Hypertoric Hitchin Systems and Kirchhoff Polynomials

Abstract

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 $\Gamma $. 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 $\Gamma $. 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.