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Analytic disks with boundaries in a maximal real submanifold of 𝐂 2

Franc Forstneric (1987)

Annales de l'institut Fourier

Let M be a two dimensional totally real submanifold of class C 2 in C 2 . A continuous map F : Δ C 2 of the closed unit disk Δ C into C 2 that is holomorphic on the open disk Δ and maps its boundary b Δ into M is called an analytic disk with boundary in M . Given an initial immersed analytic disk F 0 with boundary in M , we describe the existence and behavior of analytic disks near F 0 with boundaries in small perturbations of M in terms of the homology class of the closed curve F 0 ( b Δ ) in M . We also prove a regularity theorem...

Deformations of coherent foliations on a compact normal space

Geneviève Pourcin (1987)

Annales de l'institut Fourier

An universal analytic structure is construted on the set of (singular) holomorphic foliations on a normal compact space. Such a foliation is by definition a coherent subsheaf of the holomorphic tangent sheaf stable by the Lie-bracket

Invariance for multiples of the twisted canonical bundle

Benoît Claudon (2007)

Annales de l’institut Fourier

Let 𝒳 Δ a smooth projective family and ( L , h ) a pseudo-effective line bundle on 𝒳 (i.e. with a non-negative curvature current Θ h L ). In its works on invariance of plurigenera, Y.-T. Siu was interested in extending sections of m K 𝒳 0 + L (defined over the central fiber of the family 𝒳 0 ) to sections of m K 𝒳 + L . In this article we consider the following problem: to extend sections of m ( K 𝒳 + L ) . More precisely, we show the following result: assuming the triviality of the multiplier ideal sheaf ( 𝒳 0 , h | 𝒳 0 ) , any section of m ( K 𝒳 0 + L ) extends to 𝒳  ; in other...

Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

Daniel Greb, Christian Lehn, Sönke Rollenske (2013)

Annales scientifiques de l'École Normale Supérieure

Let  X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L . We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair ( X , L ) , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian...

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