Lange's conjecture

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

Statement

Let C be a smooth projective curve of genus greater or equal to 2. For generic vector bundles [math]\displaystyle{ E_1 }[/math] and [math]\displaystyle{ E_2 }[/math] on C of ranks and degrees [math]\displaystyle{ (r_1, d_1) }[/math] and [math]\displaystyle{ (r_2, d_2) }[/math], respectively, a generic extension

[math]\displaystyle{ 0 \to E_1 \to E \to E_2 \to 0 }[/math]

has E stable provided that [math]\displaystyle{ \mu(E_1) \lt \mu(E_2) }[/math], where [math]\displaystyle{ \mu(E_i) = d_i/r_i }[/math] is the slope of the respective bundle. 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 [math]\displaystyle{ \operatorname{Ext}^1 }[/math][math]\displaystyle{ (E_2,E_1) }[/math].

An original formulation by Lange is that for a pair of integers [math]\displaystyle{ (r_1, d_1) }[/math] and [math]\displaystyle{ (r_2, d_2) }[/math] such that [math]\displaystyle{ d_1/ r_1 \lt d_2/r_2 }[/math], there exists a short exact sequence as above with E stable. This formulation is 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. 
• Teixidor i Bigas, Montserrat; Russo, Barbara (1999). "On a conjecture of Lange". Journal of Algebraic Geometry 8 (3): 483–496. ISSN 1056-3911. Bibcode: 1997alg.geom.10019R. 
• Ballico, Edoardo (2000). "Extensions of stable vector bundles on smooth curves: Lange's conjecture". Analele Ştiinţifice ale Universităţii "Al. I. Cuza" din Iaşi. (N.S.) 46 (1): 149–156.