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Notes on symplectic 4-manifolds with \(b_ 2^ + = 1\). II. (English) Zbl 0873.53020

[For part I, see the preceding review Zbl 0873.53019.]
The authors apply the Taubes Theorems [C. H. Taubes, Math. Res. Lett. 1, 809-822 (1994; Zbl 0853.57019); Math. Res. Lett. 2, 9-13 (1995; Zbl 0854.57019); Math. Res. Lett. 2, 221-238 (1995; Zbl 0854.57020); J. Am. Math. Soc. 9, 845-918 (1996; Zbl 0867.53025)] to study closed smooth 4-manifolds \(X\) having a symplectic structure and a metric of positive scalar curvature. The main result of the paper states that such an \(X\) must be diffeomorphic to either the complex projective plane or a ruled surface up to blow-up and down. In particular, if \(c_1(K^{-1}_X)= \lambda[\omega]\) with a positive number \(\lambda\), then \(X\) must be diffeomorphic to a Del Pezzo surface.
[See also the following review Zbl 0873.53021].

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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