0
$\begingroup$

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.

Improve this question
$\endgroup$
6
  • 7
    $\begingroup$ 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 $\endgroup$
    – Francesco Polizzi
    Commented Oct 11, 2012 at 8:40
  • $\begingroup$ You are right. I think they belong to surfaces of general types and hence I am not familiar with and haven't seen them. $\endgroup$
    – user2013
    Commented Oct 11, 2012 at 8:55
  • 2
    $\begingroup$ Any abelian threefold also satisfies this. $\endgroup$
    – J.C. Ottem
    Commented Oct 11, 2012 at 14:14
  • 1
    $\begingroup$ They are not necessarily surfaces of general type! $\endgroup$
    – diverietti
    Commented Oct 11, 2012 at 14:22
  • 1
    $\begingroup$ Ok, ok. But as soon as $g(C_1)$ and $g(C_2)$ are at least $2$ they are :-) $\endgroup$
    – Francesco Polizzi
    Commented Oct 11, 2012 at 14:45

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .