Invariants of upper motives

TL;DR

This paper explores how certain algebraic invariants derived from homology theories depend only on the upper Chow motive of a variety, leading to new insights into birational invariants, canonical dimension bounds, and cobordism.

Contribution

It generalizes classical birational invariance results to a broader class of homology theories and establishes new bounds and properties related to motives and field extensions.

Findings

The ideal associated with a regular variety depends only on its upper Chow motive.

Provides a lower bound on the canonical dimension using the Grothendieck group of coherent sheaves.

Offers a new proof of a theorem in algebraic cobordism by Levine and Morel.

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 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.