Deletion-contraction triangles for Hausel–Proudfoot varieties

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 that $\mathfrak{B}$ and $\mathfrak{D}$ are diffeomorphic, the diffeomorphism carries the weight filtration on the cohomology of $\mathfrak{B}$ to the perverse Leray filtration on the cohomology of $\mathfrak{D}$, and all these structures are compatible with the deletion-contraction triangles.