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As early as the birth of Seiberg Witten invariants [W1], the positive scalar curvature metrics on four dimensional manifolds have played a very important role. It was Witten [W] who first noticed that assuming b2 > 1 then one could easily derive the vanishing result of Seiberg Witten Invariants for those manifolds which carried the positive scalar curvature (psc) metrics. Combined with Taubes’ nonvanishing result [T2] for symplectic four manifolds, one could easily conclude that if a symplectic four manifold carries some psc metric, then its b2 must be equal to one. In addition the same vanishing result was used by P. Kronheimer and T. Mrowka [K.M.] in proving the Thom conjecture, and by C.H. Taubes in his “more constraints of symplectic forms on CP ” [T3]. Furthermore R. Friedman and J. Morgan [FM] systematically discussed which kind of Kahler manifold can carry the psc metrics. Based on surface classification theory, Friedman and Morgan could argue that the Kahler surfaces carrying psc metrics are either rational, rational ruled or irrational ruled. Recently, the same vanishing result has been used by Taubes [T4] to show that CP 2 has a unique symplectic structure. At this point it is interesting to ask about the possibilities of classifying symplectic four manifolds carrying psc metrics. One of the purposes of this paper is to address this possibility. Another goal of this paper is to address the question raised by Gompf [Gom] which states that:

Some New applications of General Wall Crossing Formula, Gompf’s Conjecture and its Applications