Differential forms and cohomology in tropical and complex geometry

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.