Splitting vector bundles over real algebraic varieties
Aravind Asok, Jean Fasel, Samuel Lerbet
TL;DR
The paper develops a real-algebraic analogue of Murthy’s splitting results using motivic obstruction theory, focusing on corank $0$ and $1$ vector bundles on smooth real affine varieties. It builds a real-realization bridge between algebraic obstructions (Euler class in Chow–Witt/Grothendieck–Witt theory) and topological obstructions (Euler class and Steenrod operations) via real cycle maps and twisted Rost–Schmid theory. The authors produce striking negative results—constructing explicit counterexamples on open subsets of $\mathbb{P}^3_{\mathbb{R}}$ where topological triviality does not ensure algebraic splitting—and simultaneously offer positive results in special geometries (no real points or open real loci) where the topological data suffice for splitting. They also introduce explicit rank-$2$ vector bundles on open subsets of $\mathbb{P}^3_{\mathbb{R}}$ with prescribed Euler/Borel invariants, highlighting intricate interactions between algebraic and topological obstructions and revealing a nuanced landscape for splitting in the real setting.
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.