The Moser isotopy for holomorphic symplectic and C-symplectic structures

TL;DR

This paper extends Moser's isotopy theorem to C-symplectic structures, enabling new insights into Lagrangian fibrations and providing examples of complex manifolds with diverse algebraic compactifications.

Contribution

It proves an analogue of Moser's isotopy theorem for C-symplectic structures and applies it to extend results on Lagrangian fibrations beyond projective hyperk"ahler manifolds.

Findings

Degenerate twistorial deformation is locally trivial over the base.

Extended theorems on Lagrangian fibrations to non-projective cases.

Constructed new examples of complex manifolds with infinitely many algebraic compactifications.

Abstract

A C-symplectic structure is a complex-valued 2-form which is holomorphically symplectic for an appropriate complex structure. We prove an analogue of Moser's isotopy theorem for families of C-symplectic structures and list several applications of this result. We prove that the degenerate twistorial deformation associated to a holomorphic Lagrangian fibration is locally trivial over the base of this fibration. This is used to extend several theorems about Lagrangian fibrations, known for projective hyperk\"ahler manifolds, to the non-projective case. We also exhibit new examples of non-compact complex manifolds with infinitely many pairwise non-birational algebraic compactifications.