$K$-Equivalence and Integral Cohomology

TL;DR

This paper introduces an integral Hodge polynomial that captures the integral cohomology of smooth projective varieties and proves that $K$-equivalent varieties have isomorphic integral cohomology groups.

Contribution

It develops an integral version of the Hodge polynomial and establishes its invariance under $K$-equivalence for smooth projective varieties.

Findings

Integral Hodge polynomial extends to the Grothendieck ring

$K$-equivalent varieties have isomorphic integral cohomology

The polynomial encodes integral cohomology of varieties

Abstract

We introduce an integral version of the Hodge polynomial, which encodes the integral cohomology of smooth projective varieties. We prove it extends to a function which is well-defined on the Grothendieck ring of varieties and we obtain as a consequence that $K$-equivalent smooth projective varieties have isomorphic integral cohomology groups.