Hypertoric Hitchin systems and Kirchhoff polynomials

Abstract

We define a formal algebraic analogue of hypertoric Hitchin systems, whose complex-analytic counterparts were defined by Hausel-Proudfoot. These are algebraic completely integrable systems associated to a graph Γ. We study the variation of the Tamagawa number of the resulting family of abelian varieties, and show that it is described by the Kirchhoff polynomial of the graph Γ. In particular, this allows us to compute their p-adic volumes. We conclude the article by remarking that these spaces admit a volume preserving tropicalisation. * M.G. was supported by an NSERC discovery grant. † M.M. was supported by NSERC and the starting grant of Professor Artan Sheshmani at Aarhus University.