Invariants of upper motives

  • Olivier Haution

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Let HH be a homology theory for algebraic varieties over a field kk. To a complete kk-variety XX, one naturally attaches an ideal \MHX\MH{X} of the coefficient ring H(k)H(k). We show that, when XX is regular, this ideal depends only on the upper Chow motive of XX. This generalises the classical results asserting that this ideal is a birational invariant of smooth varieties for particular choices of HH, such as the Chow group. When HH is the Grothendieck group of coherent sheaves, we obtain a lower bound on the canonical dimension of varieties. When HH 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