Tropical Matsusaka and Matsusaka--Ran Criteria

Abstract

The tropical Schottky problem asks which principally polarized tropical abelian varieties are isomorphic to tropical Jacobian varieties equipped with their canonical principal polarizations. We prove the tropical Matsusaka criterion and the tropical Matsusaka--Ran criterion, which are the main results of this paper, and show that their hypotheses are equivalent. Together with the tropical Poincaré formula proved by Gross--Shokrieh, these results give a geometric solution to the tropical Schottky problem. In particular, we give an affirmative answer to a question posed by Brannetti--Melo--Viviani. Several tools are developed for the proofs of the two criteria. In particular, we establish Hodge-type inequalities for ample tropical Cartier divisors on tropical abelian varieties and prove a splitting criterion for principally polarized tropical abelian varieties.