Lange's conjecture

In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by [|Herbert Lange] (1983) 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 ${\displaystyle E_{1}}$ and ${\displaystyle E_{2}}$ on C of ranks and degrees ${\displaystyle (r_{1},d_{1})}$ and ${\displaystyle (r_{2},d_{2})}$, respectively, a generic extension

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

has E stable provided that ${\displaystyle \mu (E_{1})<\mu (E_{2})}$, where ${\displaystyle \mu (E_{i})=d_{i}/r_{i}}$ 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 ${\displaystyle \operatorname {Ext} ^{1}}$${\displaystyle (E_{2},E_{1})}$.

An original formulation by Lange is that for a pair of integers ${\displaystyle (r_{1},d_{1})}$ and ${\displaystyle (r_{2},d_{2})}$ such that ${\displaystyle d_{1}/r_{1}<d_{2}/r_{2}}$, 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. MR 0696517.
• Teixidor i Bigas, Montserrat; Russo, Barbara (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.
• 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. MR 1840133.