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

Let (M,I,Ω) be a holomorphically symplectic manifold equipped with holomorphic Lagrangian fibration π:M↦X, and η closed form of Hodge type (1,1)+(2,0) on X. We prove that Ω′:=Ω+π⁎η is again form, for another complex structure I′, which uniquely determined by Ω′. The corresponding deformation structures called “degenerate twistorial deformation”. map π respect to this new structure, X the fibers retain same as before. s smooth section π. there exists degenerate (M,I′,Ω′) such section.