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May 18, 2013 at 22:18 answer added Barbara timeline score: 5
May 17, 2013 at 14:21 comment added naf Any canonically polarised surface of general type with the same Hilbert polynomial as a fake projective plane is a fake projective plane (and these are all rigid).
May 17, 2013 at 13:16 comment added diverietti An irreducible compact quotient $X$ of a polydisc has ample canonical bundle but if its dimension is greater than one then $H^1(X,T_X)=0$, so that $X$ is rigid! See [Y. Matsushima and G. Shimura, "On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes". Ann. of Math. (2) 78 1963 417–449].
May 17, 2013 at 12:27 comment added Jason Starr What makes you believe that there is an irreducible component of your moduli space of positive dimension? If you are asking whether or not that is true, I suggest editing your question a bit.
May 17, 2013 at 11:30 history asked Sepehr Hamshiri CC BY-SA 3.0