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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 ℐ ( 𝒳 0 , h | 𝒳 0 ) , any section of m ( K 𝒳 0 + L ) extends to 𝒳  ; in other...

L ² h -domains of holomorphy and the Bergman kernel

Peter Pflug, Włodzimierz Zwonek (2002)

Studia Mathematica

We give a characterization of L ² h -domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.

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