Projective varieties with vanishing first cohomology for all locally free sheaves Let $k$ be an algebraically closed field, $X$ be a projective variety over $k$ with $\text{dim}>1$. Does there exists $X$ such that $H^1(X,\mathcal{F})=0$ for every locally free sheaves of finite rank?
The motivation for this problems comes from cohomological criterion for affine schemes (vanishing first cohomology  for all coherent sheaves). Also, one way to show that $\Bbb P^n_k$ $(n>1)$ has some non-split vector bundles is to notice that there exists a vector bundle with nonvanishing first cohomology, so it's natural to ask whether in general this holds.
One observation is that $H^1(X,F(n))=0$ for every $n >>0$ by projectivity in general, and $H^1(X,F(n))=0$ for every $n <<0$ if $X$ is normal (Hartshone III.7.8)

No such $X$ can exist at least in characteristic zero (and am sure it is true in all cases). Given such an $X$, we can always find a finite morphism $f:X\to P$, a projective space, by Noether Normalization. Then $f_*\mathcal{O}_X=\mathcal{O}_P\oplus G$ for some $G$, by the trace map. Now, let $E$ be any vector bundle on $P$. Then, $f_*f^*E=E\otimes f_*\mathcal{O}_X=E\oplus E\otimes G$. Thus, if $H^1(P, E)\neq 0$, nor is $H^1(P, f_*f^* E)$. But this is just $H^1(X, f^*E)$. There are plenty of such bundles $E$, for example take $E=\Omega^1_P$.
