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Quadric bundles and hyperbolic equivalence

Alexander Kuznetsov

Geometry & Topology 28 (2024) 1287–1339

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 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.

quadric bundles, quadratic forms, Witt group, hyperbolic equivalence
Mathematical Subject Classification
Primary: 14D06
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
Alexander Kuznetsov
Algebraic Geometry Section
Steklov Mathematical Institute of Russian Academy of Sciences

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