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Monodromy Filtrations and the Topology of Tropical Varieties

Published online by Cambridge University Press:  20 November 2018

David Helm  and
Eric Katz
David Helm
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712- 0257, USA email: eekatz@math.utexas.edu
Eric Katz
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712- 0257, USA email: eekatz@math.utexas.edu
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Abstract

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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 ${{\Gamma }_{X}}$.

Keywords

14T0514D06
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 4 , 01 August 2012 , pp. 845 - 868
Copyright
Copyright © Canadian Mathematical Society 2012

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