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

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.