Splitting vector bundles over real algebraic varieties

TL;DR

This paper investigates the splitting problem of vector bundles over smooth affine real varieties, exploring topological obstructions and extending known algebraic results to the real setting using motivic techniques.

Contribution

It extends Murthy's splitting theorem to real algebraic varieties and analyzes the obstructions to splitting in corank 0 and 1 cases, highlighting the complexity over the reals.

Findings

Topological obstructions dominate in corank 0 splitting.

Motivic techniques extend Murthy's theorem to real varieties.

The real case presents more complex obstructions than the algebraically closed case.

Abstract

Suppose $X$ is a smooth affine real variety and $\mathscr{E}$ is a vector bundle over $X$. We analyze the problem of splitting off a free rank one summand from $\mathscr{E}$ in corank $0$ and $1$. The problem in corank $0$ can be viewed as the search for a real analog of Murthy's celebrating splitting theorem in the algebraically closed case: to wit, beyond the vanishing of the top Chern class in Chow theory, are the obstructions to splitting ``purely topological''? In a sense, the answer in this case is yes, and we give a proof, using motivic techniques, of a mild extension of the results of Bhatwadekar-Sridharan and Bhatwadekar-Das-Mandal. In corank $1$, in the algebraically closed situation, Murthy's splitting conjecture (now a theorem in characteristic $0$) predicts that the vanishing of the top Chern class in Chow theory is the only obstruction to splitting off a free rank $1$ summand, and we can search for a suitable ``real'' analog of this assertion. We observe that several natural guesses for a ``real'' analog of Murthy's splitting conjecture cannot be true, i.e., that the situation over the real numbers is rather complicated.