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Are there any complex surface or threefold $X$ with $$ \dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2? $$ I am asking this because I don't know any example while there are complex curves of genus greater than one. I guess that there are no such example. If so, could someone kindly explain why? Any counter example is also welcome.

Edit My question turns out to be a silly question. Please ignore this.

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Take $X=C_1×C_2$ where $C_1$ and $C_2$ are smooth curves. Then $h^0(X,K_X)=g(C_1)g(C_2)$ which can be as big as you wish –  Francesco Polizzi Oct 11 '12 at 8:40
    
You are right. I think they belong to surfaces of general types and hence I am not familiar with and haven't seen them. –  Fermion Oct 11 '12 at 8:55
    
Indeed they are surfaces of general type. You can start to study them in Beauville's book or in Barth-Peters-Van de Ven –  Francesco Polizzi Oct 11 '12 at 9:05
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Any abelian threefold also satisfies this. –  J.C. Ottem Oct 11 '12 at 14:14
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Ok, ok. But as soon as $g(C_1)$ and $g(C_2)$ are at least $2$ they are :-) –  Francesco Polizzi Oct 11 '12 at 14:45
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