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 $R^{\bullet} \pi_*O_V = O_W$

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

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 $E$.

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 $R^{\bullet} \pi_*O_V = O_W$