Quadric bundles and hyperbolic equivalence

TL;DR

This paper introduces hyperbolic equivalence for quadric bundles, showing they share key properties like Brauer data and birationality, and characterizes this equivalence over projective spaces with new cohomological methods.

Contribution

It defines hyperbolic equivalence for quadric bundles and quadratic forms, establishing their shared properties and providing a new classification criterion over projective spaces.

Findings

Hyperbolic equivalent quadric bundles share Brauer data.

They are birational over the base if they have the same dimension.

Quadratic forms over projective space are hyperbolic equivalent if their cokernel sheaves are isomorphic up to twist.

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 $\mathbb{P}^n$ 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.