# CSASC 2013

9-13 June 2013
Koper, Slovenia
UTC timezone
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# $L^2$-extension of cohomology classes from a non-smooth divisor

Presented by Dr. Jean RUPPENTHAL
Type: Oral presentation
Track: Several Complex Variables

## Content

I will report on a joint project with Elizabeth Wulcan and Ha*kan Samuelsson-Kalm. Recently, Berndtsson generalized the Ohsawa-Takegoshi-Manivel $L^2$-extension theorem for holomorphic functions to the case of $\overline{\partial}$-closed forms of higher degree. He proved an $L^2$-extension theorem for $\overline{\partial}$-closed forms from a smooth divisor in a compact manifold with good (i.e. universal) $L^2$-estimates under very mild natural positivity assumptions. Similar work had been done before by Manivel, Demailly, Koziarz and others. We are now interested in the extension of $L^2$-cohomology classes also from a non-smooth divisor $Y$ (in a compact manifold $X$). In that case, one first needs to address the question what kind of forms shall be extended and how forms on $X$ shall be linked to the given form on $Y$. For this purpose, we develop an adjunction formula for the Grauert-Riemenschneider canonical sheaf of the singular variety $Y$. This formula can be used to set up a bimeromorphically invariant form of the extension problem. By a resolution of singularities, we can thus reduce the problem to the smooth case (treated by Berndtsson), and obtain an $L^2$-extension theorem under quite mild positivity assumptions (but without universal estimates).

## Place

Location: Koper, Slovenia

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