The Moser isotopy for holomorphic symplectic and C-symplectic structures
[L’isotopie de Moser pour les structures symplectiques holomorphes et C-symplectiques]
Soldatenkov, Andrey1, 2 ; Verbitsky, Misha2, 3
1 Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkina Street, Moscow 119991 (Russia)
2 Instituto Nacional de Matemática Pura e Aplicada Estrada Dona Castorina, 110 Jardim Botânico, CEP 22460-320 Rio de Janeiro, RJ (Brasil)
3 Laboratory of Algebraic Geometry National Research University HSE Department of Mathematics, 6 Usacheva Street Moscow (Russia)
Annales de l'Institut Fourier, Online first, 18 p.

Une structure C-symplectique est une 2-forme à valeurs complexes qui est holomorphiquement symplectique pour une structure complexe appropriée. Nous prouvons un analogue du théorème d’isotopie de Moser pour les familles de structures C-symplectiques et donnons plusieurs applications de ce résultat. Nous prouvons que la déformation twistorielle dégénérée associée à une fibration lagrangienne holomorphe est localement triviale sur la base de cette fibration. Ceci est utilisé pour étendre plusieurs théorèmes sur les fibrations lagrangiennes, connues pour les variétés hyperkähler projectives, au cas non projectif. Nous présentons également de nouveaux exemples de variétés complexes non compactes avec une infinité de compactifications algébriques non birationnelles deux-à-deux.

A C-symplectic structure is a complex-valued 2-form which is holomorphically symplectic for an appropriate complex structure. We prove an analogue of Moser’s isotopy theorem for families of C-symplectic structures and list several applications of this result. We prove that the degenerate twistorial deformation associated to a holomorphic Lagrangian fibration is locally trivial over the base of this fibration. This is used to extend several theorems about Lagrangian fibrations, known for projective hyperkähler manifolds, to the non-projective case. We also exhibit new examples of non-compact complex manifolds with infinitely many pairwise non-birational algebraic compactifications.

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DOI : 10.5802/aif.3684
Classification : 14J32, 14D06, 53D05
Keywords: Symplectic manifold, Lagrangian fibration, twistor family
Mots-clés : Variété symplectique, fibration lagrangienne, famille twistorielle

Soldatenkov, Andrey 1, 2 ; Verbitsky, Misha 2, 3

1 Steklov Mathematical Institute of Russian Academy of Sciences 8 Gubkina Street, Moscow 119991 (Russia)
2 Instituto Nacional de Matemática Pura e Aplicada Estrada Dona Castorina, 110 Jardim Botânico, CEP 22460-320 Rio de Janeiro, RJ (Brasil)
3 Laboratory of Algebraic Geometry National Research University HSE Department of Mathematics, 6 Usacheva Street Moscow (Russia) 
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Soldatenkov, Andrey; Verbitsky, Misha. The Moser isotopy for holomorphic symplectic and C-symplectic structures. Annales de l'Institut Fourier, Online first, 18 p.

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