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A survey of symplectic 4-manifolds with \(b^ + =1\). (English) Zbl 0870.57023

The authors partially classify minimal symplectic 4-manifolds \(X\) with \(b^+=1\), where minimal means that \(X\) contains no symplectically embedded \(S^2\) with self-intersection number \(-1\). Let \(K=-c_1(TX,J)\) be the canonical class of \(X\) with respect to a compatible almost complex structure \(J\) on \(X\), then \(K^2=2\chi+3\sigma\). For \(K^2>0\), only a few examples are known, and the case \(K^2<0\) has been solved by Ai-Ko Liu [Some new applications of general wall-crossing formula, Gompf’s conjecture and its applications, Math. Res. Lett. 3, No. 5, 569-585 (1996)]. This paper classifies minimal symplectic 4-manifolds with \(b^+=1\) and \(K^2=0\), unless \(K\) is not a torsion class and \(b_1=2\).
The partial classification relies on Taubes’ result on Seiberg-Witten and Gromov invariants, and on a wall crossing formula due to T. J. Li and A. Liu [General wall crossing formula, Math. Res. Lett. 2, No. 6, 797-810 (1995)] and H. Ohta and K. Ono [Notes on symplectic 4-manifolds with \(b^+_2=1\). II., Int. J. Math. 7, No. 6, 755-770 (1996)], which gives for any \(\beta\in H^2(X,\mathbb Z)\) a formula for the difference \(\left|\operatorname{Gr}(\beta)+(-1)^m\operatorname{Gr}(K-\beta)\right|\) when \(b^+=1\) and \(b_1=2m\). The authors conclude by giving various examples.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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