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slope of a coherent sheaf
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Definition
A rational number associated with a coherent sheaf $E$ over a curve : the degree divided by the rank :

$\mu(E)
\;\coloneqq\;
\frac{deg(E)}{rank(E)}
\,.$

For vector bundles : “normalized Chern character ” (Shatz 77 )

e.g. (Huybrechts-Lehn 96, def. 1.2.11 )

This is what enters the condition on stable coherent sheaves .

References
Shatz 1977

Michael Atiyah , Raoul Bott , section 7 of The Yang-Mills equations over Riemann surfaces , Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences

Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor , lighning summary )

Daniel Huybrechts , Manfred Lehn , The Geometry of the Moduli Spaces of Sheaves , 1996 (pdf )

Last revised on October 3, 2018 at 10:49:12.
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