PreprintDeletioncontraction triangles for HauselProudfoot varietiesZsuzsanna Dancso, Michael McBreen, Vivek ShendeAbstractTo 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 pointcount over a finite field for \(\mathfrak{B}\) is a generating polynomial for spanning subgraphs. This polynomial satisfies a deletioncontraction relation, which we lift to a deletioncontraction 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 deletioncontraction triangles. This paper is available as a pdf (752kB) file. It is also on the arXiv: arxiv.org/abs/1910.00979.
