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

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.