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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field and let $C$ be a smooth projective connected [[algebraic curve|XJM2MT4S3Y]] over $k$.\nFor a nonzero [[vector bundle|P4DZM975BV]] $E$ on $C$ of rank $r$, let $\\deg E$ be the degree of a [[divisor|Z9DN23DAE2]] of its determinant line bundle $\\det E = \\Lambda^r E$.\nThe *slope* of $E$ is $\\mu(E) = \\deg E / \\rank E$.\nThe bundle $E$ is *stable* if\n$$\\mu(F) \u003c \\mu(E)$$\nfor every subbundle $F \\subset E$ with $0 \u003c \\rank F \u003c \\rank E$.\nIt is *semistable* if $\\mu(F) \\le \\mu(E)$ for every such $F$.\n\n## Remarks\n\nExample: every line bundle is stable, since it has no subbundle of intermediate rank.\nNon-example: let $L_1, L_2$ be line bundles with $\\deg L_1 \u003e \\deg L_2$.\nThen $L_1$ is a subbundle of $E = L_1 \\oplus L_2$ with $\\mu(L_1) = \\deg L_1 \u003e (\\deg L_1 + \\deg L_2)/2 = \\mu(E)$, so $E$ is not even semistable.\nThe bundle $\\mathcal{O}_C \\oplus \\mathcal{O}_C$ is semistable but not stable, since $\\mathcal{O}_C$ has the same slope $0$.\nOne may test with all nonzero proper coherent subsheaves instead of subbundles; the condition is the same.\nA stable bundle $E$ is simple: its only endomorphisms are the scalars.\nOn a smooth projective variety of higher dimension with an ample class $H$, slope stability (Mumford–Takemoto stability) uses torsion-free subsheaves and the degree $c_1(F) \\cdot H^{n-1}$.\n",
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    "name": "Stable vector bundle",
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  "name": "Stable vector bundle",
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