Sections of Lagrangian fibrations on holomorphically symplectic manifolds and degenerate twistorial deformations

TL;DR

The paper studies how holomorphic Lagrangian fibrations on holomorphically symplectic manifolds can be deformed via degenerate twistorial deformations, preserving certain structures and allowing sections to become holomorphic.

Contribution

It introduces the concept of degenerate twistorial deformations, showing how they preserve the fibration structure and enable sections to be holomorphic.

Findings

Deformation of complex structures via closed (1,1)+(2,0) forms preserves holomorphic symplectic structure.

Existence of degenerate twistorial deformation making a given section holomorphic.

Preservation of the fibration's complex structure and fibers under deformation.

Abstract

Let $(M,I, \Omega)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $\pi:\; M \mapsto X$, and $\eta$ a closed form of Hodge type (1,1)+(2,0) on $X$. We prove that $\Omega':=\Omega+\pi^* \eta$ is again a holomorphically symplectic form, for another complex structure $I'$, which is uniquely determined by $\Omega'$. The corresponding deformation of complex structures is called "degenerate twistorial deformation". The map $\pi$ is holomorphic with respect to this new complex structure, and $X$ and the fibers of $\pi$ retain the same complex structure as before. Let $s$ be a smooth section of of $\pi$. We prove that there exists a degenerate twistorial deformation $(M,I', \Omega')$ such that $s$ is a holomorphic section.