2019/10/02 by Zsuzsanna Dancso, Dancso, Zsuzsanna, Michael McBreen +3 · 2 citations
Mathematics · #05C31 (Secondary) #14N99 (Primary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1910.00979
openalex publication_date 2019/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively like moduli of local systems on a Riemann surface, and moduli of Higgs bundles. For instance, B is a moduli space of microlocal sheaves, which generalize local systems, and 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 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 B. There is a corresponding triangle for D. Finally, we prove B and D are diffeomorphic, that the diffeomorphism carries the weight filtration on the cohomology of B to the perverse Leray filtration on the cohomology of D, and that all these structures are compatible with the deletion-contraction triangles.