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

Abstract

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 ${\Omega }^{\prime }:=\Omega +{\pi }^{\ast }\eta$ is again a holomorphically symplectic form, for another complex structure $I^{\prime }$, which is uniquely determined by ${\Omega }^{\prime }$. 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 $(M,I^{\prime },{\Omega }^{\prime })$ such that $s$ is a holomorphic section.

arXiv comment16 pages, Latex

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Cited by in corpus (4)

• Shafarevich-Tate groups of holomorphic Lagrangian fibrations
• Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids
• Haupt--Kapovich theorem revisited
• The Moser isotopy for holomorphic symplectic and C-symplectic structures