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Monodromy Filtrations and the Topology of Tropical Varieties

Published:2011-09-22
Printed: Aug 2012
• David Helm,
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA
• Eric Katz,
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA
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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 $\operatorname{Trop}(X)$ 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 $\operatorname{Trop}(X)$. 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 $\Gamma_X$.
 MSC Classifications: 14T05 - Tropical geometry [See also 12K10, 14M25, 14N10, 52B20] 14D06 - Fibrations, degenerations