Stable vector bundle

Builds on Algebraic curve Vector bundle Weil divisor

Definition

Let $k$ be an algebraically closed field and let $C$ be a smooth projective connected algebraic curve over $k$ . For a nonzero vector bundle $E$ on $C$ of rank $r$ , let $\deg E$ be the degree of a divisor of its determinant line bundle $\det E = \Lambda^r E$ . The slope of $E$ is $\mu(E) = \deg E / \rank E$ . The bundle $E$ is stable if $\mu(F) < \mu(E)$ for every subbundle $F \subset E$ with $0 < \rank F < \rank E$ . It is semistable if $\mu(F) \le \mu(E)$ for every such $F$ .

Remarks

Example: every line bundle is stable, since it has no subbundle of intermediate rank. Non-example: let $L_1, L_2$ be line bundles with $\deg L_1 > \deg L_2$ . Then $L_1$ is a subbundle of $E = L_1 \oplus L_2$ with $\mu(L_1) = \deg L_1 > (\deg L_1 + \deg L_2)/2 = \mu(E)$ , so $E$ is not even semistable. The bundle $\mathcal{O}_C \oplus \mathcal{O}_C$ is semistable but not stable, since $\mathcal{O}_C$ has the same slope $0$ . One may test with all nonzero proper coherent subsheaves instead of subbundles; the condition is the same. A stable bundle $E$ is simple: its only endomorphisms are the scalars. On a smooth projective variety of higher dimension with an ample class $H$ , slope stability (Mumford–Takemoto stability) uses torsion-free subsheaves and the degree $c_1(F) \cdot H^{n-1}$ .

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Algebraic curve concept
Vector bundle concept
Weil divisor concept
Stable vector bundle definition

Tag	`TP1RQPCETV`
URI	`m2://TP1RQPCETV`
Kind	definition
Writing	W0 stub
MSC	14H60, 14D20
Defines	Stable vector bundle
Uses	Algebraic curve, Vector bundle, Weil divisor

Stable vector bundle

Definition

Remarks

Reading path

Further reading

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