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

Advances in Mathematics. 2022. Vol. 405. Article 108479.

Bogomolov F. A., Déev R., Verbitsky M.

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

Keywords: holomorphic symplectic manifolds, Degenerate twistorial deformation, Complex Lagrangian fibrations

Publication based on the results of: Algebraic varieties from the point of view of derived categories, homological algebra, special holonomy metrics and classical algebraic geometry (2022)