Let $X$ be a variety say over $\mathbf{C}$, $L$ a very ample line bundle and $\mathcal{E}$ a coherent sheaf on $X$.
It is well known that $H^0(L^*)=0$. However, I'm curious what there can be said about $H^0(L^{-n}\otimes\mathcal{E})=0$, i.e. which sheaves $\mathcal{E}$ satisfy this vanishing for $n$ big enough?
I think one is tempted to think that must be zero for big $n$, but if e.g. $\mathcal{E}$ is of dimension $0$, for example the skyscraper sheaf of a point $\mathbf{C}_p$ then $H^0(L^{-n}\otimes \mathbf{C}_p)\cong \mathbf{C}$! What can there be said for maybe torsion-free sheaves $\mathcal{E}$?
Any help appreciated!
Best Answer
You can always take $\mathcal{E}$ to be $\mathcal{E}' \otimes L$, then $$ L^* \otimes \mathcal{E} \cong \mathcal{E}', $$ so this can be arbitrary torsion sheaf, and you can't say anything about its $H^0$ in general.
EDIT. For the modified question, it is true that $$ H^0(L^{-n} \otimes \mathcal{E}) = 0 \qquad \text{for $n \gg 0$} $$ unless $\mathcal{E}$ has 0-dimensional associated points (i.e., subsheaves supported on 0-dimensional subschemes). This follows from Serre duality and Serre vanishing.