Refined Tropicalizations for Sch\"on Subvarieties of Tori

TL;DR

This paper introduces a refined tropicalization for sch"on subvarieties of algebraic tori, connecting tropical geometry with classical invariants and providing new tools for understanding their intersection theory.

Contribution

It develops a relative refined $hi_y$-genus for sch"on subvarieties, linking it to tropical cycles and enabling the recovery of classical invariants via tropical methods.

Findings

The refined $hi_y$-genus turns generic intersections into products.

Lattice point counting recovers the ordinary $hi_y$-genus.

Refined tropicalization specializes to the unrefined version when $y=0$.

Abstract

We introduce a relative refined $\chi_y$-genus for sch\"on subvarieties of algebraic tori. These are rational functions of degree minus the codimension with coefficients in the ring of lattice polytopes. We prove that the relative refined $\chi_y$ turns sufficiently generic intersections into products, and that we can recover the ordinary $\chi_y$-genus by counting lattice points. Applying the tropical Chern character to the relative refined $\chi_y$-genus we obtain a refined tropicalization which is a tropical cycle having rational functions with $\mathbb Q$-coefficients as weights. We prove that the top-dimensional component of the refined tropicalization specializes to the unrefined tropicalization up to sign when setting $y=0$ and show that we can recover the $\chi_y$-genus by integrating the refined tropicalization with respect to a Todd measure.