Ample sheaf

A generalization of the concept of an ample invertible sheaf. Let be a Noetherian scheme over a field , and let be a locally free sheaf on (that is, the sheaf of sections of some algebraic vector bundle ). The sheaf is called ample if for each coherent sheaf on there exists an integer , depending on , such that the sheaf for is generated by its global sections (here denotes the -th symmetric power of ).

A locally free sheaf on is ample if and only if the invertible tautological sheaf on the projectivization of the bundle is ample. Another criterion of ampleness is that for each coherent sheaf on there must exist an integer , depending on , such that the cohomology group is zero for and . If the sheaves and are ample then is an ample sheaf [1]. If is a non-singular projective curve, then a sheaf on is ample if and only if and all its quotient sheaves have positive degree [2]. The tangent sheaf on is ample for any (see [1]). The converse also holds: Any non-singular -dimensional algebraic variety with an ample tangent sheaf is isomorphic to (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, "Charactérisations de l'espace projectif (conjectures de Hartshorne et de Frankel)" , Sem. 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