Deletion-contraction triangles for Hausel-Proudfoot varieties

Zsuzsanna Dancso, Michael McBreen, Vivek Shende

Abstract

To a graph, Hausel and Proudfoot associate two complex manifolds, \(\mathfrak{B}\) and \(\mathfrak{D}\), which behave, respectively like moduli of local systems on a Riemann surface, and moduli of Higgs bundles. For instance, \(\mathfrak{B}\) is a moduli space of microlocal sheaves, which generalize local systems, and \(\mathfrak{D}\) 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 \(\mathfrak{B}\) 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 \(\mathfrak{B}\). There is a corresponding triangle for \(\mathfrak{D}\). Finally, we prove \(\mathfrak{B}\) and \(\mathfrak{D}\) are diffeomorphic, that the diffeomorphism carries the weight filtration on the cohomology of \(\mathfrak{B}\) to the perverse Leray filtration on the cohomology of \(\mathfrak{D}\), and that all these structures are compatible with the deletion-contraction triangles.