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Extension of Holomorphic Functions and Cohomology Classes from Non Reduced Analytic Subvarieties

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Geometric Complex Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 246))

Abstract

The goal of this survey is to describe some recent results concerning the \(L^{2}\) extension of holomorphic sections or cohomology classes with values in vector bundles satisfying weak semi-positivity properties. The results presented here are generalized versions of the Ohsawa–Takegoshi extension theorem, and borrow many techniques from the long series of papers by T. Ohsawa. The recent achievement that we want to point out is that the surjectivity property holds true for restriction morphisms to non necessarily reduced subvarieties, provided these are defined as zero varieties of multiplier ideal sheaves. The new idea involved to approach the existence problem is to make use of \(L^{2}\) approximation in the Bochner-Kodaira technique. The extension results hold under curvature conditions that look pretty optimal. However, a major unsolved problem is to obtain natural (and hopefully best possible) \(L^{2}\) estimates for the extension in the case of non reduced subvarieties—the case when Y has singularities or several irreducible components is also a substantial issue.

J.-P. Demailly—In honor of Professor Kang-Tae Kim on the occasion of his 60th birthday.

Supported by the European Research Council project “Algebraic and Kähler Geometr” (ERC-ALKAGE, grant No. 670846 from September 2015).

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Acknowledgements

Supported by the European Research Council project “Algebraic and Kähler Geometry” (ERC-ALKAGE, grant No. 670846 from September 2015).

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Authors and Affiliations

  1. Institut Fourier, Université Grenoble Alpes, 100 rue des Maths, 38610, Gières, France

    Jean-Pierre Demailly

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  1. Jean-Pierre Demailly
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Correspondence to Jean-Pierre Demailly .

Editor information

Editors and Affiliations

  1. Department of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  2. Department of Mathematics, Pusan National University, Busan, Korea (Republic of)

    Hong Rae Cho

  3. Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, Korea (Republic of)

    Sung Yeon Kim

  4. Department of Mathematics, Gyeongsang National University, Jinju, Korea (Republic of)

    Kang-Hyurk Lee

  5. Department of Mathematics, Kyung Hee University, Seoul, Korea (Republic of)

    Jong-Do Park

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Demailly, JP. (2018). Extension of Holomorphic Functions and Cohomology Classes from Non Reduced Analytic Subvarieties. In: Byun, J., Cho, H., Kim, S., Lee, KH., Park, JD. (eds) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol 246. Springer, Singapore. https://doi.org/10.1007/978-981-13-1672-2_8

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