Lange's conjecture

In algebraic geometry, Lange's conjecture is a theorem about stability of vector bundles over curves, introduced by Herbet Lange 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 $\operatorname {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. 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.