The line bundle $O(-1)$ on a projective space or $O(-\rho)$ on a flag variety has a property that all its cohomology vanish. Is there a story behind such sheaves?


Here are more precise questions. Let $X$ be a smooth complex projective surface (say, a nice one like Del Pezzo or K3). Does there always exist a coherent locally free sheaf $M$ whose derived global sections vanish? Can one describe all such sheaves? Is there a coarse moduli space of such sheaves?



Sheaves without global sections
The bundles with no derived global sections (more generally the objects $F$ of the derived category $D^b(coh X)$ such that $Ext^\bullet(O\_X,F) = 0$) form the left orthogonal complement to the structure sheaf $O\_X$. It is denoted $O\_X^\perp$. This is quite an interesting subcategory of the derived category. 


For example, if $O\_X$ itself has no higher cohomology (i.e. it is exceptional) then there is a semiorthogonal decomposition $D^b(coh X) =< O\_X^\perp, O\_X >$. Then every object can be split into components with respect to this decomposition and so many questions about $D^b(coh X)$ can be reduced to $O\_X^\perp$ which is smaller. Further, if you have an object $E$ in $O\_X^\perp$ which has no higher self-exts (like $O(-1)$ on $P^2$), you can continue simplifying your category --- considering a semiorthogonal decomposition $O\_X^\perp = < E^\perp, E >$. For example if $X = P^2$ and $E = O(-1)$ then $E^\perp$ is generated by $O(-2)$, so there is a semiorthogonal decomposition $D^b(coh P^2) = < O\_X(-2), O\_X(-1), O\_X >$ also known as a full exceptional collection on $P^2$. It allows a reduction of many problems about $D^b(coh P^2)$ to linear algebra.


Another interesting question is when $O\_X$ is spherical (i.e. its cohomology algebra is isomorphic to the cohomology of a topological sphere). This holds for example for K3 surfaces. Then there is a so called spherical twist functor for which $O\_X^\perp$ is the fixed subcategory. 


Thus, as you see, the importance of the category $O\_X^\perp$ depends on the properties of the sheaf $O\_X$.


