Topology of projective Tate-Shafarevich twists

TL;DR

This paper investigates how Tate-Shafarevich twists affect the topology and Hodge structures of projective Lagrangian fibrations, providing evidence for Sacca's conjecture under certain conditions.

Contribution

It proves that torsion Tate-Shafarevich twists preserve rational cohomology and Hodge structures, supporting Sacca's conjecture for smooth bases and sections.

Findings

Rational cohomology groups are isomorphic for original and twisted fibrations.

Hodge structures and pairings are preserved under torsion twists.

Sacca's conjecture is confirmed for smooth bases with sections using degenerate twistor deformations.

Abstract

A Tate-Shafarevich twist $X^\phi\to B$ 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. Sacc\`a 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 \'etale topology), we show that there is an isomorphism $H^\ast(X;\mathbb Q)\cong H^\ast(X^\phi;\mathbb Q)$ 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, and both the original fibration and its twist admit $C^\infty$-sections, we show Sacc\`a's conjecture using the theory of degenerate twistor deformations.