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Tropical geometry and the motivic nearby fiber

Part of: Families, fibrations

Published online by Cambridge University Press:  09 November 2011

Eric Katz  and
Alan Stapledon
Eric Katz
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (email: eekatz@math.uwaterloo.ca)
Alan Stapledon
Affiliation:
Department of Mathematics, University of British Columbia, BC, Canada V6T 1Z2 (email: astapldn@gmail.com)
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Abstract

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We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

Keywords

tropical geometrymonodromyHodge theory

MSC classification

Secondary: 14T05: Tropical geometry 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 269 - 294
Copyright
Copyright © Foundation Compositio Mathematica 2011

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