Beschreibung:
<jats:title>Abstract</jats:title><jats:p>We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, “multiplicity-free” parameterization of Trop(<jats:italic>X</jats:italic>) by a topological space Г<jats:italic><jats:sub>X</jats:sub></jats:italic>and give a geometric interpretation of the cohomology of Г<jats:italic><jats:sub>X</jats:sub></jats:italic>in terms of the action of a monodromy operator on the cohomology of<jats:italic>X</jats:italic>. This gives bounds on the Betti numbers of Г<jats:italic><jats:sub>X</jats:sub></jats:italic>in terms of the Betti numbers of<jats:italic>X</jats:italic>which constrain the topology of Trop(<jats:italic>X</jats:italic>). We also obtain a description of the top power of the monodromy operator acting on middle cohomology of<jats:italic>X</jats:italic>in terms of the volume pairing on Г<jats:italic><jats:sub>X</jats:sub></jats:italic>.</jats:p>