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.