Tropical Geometry and the Motivic Nearby Fiber

TL;DR

This paper introduces a new motivic invariant derived from tropical geometry that captures complex geometric information about subvarieties of algebraic tori, linking tropicalization with Hodge theory and providing combinatorial formulas.

Contribution

It constructs the tropical motivic nearby fiber invariant and relates it to Hodge-Deligne polynomials and Euler characteristics, offering new combinatorial expressions and formulas.

Findings

Invariants specialize to Hodge-Deligne polynomials in schön cases

Provides combinatorial formulas for hypersurfaces and tropical varieties

Deduces Euler characteristic formulas for degenerations

Abstract

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.