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Category of vector bundles on algebraic curves and infinite dimensional Grassmannians. (English) Zbl 0723.14010
It 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 √Čtud. 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.

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
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