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Towards the Classification of Class VII Surfaces

  • Andrei Teleman
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 21)

Abstract

This article follows the ideas of my talk “A duality theorem for instanton moduli spaces on class VII surfaces” given at the workshop “INdAM Meeting: Complex and Symplectic Geometry” which took place in June 2016 in Cortona; it gives a survey on my recent results concerning the existence of a cycle of curves on class VII surfaces with small b2. The main problem in the classification of class VII surfaces is the existence of holomorphic curves. My approach uses a combination of techniques coming from complex geometry and gauge theory. The main object used in the proofs is a moduli space \(\mathcal{M}\) of polystable holomorphic bundles on the considered surface. This moduli space is identified with an instanton moduli space via the Kobayashi-Hitchin correspondence. The existence (non-existence) of curves on the base surface is related to geometric properties of the corresponding moduli space.

Notes

Acknowledgements

The author thanks Daniele Angella, Paolo De Bartolomeis, Costantino Medori, and Adriano Tomassini, the organisers of the “INdAM Meeting: Complex and Symplectic Geometry”, for the invitation to give a talk and to submit an article for the proceedings of the conference. The author is grateful to the referee for careful reading of the paper, and for his useful suggestions and comments.

References

  1. 1.
    W. Barth, K. Hulek, Ch. Peters, A. Van de Ven, Compact Complex Surfaces (Springer, New York, 2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    N. Buchdahl, Hermitian-Einstein connections and stable vector bundles over compact complex surfaces. Math. Ann. 280, 625–648 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    N. Buchdahl, A Nakai-Moishezon criterion for non-Kähler surfaces. Ann. Inst. Fourier 50, 1533–1538 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. Dloussky, Structure des surfaces de Kato. Mémoires de la Société Mathématique de France 14, 1–120 (1984)CrossRefzbMATHGoogle Scholar
  5. 5.
    G. Dloussky, Une construction élémentaire des surfaces d’Inoue-Hirzebruch. Math. Ann. 280(4), 663–682 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    G. Dloussky, On surfaces of class VII0+ with numerically anti-canonical divisor. Am. J. Math. 128(3), 639–670 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    G. Dloussky, K. Oeljeklaus, Vector fields and foliations on compact surfaces of class VII0. Ann. Inst. Fourier 49(5), 1503–1545 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    G. Dloussky, K. Oeljeklaus, M. Toma, Surfaces de la classe VII0 admettant un champ de vecteurs. Comment. Math. Helvet. 76, 640–664 (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    G. Dloussky, K. Oeljeklaus, M. Toma, Class VII0 surfaces with b 2 curves. Tohoku Math. J. 55, 283–309 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differ. Geom. 26, 397–428 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    S. Donaldson, P. Kronheimer, The Geometry of Four-Manifolds (Oxford University Press, Oxford, 1990)zbMATHGoogle Scholar
  12. 12.
    I. Enoki, Surfaces of class VII0 with curves. Tohoku Math. J. 33, 453–492 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. Gauduchon, Sur la 1-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267, 495–518 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    L. Gerstein, Basic Quadratic Forms. Graduate Studies in Mathematics, vol. 90 (AMS, Providence, 2008)Google Scholar
  15. 15.
    M. Kato, Compact complex manifolds containing “global” spherical shells. Proc. Jpn. Acad. 53(1), 15–16 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. Kato, Compact complex manifolds containing “global” spherical shells. I, in Proceedings of the International Symposium on Algebraic Geometry (Kyoto University, Kyoto, 1977) (Kinokuniya Book Store, Tokyo, 1978), pp. 45–84Google Scholar
  17. 17.
    M. Kato, On a certain class of nonalgebraic non-Kähler compact complex manifolds, in Recent Progress of Algebraic Geometry in Japan. North-Holland Mathematics Studies, vol. 73 (North-Holland, Amsterdam, 1983), pp. 28–50Google Scholar
  18. 18.
    A. Lamari, Le cône kählérien d’une surface. J. Math. Pures Appl. 78, 249–263 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. Li, S.T. Yau, Hermitian Yang-Mills connections on non-Kähler manifolds, in Mathematical Aspects of String Theory (San Diego, CA, 1986). Advanced Series in Mathematical Physics, vol. 1 (World Scientific Publishing, Singapore, 1987), pp. 560–573Google Scholar
  20. 20.
    M. Lübke, A. Teleman, The Kobayashi-Hitchin Correspondence (World Scientific Publishing Co., Singapore, 1995)CrossRefzbMATHGoogle Scholar
  21. 21.
    I. Nakamura, On surfaces of class VII0 surfaces with curves. Invent. Math. 78, 393–443 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    I. Nakamura, Towards classification of non-Kählerian surfaces. Sugaku Expositions 2(2), 209–229 (1989)zbMATHGoogle Scholar
  23. 23.
    I. Nakamura, On surfaces of class VII0 surfaces with curves II. Tohoku Math. J. 42(4), 475–516 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    K. Oeljeklaus, M. Toma, Logarithmic moduli spaces for surfaces of class VII. Math. Ann. 341(2), 323–345 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. Teleman, Projectively flat surfaces and Bogomolov’s theorem on class VII0 surfaces. Int. J. Math. 5(2), 253–264 (1994)CrossRefzbMATHGoogle Scholar
  26. 26.
    A. Teleman, Donaldson theory on non-Kählerian surfaces and class VII surfaces with b 2 = 1. Invent. Math. 162, 493–521 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    A. Teleman, The pseudo-effective cone of a non-Kählerian surface and applications. Math. Ann. 335(4), 965–989 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    A. Teleman, Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds. Geom. Topol. 11, 1681–1730 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A. Teleman, Instantons and holomorphic curves on class VII surfaces. Ann. Math. 172, 1749–1804 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    A. Teleman, A variation formula for the determinant line bundle. Compact subspaces of moduli spaces of stable bundles over class VII surfaces, in Geometry, Analysis and Probability: In Honor of Jean-Michel Bismut, ed. by F. Labourie, Y.L. Jan, X. Ma, W. Zhang. Progress in Mathematics, vol. 310 (2017), pp. 217–243Google Scholar
  31. 31.
    A. Teleman, Instanton moduli spaces on non-Kahlerian surfaces. Holomorphic models around the reduction loci. J. Geom. Phys. 91, 66–87 (2015)zbMATHGoogle Scholar
  32. 32.
    A. Teleman, Donaldson theory in non-Kählerian geometry, in Modern Geometry: A Celebration of the Work of Simon Donaldson, Proceedings of Symposia in Pure Mathematics, ed. by V. Munoz, I. Smith, R. Thomas (2017, to appear)Google Scholar

Copyright information

© Springer International Publishing AG, a part of Springer Nature 2017

Authors and Affiliations

  1. 1.Aix Marseille Université, CNRS, Centrale MarseilleMarseilleFrance

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