Deletion-contraction triangles for Hausel-Proudfoot varieties

TL;DR

This paper explores the geometric and topological properties of Hausel-Proudfoot varieties associated with graphs, establishing deletion-contraction relations and showing their cohomological structures are deeply interconnected.

Contribution

It introduces deletion-contraction exact triangles for the cohomology of these varieties and proves their diffeomorphism, linking filtrations on cohomology in a novel way.

Findings

Euler characteristics count spanning subtrees

Point-count polynomial satisfies deletion-contraction

Varieties are diffeomorphic with compatible filtrations

Abstract

To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively like moduli of local systems on a Riemann surface, and moduli of Higgs bundles. For instance, B is a moduli space of microlocal sheaves, which generalize local systems, and D carries the structure of a complex integrable system. We show the Euler characteristics of these varieties count spanning subtrees of the graph, and the point-count over a finite field for B is a generating polynomial for spanning subgraphs. This polynomial satisfies a deletion-contraction relation, which we lift to a deletion-contraction exact triangle for the cohomology of B. There is a corresponding triangle for D. Finally, we prove B and D are diffeomorphic, that the diffeomorphism carries the weight filtration on the cohomology of B to the perverse Leray filtration on the cohomology of D, and that all these structures are compatible with the deletion-contraction triangles.