Invariants of upper motives

Abstract

Let $H$ be a homology theory for algebraic varieties over a field $k$. To a complete $k$-variety $X$, one naturally attaches an ideal $\MH{X}$ of the coefficient ring $H(k)$. We show that, when $X$ is regular, this ideal depends only on the upper Chow motive of $X$. This generalises the classical results asserting that this ideal is a birational invariant of smooth varieties for particular choices of $H$, such as the Chow group. When $H$ is the Grothendieck group of coherent sheaves, we obtain a lower bound on the canonical dimension of varieties. When $H$ is the algebraic cobordism, we give a new proof of a theorem of Levine and Morel. Finally we discuss some splitting properties of geometrically unirational field extensions of small transcendence degree.