Invariants of upper motives
Let be a homology theory for algebraic varieties over a field . To a complete -variety , one naturally attaches an ideal of the coefficient ring . We show that, when is regular, this ideal depends only on the upper Chow motive of . This generalises the classical results asserting that this ideal is a birational invariant of smooth varieties for particular choices of , such as the Chow group. When is the Grothendieck group of coherent sheaves, we obtain a lower bound on the canonical dimension of varieties. When 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.
Cite this article
Olivier Haution, Invariants of upper motives. Doc. Math. 18 (2013), pp. 1555–1572DOI 10.4171/DM/436