5
$\begingroup$

I need to compute the euler characteristic $\chi(\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1})$ of the structure sheaf of $\mathbb{P}^1 \times \mathbb{P}^1$ (the product of the projective complex line with itself). I suppose that to do so, I could use Kunneth formula and a bit of Hodge theory, but I'm not familiar with this subject.

Can somebody help me? Thanks!

$\endgroup$
1
  • $\begingroup$ Do you know Serre duality and the canonical bundle? $\endgroup$
    – Matt
    Mar 27, 2014 at 19:30

1 Answer 1

6
$\begingroup$

The Euler-Poincaré characteristic $\chi(V)=\chi(V,\mathcal O_V)$ of the structure sheaf of a smooth projective variety is called the Hirzebruch arithmetic genus and it is multiplicative : $$\chi(V\times W)=\chi(V)\times \chi(W)$$ This immediately solves your problem: $\chi(\mathbb P^1\times \mathbb P^1)=\chi(\mathbb P^1)\times \chi(\mathbb P^1)=1\times 1=1$ .

Proofs can be found in Hartshorne: Exercise I, 7.2 page54 and Exercise III, 5.3 page230.
And here is a related post.

Edit: a variant
The Segre morphism embeds the variety $\mathbb P^1\times \mathbb P^1$ as a quadric $Q\subset\mathbb P^3$.
The Severi arithmetic genus of $Q$ is given by $p_a(Q)=\binom{2-1}{3}=0$ (Hartshorne, Chapter I, Exercise 7.2, page54) .
Since $\chi(Q)=1+(-1)^{dim Q}\cdot p_a(Q)$, we get again $\chi(Q)=1+(-1)^2 \cdot 0=1$

$\endgroup$

This site is temporarily in read-only mode and not accepting new answers.

Not the answer you're looking for? Browse other questions tagged .