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Contraction centers in families of hyperkähler manifolds

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Abstract

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperkähler manifold M. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the Kähler cone. Cohomology classes which can possibly be orthogonal to a wall of the Kähler cone of some deformation of M are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. Any MBM curve can be contracted on an appropriate birational model of M, unless \(b_2(M) \leqslant 5\). When \(b_2(M)>5\), this property can be used as an alternative definition of an MBM class and an MBM curve. Using the results of Bakker and Lehn, we prove that the stratified diffeomorphism type of a contraction locus remains stable under all deformations for which these classes remains of type (1,1), unless the contracted variety has \(b_2\leqslant 4\). Moreover, these diffeomorphisms preserve the MBM curves, and induce biholomorphic maps on the contraction fibers, if they are normal.

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Notes

  1. By the Calabi–Yau theorem, this is the same as a hyperkähler manifold.

  2. The latter has only finitely many connected components, see [22]

  3. See e.g. [19, Proposition 13], for local finiteness issues.

  4. We shall use the term “birational contraction” in the non-algebraic setting too, meaning “bimeromorphic contraction”.

  5. [40, Theorem 1.6]; see also [38, 39].

  6. A “wall” shall always mean a face of maximal dimension, that is \(h^{1,1}-1\).

  7. In the non-projective case without assumptions on \(b_2\),this statement can be shown using the density of complex structures corresponding to projective manifolds, but we shall not need it.

  8. By a slight abuse of terminology, we say that “F can be contracted”.

  9. G. Mongardi has introduced the notion of wall divisors in [34], the two notions turned out to be equivalent.

  10. Note added in proof: it is in the last version, with a reference to the present paper: Corollary 5.9.

  11. The mixed formal-analytic setting is natural for the deformation theory of complex analytic varieties, such as in [23] or in [8]. The objects of the relevant category are complex varieties formally completed in some directions. To be more rigorous, an object \({{\mathcal {X}}}\) of this catefory is a pro-scheme obtained as an inverse limit of complex analytic spaces with the same reduction X. The formal deformation space of X is obtained as such an inverse limit, hence it belongs to this category. If, instead of complex analytic, we start in the category of algebraic (Noetherian) schemes, the same approach gives the usual formal schemes. The analytification of a formal deformation is a complex analytic space containing X as a closed complex analytic subvariety, with the formal completion along X identified with \({{\mathcal {X}}}\).

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Acknowledgements

We are grateful to Fedor Bogomolov for pointing out a potential error in an earlier version of this work, and to Jean–Pierre Demailly, Patrick Popescu, Lev Birbrair and Daniel Barlet for useful discussions. We are especially grateful to Fabrizio Catanese who explained to us the basics of Thom–Mather theory and gave the relevant reference, and to A. Rapagnetta and the anonymous referee of the superseded version of the paper for bringing Bakker and Lehn’s paper to our attention and insisting on its importance for our subject. Much gratitude is due to Grigori Papayanov for insightful comments and the reference in Mathoverflow [37]. The referee of the present version has done a considerable work pointing out our many inaccuracies, we thank him/her very much. Remark 1.7 is inspired by a conversation with Emanuele Macri.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Université Paris-11, Campus d’Orsay, Bâtiment 307, 91405, Orsay, France

    Ekaterina Amerik

  2. Laboratory of Algebraic Geometry, Department of Mathematics, National Research University HSE, 6 Usacheva Str., Moscow, Russia

    Ekaterina Amerik & Misha Verbitsky

  3. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, 110 Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil

    Misha Verbitsky

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Ekaterina Amerik: Partially supported by the HSE University Basic Research Program and by French-Brasilian Research Network.

Misha Verbitsky: Partially supported by the HSE University Basic Research Program, FAPERJ E-26/202.912/2018 and CNPq-Process 313608/2017-2.

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Amerik, E., Verbitsky, M. Contraction centers in families of hyperkähler manifolds. Sel. Math. New Ser. 27, 60 (2021). https://doi.org/10.1007/s00029-021-00677-8

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