Monodromy filtrations and the topology of tropical varieties

Abstract

We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropica] degenerations to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space Tx and give a geometric interpretation of the cohomology of Γx in terms of the action of a monodromy operator on the cohomology of Γx. This gives bounds on the Betti numbers of Γx in terms of the Betti numbers of X which constrain the topology of Trop(X). We also obtain a description of the top power of the monodromy operator acting on middle cohomology of X in terms of the volume pairing on Γx. © Canadian Mathematical Society 2012.