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Selecta Math. (N.S.), 2014, Volume 20, Issue 4, Pages 1159–1195 (Mi smath5)  

Commuting differential operators and higher-dimensional algebraic varieties

H. Kurkea, D. Osipovb, A. Zheglovc

a Department of Mathematics, Faculty of Mathematics and Natural Sciences II, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany
b Algebra and Number Theory Department, Steklov Mathematical Institute, Gubkina str. 8, 119991 Moscow, Russia
c Department of Differential Geometry and Applications, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory, GSP, 119899 Moscow, Russia

Abstract: Several algebro-geometric properties of commutative rings of partial differential operators (PDOs) as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of PDOs, and we investigate the properties of these geometric data. This construction is in some sense similar to the construction of a formal module of Baker–Akhieser functions. On the other hand, there is a recent generalization of Satos theory, which belongs to the third author of this paper. We compare both approaches to the commutative rings of PDOs in two variables. As a by-product, we get several necessary conditions on geometric data describing commutative rings of PDOs.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00178-a
12-01-33024 mol_a_ved
13-01-00664-a
Ministry of Education and Science of the Russian Federation NSh-2998.2014.1
Nsh-581.2014.1
The second author was partially supported by Russian Foundation for Basic Research (Grant Nos. 14-01-00178-a and 12-01-33024 mol_a_ved) and by the Programme for the Support of Leading Scientific Schools of the Russian Federation (Grant No. NSh-2998.2014.1). The third author was partially supported by the RFBR Grant Nos. 14-01-00178-a, 13-01-00664 and by Grant NSh No. 581.2014.1.


DOI: https://doi.org/10.1007/s00029-014-0155-9


Bibliographic databases:

Document Type: Article
MSC: Primary 37K10, 14J60; Secondary 35S99
Language: English

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