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Oct 20, 2010 at 18:02 comment added Misha Verbitsky You are right - thanks. Sorry for the confusion.
Oct 19, 2010 at 22:57 comment added Mike Usher If J is the standard complex structure on CP2, isn't -J a complex structure with respect to which $-\omega$ is Kahler? Yau's theorem in the form that I know it (see pnas.org/content/74/5/1798.full.pdf) says that any complex surface homotopy equivalent to CP2 is biholomorphic to it. In the case of the complex structure -J, the biholomorphism is just $[z_0:z_1:z_2]\mapsto [\bar{z_0}:\bar{z_1}:\bar{z_2}]$.
Oct 19, 2010 at 19:54 history answered Misha Verbitsky CC BY-SA 2.5