View source for Harder–Narasimhan stratification

In algebraic geometry and [[complex geometry]], the '''Harder–Narasimhan stratification''' is any of a stratification of the [[moduli stack of principal bundles|moduli stack of principal ''G''-bundles]] by locally closed substacks in terms of "loci of instabilities". In the original form due to [[Stable vector bundle|Harder and Narasimhan]], ''G'' was the [[general linear group]]; i.e., the moduli stack was the [[moduli stack of vector bundles]], but, today, the term refers to any of generalizations. The scheme-theoretic version is due to Shatz and so the term "Shatz stratification" is also used synonymously. The general case is due to Behrend.<ref>{{harvnb|Behrend|2003}}</ref><ref>{{cite web|url=http://www.math.harvard.edu/~lurie/282ynotes/LectureIII-Cohomology.pdf|first=Jacob|last=Lurie|author-link=Jacob Lurie|publisher=Harvard University|title=Cohomological Formulation (Lecture 3)|date=February 5, 2014}}</ref>

<!-- == Definitions ==
Let ''X'' be a smooth projective curve over <math>\mathbf{F}_q</math> and ''G'' a smooth affine group scheme over <math>\mathbf{F}_q</math> whose fibers are connected and generic fiber is a semisimple group. Given a ''G''-bundle ''P'' on ''X'', the homomorphism <math>d(P): X(G) \to G</math> that maps a character <math>\chi</math> to the degree of the line bundle <math>P \times_{G, \chi} \mathcal{O}_X</math> is called the degree of <math>P</math>.

Let <math>\operatorname{Bun}_G</math> be the [[moduli stack of principal bundles]] on ''X'', and <math>\operatorname{Bun}_G^d</math> locally closed substack consisting of bundles having degree <math>d: X(G) \to \mathbb{Z}</math>.
If , then

Let <math>\operatorname{Bun}^d_{G, \le m} \subset \operatorname{Bun}^d_G</math> denote the substack of principal G-bundles of instability degree <math>\le m</math>. It is an open substack of <math>\operatorname{Bun}^d_G</math> since the degree is upper semicontinuous. For example, those in the complement of <math>\operatorname{Bun}_{G, \le 0}</math> is [[semi-stable bundle]]s.
-->

==References==

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*{{cite thesis |last=Behrend |first=Kai A. |url=http://www.math.ubc.ca/~behrend/thesis.pdf |title=The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles |publisher=University of British Columbia |year=2003 |type=PhD }}

==Further reading==

* Nitin Nitsure, [https://arxiv.org/abs/0909.0891 Schematic Harder-Narasimhan Stratification]

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[[Category:Algebraic geometry]]
[[Category:Stratifications]]