Blow up and Higher Direct Image

Question

Let $X$ and $Y$ be smooth projective varieties and $Y \subset X$. Let $\pi : \widetilde{X} \longrightarrow X$ be the blowing up of $X$ along $Y$ with exceptional divisor $E$. Here (Direct Image by a Blow up) it was shown that $$\pi_{*}\mathcal{O}_{\widetilde{X}}(-nE) = I_{Y/X}^{n}$$ for $n \geq 1$.

From short exact sequence $$0 \longrightarrow \mathcal{O}_{\widetilde{X}}(-E) \longrightarrow \mathcal{O}_{\widetilde{X}} \longrightarrow \mathcal{O}_{E} \longrightarrow 0$$ we get a long exact sequence

\begin{align*} 0 & \longrightarrow \pi_{*}\mathcal{O}_{\widetilde{X}}(-E) \longrightarrow \pi_{*}\mathcal{O}_{\widetilde{X}} \longrightarrow \pi_{*}\mathcal{O}_{E} \longrightarrow \\ & \longrightarrow R^{1}\pi_{*}\mathcal{O}_{\widetilde{X}}(-E) \longrightarrow R^{1}\pi_{*}\mathcal{O}_{\widetilde{X}} \longrightarrow R^{1}\pi_{*}\mathcal{O}_{E} \longrightarrow \\ & \longrightarrow R^{2}\pi_{*}\mathcal{O}_{\widetilde{X}}(-E) \longrightarrow R^{2}\pi_{*}\mathcal{O}_{\widetilde{X}} \longrightarrow R^{2}\pi_{*}\mathcal{O}_{E} \longrightarrow \cdots \tag{$*$} \end{align*}

By this answer ( Direct image of structure sheaf under blow-up along non-singular subvariety) we have $$R^{i}\pi_{*}\mathcal{O}_{\widetilde{X}} = 0 \tag{$**$}$$ for all $i > 0$.

Also we have

1) $\pi_{*}\mathcal{O}_{\widetilde{X}} = \mathcal{O}_{X}$,

2) $\pi_{*}\mathcal{O}_{E} = \mathcal{O}_{Y}$

Thus for items (1) and (2) above, we have that $\mathcal{O}_{X} \longrightarrow \mathcal{O}_{Y}$ is surjective and, therefore $R^{1}\pi_{*}\mathcal{O}_{\widetilde{X}}(-E) = 0$.

Now, using $(**)$ in $(*)$ we get the following isomorphism $$R^{1}\pi_{*}\mathcal{O}_{E} \longrightarrow R^{2}\pi_{*}\mathcal{O}_{\widetilde{X}}(-E)$$

Question What would it be $R^{j}\pi_{*}\mathcal{O}_{E}$? with $j \geq 1$. It's true that $R^{i}\pi_{*}\mathcal{O}_{\widetilde{X}}(-E) = 0$ for $i \geq 2?$ If so, it is also true that $R^{i}\pi_{*}\mathcal{O}_{\widetilde{X}}(-nE) = 0$ for $i > 0$ and $n \geq 1$?

All help is very welcome.

Thank you.

Answer 1

Let $i:Y\to X$ and $\widetilde{i}:E\to \widetilde{X}$ be the closed immersions. First, we note that $i_*R^i(\pi|_E)_*\mathcal{O}_E = R^i(\pi_*\widetilde{i}_*\mathcal{O}_E)$ by looking at the definition of $R^i\pi_*\mathcal{F}$ as the sheaf associated to the presheaf $U\mapsto H^i(U,\mathcal{F}(U))$. So it's enough to compute the higher direct images along the map $\pi|_E: E\to Y$.

If $Y\subset X$ is smooth, then the restriction $\pi|_E: E\to Y$ is isomorphic to $\Bbb P(\mathcal{N}_{Y/X})\to Y$, the projectivization of the normal bundle $\mathcal{N}_{Y/X}$, a locally free sheaf of rank $\dim X-\dim Y$. We may find an affine cover $\{\operatorname{Spec} A_i\}_{i\in I}$ of $Y$ which trivializes $\mathcal{N}_{Y/X}$, so that on this cover we are in the situation of $\phi:\Bbb P^n_{A_i}\to \operatorname{Spec} A_i$. By the well-known calculation of the cohomology of line bundles on projective space, we see that all higher direct images $R^i\phi_*\mathcal{O}(d)=0$ for $i\neq 0,n$ and $d > -n-1$ in the case $i=n$.

This shows that all higher direct images of $\mathcal{O}_E$ vanish, answering the first question. Combining this with $R^i\pi_*\mathcal{O}_{\widetilde{X}}=0$ for all $i>0$, we see that $R^i\pi_*\mathcal{O}_\widetilde{X}(-E)=0$ for all $i>0$.

To get the result for $R^i\pi_*\mathcal{O}_\widetilde{X}(-nE)$ with $n>1$, twist the exact sequence $0\to \mathcal{O}_\widetilde{X}(-E) \to \mathcal{O}_\widetilde{X} \to \mathcal{O}_E \to 0$ by $-E$, note that $\mathcal{O}_E(-nE)=\mathcal{O}_E(n)$, and then by induction we see the result for all $n>0$.