Combing a hedgehog over a field

Ananyevskiy, Alexey; Levine, Marc

doi:10.2140/ant.2025.19.2409

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Combing a hedgehog over a field

Alexey Ananyevskiy and Marc Levine

Vol. 19 (2025), No. 12, 2409–2431

DOI: 10.2140/ant.2025.19.2409

Abstract

We investigate the question of the existence of a nonvanishing section of the tangent bundle on a smooth affine quadric hypersurface $Q^{o}$ over a given perfect field $k$ . If $Q^{o}$ admits a $k$ -rational point, we give a number of necessary and sufficient conditions for such existence. We apply these conditions in a number of examples, including the case of the algebraic $n$ -sphere over $k$ , $S_{k}^{n} \subset 𝔸_{k}^{n + 1}$ , defined by the equation $\sum_{i = 1}^{n + 1} x_{i}^{2} = 1$ .

Keywords

splitting vector bundles, Euler class, Chow–Witt ring, Grothendieck–Witt group

Mathematical Subject Classification

Primary: 11E81, 14J60

Milestones

Received: 11 February 2024

Revised: 5 July 2024

Accepted: 18 October 2024

Published: 20 October 2025

Authors

Alexey Ananyevskiy
Mathematisches Institut LMU München München Germany

Marc Levine
Fakultät Mathematik Universität Duisburg-Essen Essen Germany

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 19, No. 12, 2025