Deletion-contraction triangles for Hausel-Proudfoot varieties

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 o...

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Bibliographic Details
Published in:arXiv.org 2022-10
Main Authors: Dancso, Zsuzsanna, McBreen, Michael, Shende, Vivek
Format: Article
Language:English
Subjects:
Deletion
Fields (mathematics)
Filtration
Graph theory
Homology
Isomorphism
Polynomials
Riemann surfaces
Sheaves
Online Access:Get full text
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Summary: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.
ISSN:2331-8422