Harmonic covers of skeleta

Abstract

The geometry of a toroidal scheme over a DVR is encoded in a $\mathbb{Z}$-PL space known as the dual polyhedral complex. Any such dual complex is a skeleton, i.e. a nonarchimedean analytic retract, and admits a combinatorial divisor theory via specialisation. These structures on the dual complex interact via a Poincar\'e-Lelong slope formula, which interprets specialisations of divisors as Laplacians of PL functions. The main result presented here shows that finite covers of toroidal schemes give harmonic morphisms of dual complexes, i.e. morphisms that preserve the tropical Laplace equation. A crucial ingredient is a balancing condition which is a variant of the tropical multiplicity formula for dual complexes. We apply these results to obtain a Riemann-Hurwitz formula for covers of skeleta in any dimension: the Laplacian of the different function is the tropical relative canonical divisor.