Monodromy Filtrations and the Topology of Tropical Varieties

TL;DR

This paper develops a theory linking the topology of tropical varieties to monodromy actions, providing bounds on Betti numbers and insights into the cohomology of tropical degenerations of algebraic varieties.

Contribution

It introduces a tropical degeneration framework analogous to Tevelev's compactifications, enabling a topological parameterization and cohomological analysis of tropical varieties.

Findings

Constructed normal crossings degenerations of subvarieties in a torus.

Established bounds on Betti numbers of tropical varieties.

Described the monodromy action on cohomology in terms of volume pairing.

Abstract

We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space \Gamma_X. We give a geometric interpretation of the cohomology of \Gamma_X in terms of the action of a monodromy operator on the cohomology of X. This gives bounds on the Betti numbers of $\Gamma_X$ in terms of the Betti numbers of $X$. When $X$ is a sufficiently general complete intersection, this allows us to show that the cohomology of Trop(X) vanishes in degree less than dim(X). In addition, we give a description for the top power of the monodromy operator acting on middle cohomology in terms of the volume pairing on $\Gamma_X$.