Haution, Olivier

Invariants of upper motives

Doc. Math., J. DMV 18, 1555-1572 (2013)


Summary: 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.

Mathematics Subject Classification



upper motives, canonical dimension, Grothendieck group, algebraic cobordism