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Topology of projective Tate-Shafarevich twists

David Zhiyuan Bai

Abstract

A Tate-Shafarevich twist $X^φ\to B$ of a fibration $X\to B$ modifies it by a $1$-cocycle of flows of vector fields relative to the base, locally in the analytic topology. Saccà conjectured that the total spaces of two projective Lagrangian fibrations related by such a twist are deformation-equivalent. Assuming that the class of the twist is torsion (which is often equivalent to the twist being realizable in the étale topology), we show that there is an isomorphism $H^\ast(X;\mathbb Q)\cong H^\ast(X^φ;\mathbb Q)$ of graded vector spaces that respects (1) the Hodge structures and (2) the Hodge-Riemann pairing. Consequently, the rational Beauville-Bogomolov-Fujiki lattices of these two spaces are Hodge-similar. Assuming further that $B$ is smooth, that the fibers of the fibrations are reduced outside of a locus of codimension $2$ in $B$, and that the integral homology classes of a general fiber in both spaces are primitive, we show Saccà's conjecture using a recent result of Bogomolov-Kamenova-Verbitsky. We also show that Beauville-Mukai systems for primitive classes satisfy the last condition.

Topology of projective Tate-Shafarevich twists

Abstract

A Tate-Shafarevich twist of a fibration modifies it by a -cocycle of flows of vector fields relative to the base, locally in the analytic topology. Saccà conjectured that the total spaces of two projective Lagrangian fibrations related by such a twist are deformation-equivalent. Assuming that the class of the twist is torsion (which is often equivalent to the twist being realizable in the étale topology), we show that there is an isomorphism of graded vector spaces that respects (1) the Hodge structures and (2) the Hodge-Riemann pairing. Consequently, the rational Beauville-Bogomolov-Fujiki lattices of these two spaces are Hodge-similar. Assuming further that is smooth, that the fibers of the fibrations are reduced outside of a locus of codimension in , and that the integral homology classes of a general fiber in both spaces are primitive, we show Saccà's conjecture using a recent result of Bogomolov-Kamenova-Verbitsky. We also show that Beauville-Mukai systems for primitive classes satisfy the last condition.
Paper Structure (21 sections, 39 theorems, 51 equations)

This paper contains 21 sections, 39 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. A conjecture of Saccà
  3. Cohomology groups
  4. Lagrangian fibrations
  5. Fibrations of Bogomolov-Kamenova-Verbitsky type
  6. Tate-Shafarevich twists and the decomposition theorem
  7. Recollections on Tate-Shafarevich twists
  8. Perverse sheaves
  9. Cohomology rings of twists
  10. Good twists
  11. The associated graded ring under a twist
  12. The relative Lefchetz action
  13. Lagrangian fibrations
  14. Recollections on Lagrangian fibrations
  15. Topological invariants under twists
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose $X\to B$ is a projective fibration and $X^\phi\to B$ is a Tate-Shafarevich twist of it. Suppose in addition that $X^\phi$ is also projective. Then we have $H^k(X;\mathbb Q)\cong H^k(X^\phi;\mathbb Q)$ as $\mathbb Q$-Hodge structures for each $k$.

Theorems & Definitions (83)

  • Conjecture 1
  • Conjecture 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Definition 1.1
  • ...and 73 more