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The geography of simply-connected symplectic manifolds. (English) Zbl 1039.53097
Summary: By using the Seiberg-Witten invariants we show that the region under the Noether line in the lattice domain \(\mathbb Z\times \mathbb Z\) is covered by minimal, simply connected, symplectic 4-manifolds.

53D35 Global theory of symplectic and contact manifolds
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
53D05 Symplectic manifolds (general theory)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
57R17 Symplectic and contact topology in high or arbitrary dimension
57R57 Applications of global analysis to structures on manifolds
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[1] D. Auckly: Surgery, knots and the Seiberg-Witten equations. Preprint.
[2] D. Auckly: Homotopy K3 surfaces and gluing results in Seiberg-Witten theory. Preprint. · Zbl 0870.57022
[3] W. Barth, C. Peter and A. Van de Ven: Compact Complex Surfaces. Ergebnisse der Mathematik. Springer-Verlag, Berlin, 1984. · Zbl 0718.14023
[4] M. S. Cho and Y. S. Cho: Genus minimizing in symplectic 4-manifolds. Chinese Ann. Math. Ser. B 21 (2000), 115-120. · Zbl 0967.53053
[5] Y. S. Cho: Finite group actions on the moduli space of self-dual connections I. Trans. Amer. Math. Soc. 323 (1991), 233-261. · Zbl 0724.57013
[6] Y. S. Cho: Cyclic group actions in gauge theory. Differential Geom. Appl. 6 (1996), 87-99. · Zbl 0846.58051
[7] Y. S. Cho: Seiberg-Witten invariants on non-symplectic 4-manifolds. Osaka J. Math. 34 (1997), 169-173. · Zbl 0882.57013
[8] Y. S. Cho: Finite group actions and Gromov-Witten invariants. Preprint.
[9] R. Fintushel and R. Stern: Surgery in cusp neighborhoods and the geography of Irreducible 4-manifolds. Invent. Math. 117 (1994), 455-523. · Zbl 0843.57021
[10] R. Fintushel and R. Stern: Rational blowdowns of smooth 4-manifolds. Preprint. · Zbl 0896.57022
[11] R. Gompf: A new construction of symplectic manifolds. Ann. Math. 142 (1995), 527-595. · Zbl 0849.53027
[12] P. Kronheimer and T. Mrowka: The genus of embedded surfaces in the projective plane. Math. Res. Lett. (1994), 794-808. · Zbl 0851.57023
[13] U. Persson: Chern invariants of surfaces of general type. Composito Math. 43 (1981), 3-58. · Zbl 0479.14018
[14] D. Salamon: Spin geometry and Seiberg-Witten invariants. University of Warwick (1995).
[15] A. Stipsicz: A note on the geography of symplectic manifolds. Preprint. · Zbl 0876.57039
[16] C. H. Taubes: The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett. 1 (1994), 809-822. · Zbl 0853.57019
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