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On an \(L^2\) extension theorem from log-canonical centres with log-canonical measures

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Abstract

With a view to prove an Ohsawa–Takegoshi type \(L^2\) extension theorem with \(L^2\) estimates given with respect to the log-canonical (lc) measures, a sequence of measures each supported on lc centres of specific codimension defined via multiplier ideal sheaves, this article is aiming at providing evidence and possible means to prove the \(L^2\) estimates on compact Kähler manifolds X. A holomorphic family of \(L^2\) norms on the ambient space X is introduced which is shown to “deform holomorphically” to an \(L^2\) norm with respect to an lc-measure. Moreover, the latter norm is shown to be invariant under a certain normalisation which leads to a “non-universal” \(L^2\) estimate on compact X. Explicit examples on \(\mathbb {P}^3\) with detailed computation are presented to verify the expected \(L^2\) estimates for extensions from lc centres of various codimensions and to provide hint for the proof of the estimates in general.

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Notes

  1. Strictly speaking, since \(\varphi _L\) is allowed to have poles along any lc centres of (X, S) and thus sections of can vanish along those centres, the precise description should be “the space of holomorphic sections of with an extra vanishing order along .”

  2. Notice that the inclusions need not be strict in general. Moreover, the equality is due to the assumption that for \(m \in [0,1]\) “jumps” along S exactly when \(m=1\).

  3. Analytic continuation of the residue function across 0 is suggested already by the study of residue currents in [2, 9] and [1]. See [5, §1.4] for a discussion.

  4. Here, f is abused to mean its image under the map .

  5. The notation is chosen by mimicking the reduced Planck constant \(\hbar = \frac{h}{2\pi }\). It is typeset with the code .

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  1. Department of Mathematics, Pusan National University, Busan, 46241, South Korea

    Tsz On Mario Chan

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  1. Tsz On Mario Chan
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Correspondence to Tsz On Mario Chan.

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The author would like to thank Young-Jun Choi for his support and encouragement on publishing this work. This work was supported by the National Research Foundation (NRF) of Korea grant funded by the Korea Government (No. 2018R1C1B3005963).

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Chan, T.O.M. On an \(L^2\) extension theorem from log-canonical centres with log-canonical measures. Math. Z. 301, 1695–1717 (2022). https://doi.org/10.1007/s00209-021-02890-9

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  • DOI: https://doi.org/10.1007/s00209-021-02890-9

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