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Selecta Mathematica

, Volume 20, Issue 4, pp 1159–1195 | Cite as

Commuting differential operators and higher-dimensional algebraic varieties

  • Herbert Kurke
  • Denis Osipov
  • Alexander ZheglovEmail author
Article

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 Sato’s 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.

Keywords

Commuting partial differential operators Algebraic integrable systems Sato theory Algebraic KP theory Algebraic surfaces Two-dimensional local fields 

Mathematics Subject Classification (2010)

Primary 37K10 14J60 Secondary 35S99 

Notes

Acknowledgments

Part of this research was done at the Mathematisches Forschungsinstitut Oberwolfach during a stay within the Research in Pairs Programme from January 23 till February 5, 2011. We would like to thank the MFO at Oberwolfach for the excellent working conditions. We are also grateful to Igor Burban and Andrey E. Mironov for many stimulating discussions and useful references. We are grateful to the referee for carefully reading the article and suggestions that made a lot of improvements for the exposition of our results. 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.

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Herbert Kurke
    • 1
  • Denis Osipov
    • 2
  • Alexander Zheglov
    • 3
    Email author
  1. 1.Department of Mathematics, Faculty of Mathematics and Natural Sciences IIHumboldt University of BerlinBerlinGermany
  2. 2.Algebra and Number Theory DepartmentSteklov Mathematical InstituteMoscowRussia
  3. 3.Department of Differential Geometry and Applications, Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia

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