Ample sheaf

A generalization of the concept of an ample invertible sheaf. Let $X$ be a Noetherian scheme over a field $k$, and let $\mathcal E$ be a locally free sheaf on $X$ (that is, the sheaf of sections of some algebraic vector bundle $E\to X$). The sheaf $\mathcal E$ is called ample if for each coherent sheaf $\mathcal F$ on $X$ there exists an integer $n_0$, depending on $\mathcal F$, such that the sheaf $\mathcal F\otimes S^n\mathcal E$ for $n\geq n_0$ is generated by its global sections (here $S^n\mathcal E$ denotes the $n$-th symmetric power of $\mathcal E$).

A locally free sheaf $\mathcal E$ on $X$ is ample if and only if the invertible tautological sheaf $\mathcal L(\mathcal E)$ on the projectivization $P(E)$ of the bundle $E$ is ample. Another criterion of ampleness is that for each coherent sheaf $\mathcal F$ on $X$ there must exist an integer $n_0$, depending on $\mathcal F$, such that the cohomology group $H^i(X,\mathcal F\otimes S^n\mathcal E)$ is zero for $n\geq n_0$ and $i>0$. If the sheaves $\mathcal E$ and $\mathcal F$ are ample then $\mathcal E\otimes\mathcal F$ is an ample sheaf [1]. If $X$ is a non-singular projective curve, then a sheaf $\mathcal E$ on $X$ is ample if and only if $\mathcal E$ and all its quotient sheaves have positive degree [2]. The tangent sheaf on $P^N$ is ample for any $N$ (see [1]). The converse also holds: Any non-singular $N$-dimensional algebraic variety with an ample tangent sheaf is isomorphic to $P^N$ (see [1], [3]).

References

[1]	R. Hartshorne, "Ample vector bundles" Publ. Math. IHES , 29 (1966) pp. 319–350
[2]	R. Hartshorne, "Ample vector bundles on curves" Nagoya Math. J. , 43 (1971) pp. 73–89
[3]	M. Demazure, "Caractérisations de l'espace projectif (conjectures de Hartshorne et de Frankel)" , Sém. Bourbaki 1979/80 , Lect. notes in math. , 842 , Springer (1981) pp. 11–19

Comments

The theorem stated in the last line of the text is due to S. Mori [a1].

References

[a1]	S. Mori, "Positive manifolds with ample tangent bundles" Ann. of Math. , 110 (1979) pp. 593–606