Chern numbers of certain Lefschetz fibrations
HTML articles powered by AMS MathViewer
- by András K. Stipsicz PDF
- Proc. Amer. Math. Soc. 128 (2000), 1845-1851 Request permission
Erratum: Proc. Amer. Math. Soc. 128 (2000), 2833-2834.
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.References
- W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
- S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR 710056
- S. K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Differential Geom. 44 (1996), no. 4, 666–705. MR 1438190
- Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066
- Ronald Fintushel and Ronald J. Stern, Immersed spheres in $4$-manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995), no. 2, 145–157. MR 1349567
- Ronald Fintushel and Ronald J. Stern, Rational blowdowns of smooth $4$-manifolds, J. Differential Geom. 46 (1997), no. 2, 181–235. MR 1484044
- R. Gompf, Lecture at MSRI, 1997.
- R. Gompf and A. Stipsicz 4-Manifolds and Kirby Calculus, book in preparation.
- A. Kas, On the handlebody decomposition associated to a Lefschetz fibration, Pacific J. Math. 89 (1980), no. 1, 89–104. MR 596919
- Dieter Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds (after C. H. Taubes), Astérisque 241 (1997), Exp. No. 812, 4, 195–220. Séminaire Bourbaki, Vol. 1995/96. MR 1472540
- P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), no. 6, 797–808. MR 1306022, DOI 10.4310/MRL.1994.v1.n6.a14
- Yukio Matsumoto, Diffeomorphism types of elliptic surfaces, Topology 25 (1986), no. 4, 549–563. MR 862439, DOI 10.1016/0040-9383(86)90031-5
- John W. Morgan, Zoltán Szabó, and Clifford Henry Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996), no. 4, 706–788. MR 1438191
- Ulf Persson, Chern invariants of surfaces of general type, Compositio Math. 43 (1981), no. 1, 3–58. MR 631426
- Ulf Persson, Chris Peters, and Gang Xiao, Geography of spin surfaces, Topology 35 (1996), no. 4, 845–862. MR 1404912, DOI 10.1016/0040-9383(95)00046-1
- András Stipsicz, A note on the geography of symplectic manifolds, Turkish J. Math. 20 (1996), no. 1, 135–139. MR 1392669
- A. Stipsicz, Simply connected symplectic 4-manifolds with positive signature, preprint.
- Clifford Henry Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), no. 6, 809–822. MR 1306023, DOI 10.4310/MRL.1994.v1.n6.a15
- Clifford H. Taubes, $\textrm {SW}\Rightarrow \textrm {Gr}$: from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996), no. 3, 845–918. MR 1362874, DOI 10.1090/S0894-0347-96-00211-1
- Clifford Henry Taubes, Counting pseudo-holomorphic submanifolds in dimension $4$, J. Differential Geom. 44 (1996), no. 4, 818–893. MR 1438194
- Edward Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769–796. MR 1306021, DOI 10.4310/MRL.1994.v1.n6.a13
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
- MR Author ID: 346634
- Email: stipsicz@cs.elte.hu
- 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
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1641113