Invariants of upper motives

  • Olivier Haution

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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.

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Olivier Haution, Invariants of upper motives. Doc. Math. 18 (2013), pp. 1555–1572

DOI 10.4171/DM/436