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↦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 π. We prove that there exists a degenerate twistorial deformation (M,I^′,Ω^′) such that s is a holomorphic section.