On the cohomological classification of vector bundles on smooth real affine surfaces and threefolds

TL;DR

This paper investigates the cohomological classification of vector bundles on smooth real affine surfaces and threefolds, revealing parallels with algebraically closed fields and providing new examples of projective modules with specific properties.

Contribution

It extends the classification framework to real affine varieties under certain conditions and constructs novel examples of projective modules with trivial Chern classes that are not stably free.

Findings

Classification mirrors that over algebraically closed fields under certain conditions

Provides an efficient proof of Kucharz's theorem on Chern classes

First known example of a non-stably free projective module with trivial Chern classes over a 3-dimensional real affine algebra

Abstract

We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and in a coming joint paper with S. Banerjee and J. Fasel, under suitable cohomological assumptions on the real locus of such varieties, this classification mirrors the one obtained on algebraically closed base fields by Mohan Kumar and Murthy and by Asok and Fasel. Using an argument due to Fasel, we also give an efficient proof of a theorem of Kucharz characterising the triples of algebraic cycles that can be realised as the Chern classes of a rank $3$ bundle on a smooth real affine threefold. We further answer the questions left open by Kucharz; to our knowledge, we give the first instance of a projective module over a smooth affine $\mathbb{R}$-algebra of dimension $3$ with trivial Chern classes which is not stably free.