Cohomologically tropical varieties

TL;DR

This paper introduces the concept of cohomologically tropical varieties, establishing conditions under which tropical and rational cohomology coincide, and explores their properties and implications in degenerations.

Contribution

It characterizes cohomologically tropical varieties as schön and wunderschön with tropical homology manifold tropicalizations, extending previous work on Hodge numbers in degenerations.

Findings

Cohomologically tropical varieties are characterized by tropical homology manifold conditions.

The cohomology map between tropical and rational cohomology is an isomorphism for these varieties.

Hodge numbers of smooth fibers are reflected in tropical cohomology during degenerations.

Abstract

Given the tropicalization of a complex subvariety of the torus, we define a morphism between the tropical cohomology and the rational cohomology of their respective tropical compactifications. We say that the subvariety of the torus is cohomologically tropical if this map is an isomorphism for all closed strata of the tropical compactification. We prove that a sch\"on subvariety of the torus is cohomologically tropical if and only if it is wundersch\"on and its tropicalization is a tropical homology manifold. The former property means that the open strata in the boundary of a tropical compactification are all connected and the mixed Hodge structures on their cohomology are pure of maximum possible weight; the latter property requires that, locally, the tropicalization verifies tropical Poincar\'e duality. We study other properties of cohomologically tropical and wundersch\"on varieties, and show that in a semistable degeneration to an arrangement of cohomologically tropical varieties, the Hodge numbers of the smooth fibers are captured in the tropical cohomology of the tropicalization. This extends the results of Itenberg, Katzarkov, Mikhalkin, and Zharkov.