$ such that $\Omega({\bf r},{\bf h})$ is sparse.

Theorems & Definitions (4)

  • Lemma 2.1
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 5.1