A finiteness theorem for algebraic cycles

TL;DR

This paper proves that certain subspaces of algebraic cycles on powers of a smooth projective variety are finite-dimensional if the variety's motive is a summand of an abelian variety's motive, extending understanding of algebraic cycles.

Contribution

It establishes a finiteness theorem for algebraic cycles generated by specific operations on powers of a variety with a motive related to an abelian variety.

Findings

Finite-dimensionality of generated cycle subspaces under given conditions

Extension of finiteness results to varieties with motives as summands of abelian varieties

Provides new tools for studying algebraic cycles and motives

Abstract

Let X be a smooth projective variety. Starting with a finite set of cycles on powers X^m of X, we consider the Q-vector subspaces of the Q-linear Chow groups of the X^m obtained by iterating the algebraic operations and pullback and push forward along those morphisms X^l -> X^m for which each component X^l -> X is a projection. It is shown that these Q-vector subspaces are all finite-dimensional, provided that the Q-linear Chow motive of X is a direct summand of that of an abelian variety.