Some new applications of general wall crossing formula, Gompf’s conjecture and its applications.

*(English)*Zbl 0872.57025In this short article, the author settled the following conjecture of Gompf: “If \(M \) is a minimal symplectic four manifold with \(c_1 (K)^2<0\), then it is irrational ruled”. Before this time, C. H. Taubes’ paper [Math. Res. Lett. 2, 221-238 (1995; Zbl 0854.57020)] had already established the conjecture in the case when \(b^+_2>1\). Thus the main contribution of the author was in the remaining case \(b^+_2 =1\), for which the author and T.-J. Li had obtained a formula for the Seiberg-Witten invariant in [Math. Res. Lett. 2, No. 6, 797-810 (1995; Zbl 0871.57017)]. Another important ingredient of the proof is Taubes’ Theorem “\(SW = GR\)”. As pointed out by the author, the conjecture can be regarded as an extension of Enriques’ Theorem in algebraic geometry “If \(M\) is a minimal Kähler surface with \(c_1 (K)^2 <0\), then it is irrational ruled”, where the use of Riemann-Roch formula is replaced by a suitable modification of “\(SW=GR\)”.

Reviewer: R.Lee (New Haven)

##### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |