TY - JOUR

T1 - Sections of Lagrangian fibrations on holomorphically symplectic manifolds and degenerate twistorial deformations

AU - Bogomolov, Fedor A.

AU - Déev, Rodion N.

AU - Verbitsky, Misha

N1 - Funding Information:
Fedor Bogomolov is partially supported by the HSE University Basic Research Program, Russian Academic Excellence Project ‘5-100’ and by EPSRC programme grant EP/M024830.Partially supported by the Russian Academic Excellence Project ‘5-100’, FAPERJ E-26/202.912/2018 and CNPq - Process 313608/2017-2.
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/8/27

Y1 - 2022/8/27

N2 - 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.

AB - 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.

KW - Complex Lagrangian fibrations

KW - Degenerate twistorial deformation

KW - Holomorphic symplectic manifolds

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U2 - 10.1016/j.aim.2022.108479

DO - 10.1016/j.aim.2022.108479

M3 - Article

AN - SCOPUS:85132228927

VL - 405

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 108479

ER -