nLab slope of a coherent sheaf

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.

Related concepts

• slope filtration

• degree of a coherent sheaf

• rank of a coherent sheaf

• stable coherent sheaf

• stable vector bundle

• Bridgeland stability condition

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)