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Stein complements in compact Kähler manifolds

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Abstract

Given a projective or compact Kähler manifold X and a (smooth) hypersurface Y, we study conditions under which \(X {\setminus } Y\) could be Stein. We apply this in particular to the case when X is the projectivization of the so-called canonical extension of the tangent bundle \(T_M\) of a projective manifold M with Y being the projectivization of \(T_M\) itself.

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Notes

  1. We say that a complex manifold U is affine if there exists an affine variety \(U_\textrm{alg}\) such that U is biholomorphic to the analytification of \(U_\textrm{alg}\).

  2. We thank John Ottem for explaining to us how to construct many more examples based on his earlier work: in [52, Ex. 5.5] he constructs a hypersurface \(X \subset \mathbb {P}^1 \times \mathbb {P}^3\) that admits a pseudoautomorphism \(f: X \dashrightarrow X\), in fact f flips a curve C. Given a sufficiently general divisor D of class \({\mathcal O}_{\mathbb {P}^1 \times \mathbb {P}^3}(1,1)\) that contains C, one can check that its strict transform \(f_* D=:Y \subset X\) is smooth. By construction Y is not nef, yet \(X {\setminus } Y \simeq X {\setminus } D\) is affine.

  3. The statement is for projective fourfolds, but the proof is local around the image of the exceptional locus, so it also works in the Kähler case.

  4. In particular one has \({{\,\textrm{Pic}\,}}(U) \simeq {{\,\textrm{Pic}\,}}(C) \simeq \mathbb {Z}\).

  5. Note that since S is smooth, the exceptional divisor is Cartier.

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Acknowledgements

For very useful discussions we thank B. Claudon, F. Gounelas, D. Greb, A. Sarti and M. Zaidenberg. The first-named author thanks the Institut Universitaire de France and the A.N.R. project Foliage (ANR-16-CE40-0008) for providing excellent working conditions. We thank the referee for very pertinent comments.

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  1. Université Côte d’Azur, CNRS, LJAD, Nice, France

    Andreas Höring

  2. Institut universitaire de France, Paris, France

    Andreas Höring

  3. Mathematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Thomas Peternell

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Höring, A., Peternell, T. Stein complements in compact Kähler manifolds. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02801-3

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