  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 $\Gamma_X$.

Algebraic Geometry