Integrable systems and local fields.

*(English)*Zbl 1014.14015The well-known Krichever corresdondence, linking algebro-geometric data with the theory of infinite-dimensional Grassmannians and algebras of differential operators, has been a subject of vast research during the past 25 years. This construction has been successfully used in the theory of integrable Hamiltonian systems, particularly in the theory of the KP and KdV equations, and it has been an essential ingredient for the final proofs of some longstanding conjectures such as the Schottky problem and the Novikov conjecture. Also, various attempts have been made, over the past two decades, to generalize Krichever’s construction in different directions.

The paper under review is devoted to a purely algebraic approach to the so-called Krichever map. Just assuming an arbitrary groundfield \(k\) of characteristic zero, the author describes a general connection between the KP hierarchy (in Lax form) and the vector fields on infinite Grassmannians. This is not only more general, but also much simpler and much more systematic and transparent than the previously known algebraic-analytic constructions. Then, using his theory of adelic complexes [cf. T. Fimmel and A. N. Parshin, “Introduction to higher adelic theory” (book to appear)], the author establishes the general Krichever correspondence in dimension one, before extending it to algebraic surfaces in the following sections of the paper.

The Krichever correspondence appears here as a quasi-isomorphism between certain adelic complexes, and it is precisely this form that suggests the possibility of further generalizations to higher-dimensional varieties. An approach to such a further generalization has been proposed, in the meantime, by D. V. Osipov in his recent paper “Krichever correspondence for algebraic varieties” [Izv. Math. 65, 941-975 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, 91-128 (2001; Zbl 1068.14053)].

Altogether, this paper reveals the apparently very general character of the Krichever correspondence as well as its crucial significance and ubiquity. Apart from establishing the Krichever correspondence in dimension two, the purely algebraic and far-reaching methodical approach represents the pioneering character of this work.

The paper under review is devoted to a purely algebraic approach to the so-called Krichever map. Just assuming an arbitrary groundfield \(k\) of characteristic zero, the author describes a general connection between the KP hierarchy (in Lax form) and the vector fields on infinite Grassmannians. This is not only more general, but also much simpler and much more systematic and transparent than the previously known algebraic-analytic constructions. Then, using his theory of adelic complexes [cf. T. Fimmel and A. N. Parshin, “Introduction to higher adelic theory” (book to appear)], the author establishes the general Krichever correspondence in dimension one, before extending it to algebraic surfaces in the following sections of the paper.

The Krichever correspondence appears here as a quasi-isomorphism between certain adelic complexes, and it is precisely this form that suggests the possibility of further generalizations to higher-dimensional varieties. An approach to such a further generalization has been proposed, in the meantime, by D. V. Osipov in his recent paper “Krichever correspondence for algebraic varieties” [Izv. Math. 65, 941-975 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, 91-128 (2001; Zbl 1068.14053)].

Altogether, this paper reveals the apparently very general character of the Krichever correspondence as well as its crucial significance and ubiquity. Apart from establishing the Krichever correspondence in dimension two, the purely algebraic and far-reaching methodical approach represents the pioneering character of this work.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14H70 | Relationships between algebraic curves and integrable systems |

37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |

13N10 | Commutative rings of differential operators and their modules |

14G20 | Local ground fields in algebraic geometry |

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