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

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