Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class

Omid Amini, Matthieu Piquerez

This is a sequel to our work in tropical Hodge theory. Our aim here is to prove a tropical analogue of the Clemens-Schmid exact sequence in asymptotic Hodge theory. As an application of this result, we prove the tropical Hodge conjecture for smooth projective tropical varieties which are rationally triangulable. This provides a partial answer to a question of Kontsevich who suggested the validity of the tropical Hodge conjecture could be used as a test for the validity of the Hodge conjecture.

https://arxiv.org/abs/2012.13142