Complex manifold $X$ with $\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2$ - MathOverflow most recent 30 from http://mathoverflow.net2013-06-19T20:52:13Zhttp://mathoverflow.net/feeds/question/109362http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/109362/complex-manifold-x-with-dim-h0x-omega-dim-mathbbc-x2Complex manifold $X$ with $\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2$Fermion2012-10-11T08:30:50Z2012-10-11T08:53:41Z
<p>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. </p>
<p><strong>Edit</strong>
My question turns out to be a silly question. Please ignore this. </p>