On the Chow ring of certain rational cohomology tori

TL;DR

This paper investigates conditions under which an abelian cover between complex algebraic varieties induces isomorphisms in rational cohomology and Chow rings, revealing a deep connection between these algebraic invariants.

Contribution

It establishes an equivalence between isomorphisms in rational cohomology rings and Chow rings for abelian covers of varieties with quotient singularities.

Findings

Isomorphism in cohomology implies isomorphism in Chow rings.

Characterization of when pullback induces ring isomorphisms.

Connection between topological and algebraic cycle invariants.

Abstract

Let $f: X \rightarrow A$ be an abelian cover from a complex algebraic variety with quotient singularities to an abelian variety. We show that $f^*$ induces an isomorphism between the rational cohomology rings $H^\bullet(A, \mathbb{Q})$ and $H^\bullet(X, \mathbb{Q})$ if and only if $f^*$ induces an isomorphism between the Chow rings with rational coefficients $\mathrm{CH}^\bullet(A)_{\mathbb{Q}}$ and $\mathrm{CH}^\bullet(X)_{\mathbb{Q}}$.