Torsion codimension 2 cycles on supersingular abelian varieties

TL;DR

This paper proves that torsion codimension 2 cycles on supersingular abelian varieties are algebraically trivial and establishes the equivalence of homological and algebraic equivalence for these cycles over algebraically closed finite fields.

Contribution

It demonstrates the algebraic triviality of torsion codimension 2 cycles and equates homological and algebraic equivalence on supersingular abelian varieties.

Findings

Torsion codimension 2 cycles are algebraically equivalent to zero.

Homological and algebraic equivalence coincide for codimension 2 cycles.

Results hold over algebraic closure of finite fields.

Abstract

We prove that torsion codimension 2 algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension 2 on supersingular abelian varieties over the algebraic closure of finite fields.