Sheaf cohomology and double covers

Question

Let $\pi:X\rightarrow Y$ be a double cover of complex varieties and take $L$ holomorphic line bundle on $Y$.

I read that there are the isomorphisms

1) $H^p(X,\mathcal{O}_X)\simeq H^p(Y,\pi_*\mathcal{O}_X)$

2) $H^p(Y,\mathcal{O}_Y\oplus L)\simeq H^p(Y,\mathcal{O}_Y)\oplus H^p(Y,L)$

How can i prove them?

The first question follows from the exactness of $\pi_\ast$ (this would be true e.g. for any affine map $\pi$). The second is just because the functors $H^p(Y,-)$ are additive. This is very general. — Sam Gunningham, Commented Jul 6, 2013 at 5:08
Additional note: Since $\pi$ is affine, a cover of $Y$ by open affines pulls back to a cover of $X$ by open affines. Consequently, the Cech complex of $\pi_* \mathcal O_X$ with respect to a cover by open affines pulls back to a Cech complex of $\mathcal O_X$. This can be used for the first isomorphism. — Charles Staats, Commented Jul 6, 2013 at 5:28
I suppose what you really want to know is why is $H^p(X,\mathcal{O}_X)\cong H^p(Y,\mathcal{O}_Y)\oplus H^p(Y, L)$ for some line bundle $L$? Answer: because of (1), (2) and (3) $\pi_*\mathcal{O}_X = \mathcal{O}_Y\oplus L$, where $L$ is the (-1)-eigenbundle of the leftside under the Galois action. — Donu Arapura, Commented Jul 6, 2013 at 15:03

Marci · Accepted Answer · 2013-07-06 05:10:03Z

1

Hints: I would show that $R^i\pi_*O_X=0$ for $i>0$, and then I would use adjunction.

(2) holds in general. Basically, you need to show that $\Gamma(Y,A\oplus B)=\Gamma(Y,A)\oplus \Gamma(Y,B)$.

answered Jul 6, 2013 at 5:10

Marci

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