## 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.

This paper is available as a pdf (752kB) file. It is also on the arXiv: arxiv.org/abs/1910.00979.

 Thursday, March 12, 2020