Birational invariants and A^1-connectedness

TL;DR

This paper explores the connection between A^1-homotopy theory and birational geometry, demonstrating how A^1-invariants relate to classical topology, motives, and cohomological properties of algebraic varieties.

Contribution

It establishes that the zeroth A^1-homology sheaf is a birational invariant for smooth proper varieties over infinite fields and links these sheaves to cohomological invariants like unramified étale cohomology.

Findings

Zeroth A^1-homology sheaf is a birational invariant for smooth proper varieties over infinite fields.

A^1-connectedness can be characterized by the vanishing of unramified invariants.

Vanishing results for cohomology of A^1-connected varieties are derived.

Abstract

We study some aspects of the relationship between A^1-homotopy theory and birational geometry. We study the so-called A^1-singular chain complex and zeroth A^1-homology sheaf of smooth algebraic varieties over a field k. We exhibit some ways in which these objects are similar to their counterparts in classical topology and similar to their motivic counterparts (the (Voevodsky) motive and zeroth Suslin homology sheaf). We show that if k is infinite the zeroth A^1-homology sheaf is a birational invariant of smooth proper varieties, and we explain how these sheaves control various cohomological invariants, e.g., unramified \'etale cohomology. In particular, we deduce a number of vanishing results for cohomology of A^1-connected varieties. Finally, we give a partial converse to these vanishing statements by giving a characterization of A^1-connectedness by means of vanishing of unramified invariants.