Monodromy Filtrations and the Topology of Tropical Varieties

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 ΓX . We give a geometric interpretation of the cohomology of ΓX in terms of the action of a monodromy operator on the cohomology of X. This gives bounds on the Betti numbers of ΓX 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 dimX.