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Mat. Sb., 2015, Volume 206, Number 5, Pages 61–106 (Mi msb8429)  

This article is cited in 2 scientific papers (total in 2 papers)

Geometric properties of commutative subalgebras of partial differential operators

A. B. Zheglova, H. Kurkeb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Humboldt University, Berlin

Abstract: We investigate further algebro-geometric properties of commutative rings of partial differential operators, continuing our research started in previous articles. In particular, we start to explore the simplest and also certain known examples of quantum algebraically completely integrable systems from the point of view of a recent generalization of Sato's theory, developed by the first author. We give a complete characterization of the spectral data for a class of ‘trivial’ commutative algebras and strengthen geometric properties known earlier for a class of known examples. We also define a kind of restriction map from the moduli space of coherent sheaves with fixed Hilbert polynomial on a surface to an analogous moduli space on a divisor (both the surface and the divisor are part of the spectral data). We give several explicit examples of spectral data and corresponding algebras of commuting (completed) operators, producing as a by-product interesting examples of surfaces that are not isomorphic to spectral surfaces of any (maximal) commutative ring of partial differential operators of rank one. Finally, we prove that any commutative ring of partial differential operators whose normalization is isomorphic to the ring of polynomials $k[u,t]$ is a Darboux transformation of a ring of operators with constant coefficients.
Bibliography: 39 titles.

Keywords: commuting differential operators, quantum integrable systems, moduli space of coherent sheaves, Darboux transformation.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00178-а
13-01-00664
Ministry of Education and Science of the Russian Federation НШ-581.2014.1
This research was partially supported by the Russian Foundation for Basic Research (grant nos. 14-01-00178-a and 13-01-00664) and the Programme of the President of the Russian Federation for the Support of Leading Scientific Schools (grant no. НШ-581.2014.1).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8429

Full text: PDF file (911 kB)
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English version:
Sbornik: Mathematics, 2015, 206:5, 676–717

Bibliographic databases:

UDC: 517.957+512.72+512.71
MSC: Primary 13N15, 37K20; Secondary 14H70
Received: 06.10.2014 and 01.02.2015

Citation: A. B. Zheglov, H. Kurke, “Geometric properties of commutative subalgebras of partial differential operators”, Mat. Sb., 206:5 (2015), 61–106; Sb. Math., 206:5 (2015), 676–717

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. B. Zheglov, “Surprising examples of nonrational smooth spectral surfaces”, Sb. Math., 209:8 (2018), 1131–1154  mathnet  crossref  crossref  adsnasa  isi  elib
    2. Vik. S. Kulikov, “On divisors of small canonical degree on Godeaux surfaces”, Sb. Math., 209:8 (2018), 1155–1163  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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