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A generalization of the concept of an ample [[Invertible sheaf|invertible sheaf]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121401.png" /> be a Noetherian scheme over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121402.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121403.png" /> be a locally free sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121404.png" /> (that is, the sheaf of sections of some algebraic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121405.png" />). The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121406.png" /> is called ample if for each coherent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121407.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121408.png" /> there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a0121409.png" />, depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214010.png" />, such that the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214012.png" /> is generated by its global sections (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214013.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214014.png" />-th symmetric power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214015.png" />).
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A generalization of the concept of an ample [[Invertible sheaf|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214017.png" /> is ample if and only if the invertible tautological sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214018.png" /> on the projectivization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214019.png" /> of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214020.png" /> is ample. Another criterion of ampleness is that for each coherent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214021.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214022.png" /> there must exist an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214023.png" />, depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214024.png" />, such that the cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214025.png" /> is zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214027.png" />. If the sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214029.png" /> are ample then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214030.png" /> is an ample sheaf [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214031.png" /> is a non-singular projective curve, then a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214033.png" /> is ample if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214034.png" /> and all its quotient sheaves have positive degree [[#References|[2]]]. The tangent sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214035.png" /> is ample for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214036.png" /> (see [[#References|[1]]]). The converse also holds: Any non-singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214037.png" />-dimensional algebraic variety with an ample tangent sheaf is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012140/a01214038.png" /> (see [[#References|[1]]], [[#References|[3]]]).
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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 [[#References|[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 [[#References|[2]]]. The tangent sheaf on $P^N$ is ample for any $N$ (see [[#References|[1]]]). The converse also holds: Any non-singular $N$-dimensional algebraic variety with an ample tangent sheaf is isomorphic to $P^N$ (see [[#References|[1]]], [[#References|[3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne,   "Ample vector bundles"  ''Publ. Math. IHES'' , '''29'''  (1966)  pp. 319–350</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne,   "Ample vector bundles on curves"  ''Nagoya Math. J.'' , '''43'''  (1971)  pp. 73–89</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne, "Ample vector bundles"  ''Publ. Math. IHES'' , '''29'''  (1966)  pp. 319–350</TD></TR>
 
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<TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Ample vector bundles on curves"  ''Nagoya Math. J.'' , '''43'''  (1971)  pp. 73–89</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR>
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</table>
  
 
====Comments====
 
====Comments====

Latest revision as of 06:50, 9 November 2023

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
How to Cite This Entry:
Ample sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ample_sheaf&oldid=14226
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article