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Stability of the symplectomorphism groups of rational surfaces

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Abstract

We apply Zhang’s almost Kähler Nakai–Moishezon theorem and Li–Zhang’s comparison of J-symplectic cones to establish a stability result for the symplectomorphism group of a rational 4-manifold M with Euler number up to 12. As a corollary, we also derive a stability result for the space of embedded symplectic balls in M. A noteworthy feature of our approach is that we systematically explore various spaces and groups associated to a symplectic cohomology class u rather than with a single symplectic form \(\omega \). To this end, we prove a weaker version of the tamed J-inflation procedures of D. McDuff and O. Buse that fixes a gap in their original formulations.

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Notes

  1. To our knowledge, the only known examples of cohomologous symplectic forms that are not diffeomorphic occur in dimensions \(\ge 6\). It may be possible that no such examples exist in dimension 4.

  2. Recently, P. Chakravarthy and M. Pinsonnault were able to restore the tamed version when \(Z\cdot Z\le 0\) ([8]).

  3. For a nice introduction to fibrant replacements and homotopy commuting diagrams, see [34] Section 7.6.

  4. Established by Buchdahl [6] and Lamari [18] in dimension 4, and by Demailly-Paun [10] in arbitrary dimension.

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Acknowledgements

We are grateful to Olguta Buse, Richard Hind, Weiwei Wu, and Weiyi Zhang for their helpful conversations. We thank Dusa McDuff for her careful reading of the first draft of this manuscript, very constructive comments, and for pointing out an oversight in Sect. 4.1. We also thank the anonymous referees for their careful reading and many constructive comments.

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Authors and Affiliations

  1. Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Sílvia Anjos

  2. Department of Mathematics, University of Dayton, Dayton, OH, 45469, USA

    Jun Li

  3. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Tian-Jun Li

  4. Department of Mathematics, University of Western Ontario, London, Canada

    Martin Pinsonnault

Authors
  1. Sílvia Anjos
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  2. Jun Li
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  3. Tian-Jun Li
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  4. Martin Pinsonnault
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Correspondence to Martin Pinsonnault.

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The authors contributed equally to this work.

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The first author is partially supported by FCT/Portugal through projects UID/MAT/04459/2020 and PTDC/MAT-PUR/29447/2017. The second author is supported by AMS-Simons Travel Grant. The third author is supported by NSF Grant. The fourth author is partially supported by NSERC Discovery Grant RGPIN-2020-06428.

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Anjos, S., Li, J., Li, TJ. et al. Stability of the symplectomorphism groups of rational surfaces. Math. Ann. 389, 85–119 (2024). https://doi.org/10.1007/s00208-023-02643-5

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