Monodromy filtrations and the topology of tropical varieties

DOI10.4153/CJM-2011-067-9zbMATH Open1312.14145arXiv0804.3651OpenAlexW2058671753MaRDI QIDQ2903308

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Publication date: 8 August 2012

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Abstract: 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 $Gamm{a}_X$ 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 $Gamm{a}_X$.

Full work available at URL: https://arxiv.org/abs/0804.3651

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