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 $E_1$ and $E_2$ on C of ranks and degrees $(r_1,d_1)$ and $(r_2,d_2)$, respectively, a generic extension

$0\to E_1\to E\to E_2\to 0$

has E stable provided that $\mu (E_1)<\mu (E_2)$, where $\mu (E_i)=d_i/r_i$ 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 ${\mathrm{Ext}}^1$$(E_2,E_1)$.

An original formulation by Lange is that for a pair of integers $(r_1,d_1)$ and $(r_2,d_2)$ such that $d_1/r_1<d_2/r_2$, 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.