Lange's conjecture

In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by [|Lange] (1983) and proved by Montserrat Teixidor i Bigas and Barbara Russo in 1999.

Statement

Let C be a smooth projective curve of genus ≥ 2. For generic vector bundles E₁ and E₂ on C of ranks and degrees (r₁, d₁) and (r₂, d₂) respectively, a generic extension

0\to E_{1}\to E\to E_{2}\to 0

has E stable provided that μ(E₁) < μ(E₂), where μ(E_i) = d_i / r_i is the slope. The notion of a generic vector bundle here is a generic point in the moduli space of semistable vector bundles on C, and a generic extension is one that corresponds to a generic point in the vector space Ext¹(E₂, E₁).

An original formulation by Lange is that for a pair of integers (r₁, d₁) and (r₂, d₂) such that d₁ / r₁ < d₂ / r₂, there exists a short exact sequence as above with E stable. It's equivalent because the existence of a short exact sequence like that is an open condition on E in the moduli space of semistable vector bundles on C.

References

Lange, Herbert (1983), "Zur Klassifikation von Regelmannigfaltigkeiten", Mathematische Annalen, 262 (4): 447–459, doi:10.1007/BF01456060, ISSN 0025-5831, MR 0696517
Montserrat Teixidor i Bigas and Barbara Russo (1999), "On a conjecture of Lange", Journal of Algebraic Geometry, 8 (3): 483–496, arXiv:alg-geom/9710019, Bibcode:1997alg.geom.10019R, ISSN 1056-3911, MR 1689352
E. Ballico (2000), "Extensions of stable vector bundles on smooth curves: Lange's conjecture", An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 46 (1): 149–156, MR 1840133

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.