Tropical Geometry and the Motivic Nearby Fiber

Details Tropical Geometry and the Motivic Nearby Fiber Tropical Geometry and the Motivic Nearby Fiber

2010-07-03

arxiv.org/abs/1007.0511v2

Mathematics

Algebraic Geometry

27 pages. Compositio Mathematica, to appear

Scientific paper

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 schon 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 schon hypersurfaces and smooth tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

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