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.

I am not familiar with varieties defined over other fields than complex numbers. There may be counter-examples in such cases.

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.