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Quadric bundles and hyperbolic equivalence

Alexander Kuznetsov

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.

Quadric bundles and hyperbolic equivalence

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 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.

Paper Structure

This paper contains 14 sections, 39 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Quadric bundles and hyperbolic equivalence
  3. Hyperbolic reduction
  4. Hyperbolic extension
  5. Hyperbolic equivalence
  6. VHC resolutions on projective spaces
  7. Complexes of split bundles
  8. VHC resolutions and uniqueness
  9. Existence of VHC resolutions
  10. Hyperbolic equivalence on projective spaces
  11. Reminder on symmetric sheaves
  12. Elementary modifications
  13. Modification theorem
  14. Proof of Theorem \ref{['thm:he-intro']} and Corollary \ref{['corollary:invariants']}

Key Result

Proposition \oldthetheorem

Let $({\mathcal{E}},q)$ and $({\mathcal{E}}',q')$ be hyperbolic equivalent generically non-degenerate quadratic forms over $X$ and let $Q \to X$ and $Q' \to X$ be the corresponding hyperbolic equivalent quadric bundles, where $X$ is a scheme over a field ${\mathsf{k}}$ of characteristic not equal to

Theorems & Definitions (94)

  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 84 more