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