Smash nilpotent cycles on products of curves

TL;DR

This paper proves Voevodsky's conjecture that numerical and smash equivalence coincide for 1-cycles on products of curves, showing that numerically trivial 1-cycles on abelian varieties are smash nilpotent.

Contribution

It establishes the conjecture for one-dimensional cycles on arbitrary products of curves, advancing understanding of cycle equivalences.

Findings

Numerical and smash equivalence coincide for 1-cycles on products of curves.

Numerically trivial 1-cycles on abelian varieties are smash nilpotent.

Voevodsky's conjecture is confirmed in this specific case.

Abstract

Voevodsky has conjectured that numerical and smash equivalence coincide on a smooth projective variety. We prove the conjecture for one dimensional cycles on an arbitrary product of curves. As a consequence we get that numerically trivial 1-cycles on an abelian variety are smash nilpotent.