Monodromy Filtrations and the Topology of Tropical Varieties

By David Helm, Eric Katz. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2008

48 citations recorded by the source corpus.

Abstract 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 $\left( X \right)$ by a topological space ${{\Gamma }_{X}}$ and 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 $X$ which constrain the topology of Trop $\left( X \right)$ . 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 ${{\Gamma }_{X}}$ .

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5d6c87cef438abe3084ba6782b70fc801a2ae2b8