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$K$-Equivalence and Integral Cohomology

Matthew Satriano, Evan Sundbo

TL;DR

The work constructs an integral invariant, $H_{ ext{vir}, obreak obreak Z}$, generalizing the Hodge polynomial to capture integral cohomology data within the Grothendieck ring of varieties. By encoding both free and torsion components via carefully designed generators and relations, and proving multiplicativity and blow-up behavior, the authors show that $K$-equivalent smooth projective varieties have isomorphic integral cohomology groups. The results extend to non-smooth/projective cases and imply independence of crepant resolutions under certain singularity hypotheses, with potential for stringy and stacky generalizations. These findings provide a robust algebraic framework for comparing integral cohomology across birational models and offer new tools for understanding topological invariants in algebraic geometry.

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.

$K$-Equivalence and Integral Cohomology

TL;DR

The work constructs an integral invariant, , generalizing the Hodge polynomial to capture integral cohomology data within the Grothendieck ring of varieties. By encoding both free and torsion components via carefully designed generators and relations, and proving multiplicativity and blow-up behavior, the authors show that -equivalent smooth projective varieties have isomorphic integral cohomology groups. The results extend to non-smooth/projective cases and imply independence of crepant resolutions under certain singularity hypotheses, with potential for stringy and stacky generalizations. These findings provide a robust algebraic framework for comparing integral cohomology across birational models and offer new tools for understanding topological invariants in algebraic geometry.

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 -equivalent smooth projective varieties have isomorphic integral cohomology groups.
Paper Structure (5 sections, 9 theorems, 39 equations)

This paper contains 5 sections, 9 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Integral Virtual Hodge Functions
  3. Integral cohomology under $K$-equivalance
  4. $H_\mathbb{Z}$ as a function on the Grothendieck ring of varieties
  5. Proof of Theorem \ref{['thm:HZ-k-equivalence']}

Key Result

Theorem 1.1

If $X$ and $X'$ are $K$-equivalent smooth projective varieties over $\mathbb{C}$ then $H^i(X,\mathbb{Z})\simeq H^i(X',\mathbb{Z})$ for all $i$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • Definition 3.1
  • Proposition 3.2
  • ...and 12 more