Deletion-contraction triangles for Hausel–Proudfoot varieties

  • Zuszsanna Dancso

    University of Sydney, Australia
  • Michael McBreen

    Chinese University of Hong Kong, Hong Kong
  • Vivek Shende

    UC Berkeley, USA
Deletion-contraction triangles for Hausel–Proudfoot varieties cover

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To a graph, Hausel and Proudfoot associate two complex manifolds, and , which behave, respectively, like moduli of local systems on a Riemann surface and moduli of Higgs bundles. For instance, is a moduli space of microlocal sheaves, which generalize local systems, and 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 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 . There is a corresponding triangle for .

Finally, we prove that and are diffeomorphic, the diffeomorphism carries the weight filtration on the cohomology of to the perverse Leray filtration on the cohomology of , and all these structures are compatible with the deletion-contraction triangles.

Cite this article

Zuszsanna Dancso, Michael McBreen, Vivek Shende, Deletion-contraction triangles for Hausel–Proudfoot varieties. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1369