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In this paper we obtain a stability theorem of generalized K¨ahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of K¨ahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact K¨ahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized K¨ahler structures on Fano surfaces and toric K¨ahler manifolds. In particular, we show that every nonzero holomorphic Poisson structure on a K¨ahler manifold induces deformations of non-trivial generalized K¨ahler structures.

In this paper, we establish an analytic foundation for a fully non-linear equation 2 1 = f on manifolds with metrics of positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equation without any condition on the sign of f and it is elliptic not only for f > 0 but also for f < 0. By defining a Yamabe constant Y2,1 with respect to this equation, we show that a manifold with metrics of positive scalar curvature admits a conformal metric of positive scalar curvature and positive 2-scalar curvature if and only if Y2,1 > 0. We give a complete solution for the corresponding Yamabe problem. Namely, let g0 be a positive scalar curvature metric, then in its conformal class there is a conformal metric with 2(g) = 1(g), for some constant . Using these analytic results, we give a rough classification of the space of manifolds with metrics of positive scalar curvature.

No abstract uploaded!

No abstract uploaded!