Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi\_\*F=0$ for $i>0$. 



> 
> Can we conclude that $R^i\pi\_\*F(-E)=0$ for $i>0$?
> 
> 
> 


The idea being to somehow use that$-E$ is nef on $E$.


EDIT:
the exceptional fibres of $\pi$ are one-dimensional and $R^{\bullet} \pi\_\*O\_V = O\_W$ 


EDIT 2: we may assume that the support of $F$ is one-dimensional



Vanishing associated to a resolution of singularities
1
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The statement is still false with the most recently added conditions. 
The problem is that $-E$ is **not** necessarily ($\pi$-)nef. I will discuss below what one can do to fix the statement so it will hold. First, here is an example:


Let $W$ be a two dimensional rational double point that's either a $D\_n$ or an $E\_n$ singularity. The point is to have an exceptional curve in the resolution that intersects at least 3 others. 


Let $\pi:V\to W$ be the minimal resolution of $W$ and $E\_0,E\_1,E\_2,E\_3$ four exceptional curves with intersection matrix
$$\left(\begin{matrix} -2 & 1 & 1 & 1 \\ 1 & -2 & 0 & 0 \\ 1 & 0 & -2 & 0 \\ 1 & 0 & 0 & -2 \end{matrix}\right),$$


and let $E$ be the reduced exceptional divisor. 
It follows that then
$$
E\_0\cdot E=E\_0\cdot (E\_0+E\_1+E\_2+E\_3)=1.
$$
In particular, $-E$ is not nef on $E\_0$.


Now, let $F=\mathscr O\_{E\_0}(-1)$. Since $E\_0\simeq \mathbb P^1$, it follows that $R^i\pi\_\*F=0$ for all $i\geq 0$ (even for $i=0$!), but since $F(-E)\simeq \mathscr O\_{E\_0}(-2)$, $R^1\pi\_\*F\neq 0$.


2
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Here is a statement in the spirit of the question which is true (notice that some assumptions are actually weakened, in particular, it is not needed that the singularities of $W$ be rational):



> 
> **Claim** Let $\pi:V\to W$ be a projective birational morphism over a field of characteristic $0$ such that every fiber of $\pi$ has dimension at most 1 and let $E$ denote the exceptional divisor of $\pi$. Let $\mathscr F$ be a coherent sheaf on $V$ such that $Z=\mathrm{supp}\mathscr F$ is 1-dimensional. Assume that $W$ has $\mathbb Q$-factorial singularities and that $R^1\pi\_\*\mathscr F=0$.
>  Then there exists an effective exceptional divisor $E'$ such that $R^i\pi\_\*\mathscr F(-aE')=0$ for all $a>0$ and $i>0$ ($a,i\in \mathbb N$). Furthermore, if $V$ and $W$ are normal, then $\mathrm{supp}E'=\mathrm{supp}E$ in a neighbourhood of $Z$.
> 
> 
> 


**Proof** First off, notice that since the maximal fiber dimension of $\pi$ is 1, all $R^i\pi\_\*=0$ for $i>1$. Furthermore, since the claim is local on $W$, we may assume that it is affine. 


Next, we claim that one may assume that $Z$ is contained in a fiber of $\pi$. Indeed let 
$Z=Z\_1\cup Z\_2$ where $Z\_1$ is contained in a union of fibers and $Z\_2$ does not have any components contained in a fiber (i.e., $\pi$ is finite on $Z\_2$). Since $W$ is affine, it follows that so is $Z\_2$. Now let $\mathscr G$ be a coherent sheaf (we will apply this with $\mathscr G=\mathscr F$ and $\mathscr G=\mathscr F(-E)$). Then we have a short exact sequence
$$
0\to \mathscr G\to \mathscr G|\_{Z\_1}\oplus \mathscr G|\_{Z\_2}\to \mathscr G|\_{Z\_1\cap Z\_2}\to 0.
$$
Since $Z\_2$ is affine, it follows that $\pi\_\*\mathscr G|\_{Z\_2}\to \pi\_\*\mathscr G|\_{Z\_1\cap Z\_2}$ is surjective and $R^i\pi\_\*\left(\mathscr G|\_{Z\_2}\right)=0$ and $R^i\pi\_\*\left(\mathscr G|\_{Z\_1\cap Z\_2}\right)=0$ for $i>0$, hence 
$R^i\pi\_\*\mathscr G\simeq R^i\pi\_\*\left(\mathscr G|\_{Z\_1}\right)$ for $i>0$. This implies that we may assume that $Z=Z\_1$ and then, since the statement is local we may assume that $Z$ is mapped to a single point, i.e., it is contained in a fiber. 


Finally let $L$ be an arbitrary effective divisor on $V$ such that $\mathrm{supp}L\cap Z\neq\emptyset$ but 
$Z\not\subseteq \mathrm{supp}L$. (As $\pi$ is projective, such an $L$ exists). 
Let $H=\pi\_\*L$ and notice that since $W$ is $\mathbb Q$-factorial, $H$ is $\mathbb Q$-Cartier. Replacing $L$ with an appropriate multiple, we may assume that $H$ is Cartier.
Compare $\pi^\*H$ and $\widetilde H$ (the strict transform of $H$) and observe that


1. $\widetilde H=L$ by construction and hence $\widetilde H|\_Z\geq 0$ and $\widetilde H|\_Z\neq 0$,
2. $\pi^\*H\sim \widetilde H+\sum\_i m\_iE\_i$, where $E\_i$ are the irreducible components of the exceptional divisor and $m\_i\geq 0$,
3. $\pi^\*H|\_Z=0$ since $Z$ is contained in a fiber.


Therefore it follows that $(-\sum m\_iE\_i)|\_Z\geq 0$ and $(-\sum m\_iE\_i)|\_Z\neq 0$. Let $d$ denote the largest common divisor of $m\_i$ and let $E'=\sum\frac{m\_i}dE\_i$. Then we still have that $(-aE')|\_Z\geq 0$ and $(-aE')|\_Z\neq 0$ for any $a>0$. It follows that then $\mathscr F\subseteq \mathscr F(-aE')$ so we have a short exact sequence
$$
0 \to \mathscr F\to \mathscr F(-aE')\to \mathscr Q\to 0
$$ 
where $\mathscr Q$ is supported at finitely many points. Then $R^i\pi\_\*\mathscr Q=0$ for $i>0$ and hence it follows that $0=R^i\pi\_\*\mathscr F\to R^i\pi\_\*\mathscr F(-aE')$ is surjective for all $i>0$ and the desired statement follows. 


The only thing left to prove is that if $V$ and $W$ are normal, then $\mathrm{supp}E'=\mathrm{supp}E$ in a neighbourhood of $Z$. The above proof actually shows that $-E'$ is $\pi$-nef and then this statement follows from 3.39 of [Birational Geometry of Algebraic Varieties](http://books.google.com/books?id=YrysxvPbBLwC&lpg=PA102&ots=_4uOxAeVw5&dq=kollar%2520mori%25203.39&pg=PA102#v=onepage&q&f=false1.2.) by János Kollár and Shigefumi Mori and the choices that ensured that $E'|\_Z\neq 0$.
$\square$


