Homological tools, intersection theory, and tropical abelian varieties

Abstract

We develop tropical homological tools and apply them to the study of certain tautological cycles in the Jacobian of a metric graph. First, in joint work with Junaid Hasan and Farbod Shokrieh, we construct an explicit tropical de Rham map from superforms to tropical cochains and prove that it induces an isomorphism of cohomology rings that is compatible with Poincaré duality on both sides. Applying this to a principally polarized tropical abelian variety (X, ϑ), we produce a canonical(1, 1)-superform νϑ whose cohomology class is Poincaré-dual to the theta divisor. Specializing to the Jacobian of a metric graph Γ, we show that integrating powers of νϑ over the effective loci Wd recovers Foster’s theorem and Kirchhoff’s matrix-tree theorem as special cases. We also give an alternative proof of the tropical Poincaré formula. Second, we study the tropical Ceresa cycle W1 - W1- in Jac(Γ). We define a graph invariant α(G) called the Ceresa period which acts as an obstruction to W1 and W1- being algebraically equivalent. We prove that α(G) vanishes if and only if G is of hyperelliptic type, which has a forbidden minor characterization in terms of the graphs K4 and L3. As a consequence, a very general metric graph overlying a non-hyperelliptic-type graph has algebraically nontrivial tropical Ceresa cycle.