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On \(L^2\) extension from singular hypersurfaces

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Abstract

In \(L^2\) extension theorems from a singular hypersurface in a complex manifold, important roles are played by certain measures such as the Ohsawa measure which determine when a given function can be extended. We show that the singularity of the Ohsawa measure can be identified in terms of singularity of pairs from algebraic geometry. Using this, we give an analytic proof of the inversion of adjunction in this setting. Then these considerations enable us to compare various positive and negative results on \(L^2\) extension from singular hypersurfaces. In particular, we generalize a recent negative result of Guan and Li which places limitations on strengthening such \(L^2\) extension by employing a less singular measure in the place of the Ohsawa measure.

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Notes

  1. Although not necessary for this paper, we remark that another aspect of comparison could be on the curvature conditions in \(L^2\) extension theorems, especially for compact X.

  2. For the purpose of this paper, we recall here only the case of Y being irreducible of codimension 1. For the full statements, see the original papers.

  3. It does not matter for our purpose even if \(B_+\) and \(B_-\) share components.

  4. Some considerations based on [13] in Sect. 3 of this paper are used in [26].

  5. Also see [11, Thm. 2.5] for a related more general statement.

  6. In particular, here we do not mean a proof which uses \(L^2\) extension just somewhere in the argument.

  7. In fact, it is likely that similar ideas may lead to answering Question B in general. An \(L^2\) extension theorem for this generality is already available from [27], see Remark 5.3.

  8. There was an earlier definition of the different from [28, 47] reducing to the dimension 2 case by taking hyperplane sections.

  9. In particular, independence of the choice of \(\widetilde{g}\) is built into the definition of the Ohsawa measure and its existence is treated in an increasing order of generalities.

  10. See [3, (1.3)] for more information on this property.

  11. using the additive notation for tensor products of line bundles.

  12. In terms of [13], ‘\(\psi \) has log canonical singularities along Y’ which is the same as the pair being lc (i.e. log canonical) in the standard terminology.

  13. Here we use the log resolution in the analytic category, cf. [2, 25, 51]. In particular, we use [51, Thm. 2.0.2, 2.0.3] as \(U'\) is taken to be relatively compact from Step 1.

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Acknowledgements

We would like to thank R. Berman, J. Kollár and T. Ohsawa for their helpful comments regarding the subject of this paper. We also thank Mario Chan for pointing out some inaccuracies in an earlier version regarding adjoint ideals (cf. [23, 7]). We would like to thank anonymous referees for useful comments. This research was supported by Basic Science Research Programs through National Research Foundation of Korea funded by the Ministry of Education: (2018R1D1A1B07049683) for D.K. and (2020R1A6A3A01099387) for H.S. Also H.S. was supported by the Institute for Basic Science (IBS-R032-D1-2022-a00).

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  1. Hoseob Seo

    Present address: Center for Complex Geometry, Institute for Basic Science (IBS), 55 Expo-ro, Yuseong-gu, Daejeon, 34126, Korea

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  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea

    Dano Kim

  2. Research Institute of Mathematics, Seoul National University, Seoul, 08826, Korea

    Hoseob Seo

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Kim, D., Seo, H. On \(L^2\) extension from singular hypersurfaces. Math. Z. 303, 89 (2023). https://doi.org/10.1007/s00209-023-03248-z

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