Differential forms and cohomology in tropical and complex geometry

TL;DR

This paper introduces modified maps connecting tropical and complex cohomology theories, demonstrating their compatibility with integrals and dualities, thus bridging tropical and complex geometry in a new way.

Contribution

It presents new maps from tropical to Dolbeault cohomology for complex projective varieties, extending previous work and establishing their compatibility with integrals and dualities.

Findings

Maps are compatible with integrals on semi-algebraic subsets.

Weighted tropicalizations induce dual maps under certain conditions.

The approach bridges tropical and complex cohomology theories.

Abstract

Ducros, Hrushovski, and Loeser gave maps from families of archimedean diffrential forms to non-archiemedean (or tropical) ones, which are compatible with integrals on algebraic varieties. In this paper, we introduce slight modifications of their maps for complex projective varieties which give natural maps from tropical to the usual Dolbeault cohomology. We also show that our maps are compatible with integrals on generic semi-algebraic subsets and those on their weighted tropicalizations. Weighted tropicalizations induce the dual maps of the above maps of Dolbeault cohomology groups under some assumptions.