Quadric bundles and hyperbolic equivalence

Kuznetsov, Alexander

doi:10.2140/gt.2024.28.1287

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Quadric bundles and hyperbolic equivalence

Alexander Kuznetsov

Geometry & Topology 28 (2024) 1287–1339

DOI: 10.2140/gt.2024.28.1287

Abstract

We introduce the notion of hyperbolic equivalence for quadric bundles and quadratic forms on vector bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties.

Furthermore, when the base is a projective space we show that two quadratic forms are hyperbolic equivalent if and only if their cokernel sheaves are isomorphic up to twist, their fibers over a fixed point of the base are Witt equivalent, and, in some cases, certain quadratic forms on intermediate cohomology groups of the underlying vector bundles are Witt equivalent. For this we show that any quadratic form over $ℙ^{n}$ is hyperbolic equivalent to a quadratic form whose underlying vector bundle has many cohomology vanishings; this class of bundles, called VLC bundles in the paper, is interesting by itself.

Keywords

quadric bundles, quadratic forms, Witt group, hyperbolic equivalence

Mathematical Subject Classification

Primary: 14D06

References

Publication

Received: 20 November 2021

Revised: 12 August 2022

Accepted: 16 September 2022

Published: 10 May 2024

Proposed: Marc Levine

Seconded: Mark Gross, Richard P Thomas

Authors

Alexander Kuznetsov
Algebraic Geometry Section Steklov Mathematical Institute of Russian Academy of Sciences Moscow Russia

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Volume 28, issue 3 (2024)