Rationality of Integral Cycles

Abstract

In this article we provide the sufficient condition for a Chow group element to be defined over the ground field. This is an integral version of the result known for $\mathbb{Z}/2$-coefficients. We also show that modulo 2 and degree $r$ cohomological invariants of algebraic varieties can not affect rationality of cycles of codimensions up to $2^{r-1}-2$.