Birational Chow-K\"unneth decompositions

TL;DR

This paper introduces the concept of birational Chow-K"unneth decompositions, explores their invariance properties, and demonstrates their existence for specific classes of algebraic varieties, refining the understanding of motives in algebraic geometry.

Contribution

It defines birational Chow-K"unneth decompositions, proves their stable birational invariance, and establishes their existence for Jacobians, Hilbert schemes on K3 surfaces, and certain rational cubic varieties.

Findings

Existence of birational Chow-K"unneth decompositions for Jacobians.

Existence for Hilbert schemes of points on K3 surfaces.

Existence for varieties of lines on stably rational cubic threefolds and fourfolds.

Abstract

We study the notion of a birational Chow-K\"unneth decomposition, which is essentially a decomposition of the integral birational motive of a variety. The existence of a birational Chow-K\"unneth decomposition is stably birationally invariant and this notion refines the Chow theoretical decomposition of the diagonal. We show that a birational Chow-K\"unneth decompostion exists for the following varieties: (a) Jacobian variety; (b) Hilbert scheme of points on a $K3$ surface and (c) The variety of lines on a stably rational cubic threefold or a stably rational cubic fourfold.