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Chern numbers of certain Lefschetz fibrations


Author: András K. Stipsicz
Journal: Proc. Amer. Math. Soc. 128 (2000), 1845-1851
MSC (1991): Primary 57R99, 57M12
DOI: https://doi.org/10.1090/S0002-9939-99-05172-2
Published electronically: October 18, 1999
Erratum: Proc. Amer. Math. Soc. 128 (2000), 2833-2834.
MathSciNet review: 1641113
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Abstract: We address the geography problem of relatively minimal Lefschetz fibrations over surfaces of nonzero genus and prove that if the fiber-genus of the fibration is positive, then $0\leq c_1^2\leq 5c_2$ (equivalently, $0\leq c_1^2 \leq 10 \chi _h $) holds for those symplectic 4-manifolds. A useful characterization of minimality of such symplectic 4-manifolds is also proved.


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  • [BPV] W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik, Spinger-Verlag Berlin, 1984. MR 86c:32026
  • [D1] S. Donaldson, An application of gauge theory to four dimensional topology, J. Diff. Geom. 18 (1983), 279-315. MR 85c:57015
  • [D2] S. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996), 666-705. MR 98h:53045
  • [F] M. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982), 357-453. MR 84b:57006
  • [FS1] R. Fintushel and R. Stern, Immersed spheres in 4-manifolds and the immersed Thom-conjecture, Turkish J. Math. 19 (1995), 27-40. MR 96j:57036
  • [FS2] R. Fintushel and R. Stern, Rational blowdowns of smooth 4-manifolds, J. Diff. Geom. 46 (1997), 181-235. MR 98j:57047
  • [G] R. Gompf, Lecture at MSRI, 1997.
  • [GS] R. Gompf and A. Stipsicz 4-Manifolds and Kirby Calculus, book in preparation.
  • [Ka] A. Kas, On the handlebody decomposition associated to a Lefschetz fibration Pacific J. Math. 89 (1980), 89-104. MR 82f:57012
  • [Ko] D. Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds [after C. H. Taubes], Seminare Bourbaki 48 ème année (1995-96) n$^o$ 812, 195-220. MR 98h:57057
  • [KM] P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Letters 1 (1994) 797-808. MR 96a:57073
  • [M] Y. Matsumoto, Diffeomorphism types of elliptic surfaces, Topology 25 (1986), 544-563. MR 88b:32061
  • [MSzT] J. Morgan, Z. Szabó and C. Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Diff. Geom. 44 (1996) 818-893. MR 97m:57052
  • [P] U. Persson, Chern invariants of surfaces of general type, Compositio Math. 43 (1981), 3-58. MR 83b:14012
  • [PPX] U. Persson, C. Peters and G. Xiao, Geography of spin surfaces, Topology 35 (1996), 845-862. MR 98h:14046
  • [S1] A. Stipsicz, A note on the geography of symplectic manifolds, Turkish J. Math. 20 (1996), 135-139. MR 97m:57035
  • [S2] A. Stipsicz, Simply connected symplectic 4-manifolds with positive signature, preprint.
  • [T1] C. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Letters 1 (1994), 809-822. MR 95j:57039
  • [T2] C. Taubes, $SW \to Gr$: From the Seiberg-Witten equations to pseudo-holomorphic curves, Journal of the AMS 9 (1996), 845-918. MR 97a:57033
  • [T3] C. Taubes, Counting pseudo-holomorphic submanifolds in dimension 4, J. Diff. Geom. 44 (1996), 818-893. MR 97k:58029
  • [W] E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769-796. MR 96d:57035

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Additional Information

András K. Stipsicz
Affiliation: Department of Analysis, ELTE TTK, 1088. Múzeum krt. 6-8., Budapest, Hungary
Address at time of publication: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: stipsicz@cs.elte.hu

DOI: https://doi.org/10.1090/S0002-9939-99-05172-2
Keywords: 4-manifolds, Lefschetz fibrations, geography problem
Received by editor(s): June 29, 1998
Received by editor(s) in revised form: July 14, 1998
Published electronically: October 18, 1999
Additional Notes: Supported by the Magyary Zoltán Foundation and OTKA
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society

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