Category of vector bundles on algebraic curves and infinite dimensional Grassmannians.

*(English)*Zbl 0723.14010It is a classical problem to determine all commutative algebras of ordinary differential equations in one variable. Work at the turn of the century showed that the problem was related to algebraic curves, but in the last decade a great deal of progress on the question has been made, spurred on by its relevance to soliton theory, for example by G. B. Segal and G. Wilson [Publ. Math., Inst. Hautes Étud. Sci. 61, 5- 65 (1985; Zbl 0592.35112)]. The Krichever construction effectively solves the problem for algebras where the greatest common divisor of the orders of the operators in the algebra is 1. An algebra is then classified by an algebraic curve of genus g, a marked point \(p\in C\) and a line bundle L of genus \((g-1)\) which is non-special.

This paper tackles the general case (up to a natural notion of equivalence) in terms of a curve C, a point p and a semi-stable vector bundle E of rank r and degree \(r(g-1)\) which is non-special in the sense of A. Beauville, M. S. Narasimhan and S. Ramanan [Reine Angew. Math. 398, 169-179 (1989; Zbl 0666.14015)], together with some local data around p. The approach is a categorical one using the well- known infinite Grassmannian. There is also a remark concerning the generation of generic vector bundles by the application of KP flows.

This paper tackles the general case (up to a natural notion of equivalence) in terms of a curve C, a point p and a semi-stable vector bundle E of rank r and degree \(r(g-1)\) which is non-special in the sense of A. Beauville, M. S. Narasimhan and S. Ramanan [Reine Angew. Math. 398, 169-179 (1989; Zbl 0666.14015)], together with some local data around p. The approach is a categorical one using the well- known infinite Grassmannian. There is also a remark concerning the generation of generic vector bundles by the application of KP flows.

Reviewer: N.J.Hitchin (Coventry)

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14H60 | Vector bundles on curves and their moduli |

13N99 | Differential algebra |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |