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Some new applications of general wall crossing formula, Gompf’s conjecture and its applications. (English) Zbl 0872.57025
In 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)
##### Keywords:
4-manifolds; Kähler surface; Riemann-Roch formula
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