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L² Approaches in Several Complex Variables pp 205–237Cite as

L2 Approaches to Holomorphic Foliations

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Abstract

Results on the L 2 \(\bar {\partial }\)-cohomology groups are applied to holomorphic foliations with an emphasis on the cases with Levi flat hypersurfaces as stable sets. Nonexistence theorems are discussed for holomorphic foliations of codimension one on compact Kähler manifolds under some assumptions on geometric properties of the complement of stable sets. For the special cases such as \(\mathbb {C}\mathbb {P}^n\), complex tori and Hopf surfaces, nonexistence, reduction and classification theorems will be proved. Closely related materials have been already discussed in Sect. 2.4., e.g. Theorem 2.79.

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Notes

  1. 1.

    These example can be counted also as trivial ones. BEDFORD once warned the author not to talk about them anymore.

  2. 2.

    The proof of Theorem 0.2 in [Oh’06] is incomplete.

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Ohsawa, T. (2018). L2 Approaches to Holomorphic Foliations. In: L² Approaches in Several Complex Variables. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56852-0_5

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