Far Eastern Mathematical Journal

To content of the issue


Baker – Akhiezer modules, Krichever sheaves, and commuting rings of partial differential operators


A. B. Zheglov, A. E. Mironov

2012, issue 1, Ñ. 20–34


Abstract
In this work we give a review of several results about commutative subrings of partial differential operators. We show that $n$-dimensional commutative ring of partial differential operators with scalar (not matrix) coefficients (with certain mild conditions) corresponds to a Baker – Akhiezer module on the spectral algebraic variety. We also show that there is a family of coherent torsion free sheaves of special type. The existence of such sheaves gives a strong restriction on the structure of the spectral variety, in particular, it is possible to find the selfintersection index of a divisor at infinity.

Keywords:
commuting partial differential operators, spectral varieties, Baker-Akhieser modules

Download the article (PDF-file)

References

[1] I. M. Krichever, “Integrirovanie nelineinykh uravnenii metodami algebraicheskoi geometrii”, Funkts. analiz i ego prilozh., 11:1 (1977), 15–31.
[2] I. M. Krichever, “Kommutativnye kol'tsa lineinykh obyknovennykh differentsial'nykh operatorov”, Funkts. analiz. i ego prilozh., 12:3 (1978), 20–31.
[3] I. M. Krichever, S. P. Novikov, “Golomorfnye rassloeniia nad algebraicheskimi krivymi i nelineinye uravneniia”, Uspekhi matem. nauk, 35:6 (1980), 47–68.
[4] O. I. Mokhov, “Kommutiruiushchie differentsial'nye operatory ranga 3 i nelineinye uravneniia”, Izvestiia AN SSSR. Seriia matem., 53:6 (1989), 1291–1314.
[5] A. E. Mironov, “Ob odnom kol'tse kommutiruiushchikh differentsial'nykh operatorov ranga dva, otvechaiushchem krivoi roda dva”, Matem. sb., 195:5 (2004), 103–-114.
[6] A. E. Mironov, “Kommutiruiushchie differentsial'nye operatory ranga 2, otvechaiushchie krivoi roda 2”, Funkts. analiz i ego pril., 39:3 (2005), 91-94.
[7] D. Zuo, “Commuting differential operators of rank 3 associated to a curve of genus 2”, arXiv: 1105.5774.
[8] I. M. Krichever, “Algebraicheskie krivye i kommutiruiushchie matrichnye differentsial'nye operatory”, Funkts. analiz i ego prilozh., 10:2 (1976), 75–77.
[9] B. A. Dubrovin, “Matrichnye konechnozonnye operatory.”, Sovremennye problemy matematiki., 23, Itogi nauki i tekhniki (1983), 33–78.
[10] P. G. Grinevich, “Vektornyi rang kommutiruiushchikh matrichnykh differentsial'nykh operatorov. Dokazatel'stvo kriteriia S. P. Novikova”, Izv. AN SSSR. Ser. matem., 50:3 (1986), 458–478.
[11] I. M. Krichever, “Metody algebraicheskoi geometrii v teorii nelineinykh uravnenii”, Uspekhi matem. nauk, 32:6 (1977), 183–208.
[12] A. Kasman, E. Previato, “Commutative partial differential operators”, Physica D., 152–153 (2001), 66–77.
[13] O. Chalykh, A. Veselov, “Commutative rings of partial differential operators and Lie algebras”, Comm. Math. Phys., 126:3 (1990), 597–611.
[14] A. P. Veselov, M. V. Feigin, O. A. Chalykh, “Novye integriruemye deformatsii kvantovoi zadachi Kalodzhero – Mozera”, Uspekhi matem. nauk, 51:3 (1996), 185–186.
[15] V. M. Bukhshtaber, S. Iu. Shorina, “Kommutiruiushchie differentsial'nye mnogomernye operatory tret'ego poriadka, zadaiushchie KdF-ierarkhiiu”, Uspekhi matem. nauk, 58:3 (2003), 187–188.
[16] S. L. Kleiman, “Toward a numerical theory of ampleness”, Annals of Math., 84 (1966), 293–344.
[17] U. Fulton, Teoriia peresechenii, Mir, Moskva, 1989.
[18] R. Hartshorne, Algebraic geometry, Springer, New York – Berlin – Heilderberg, 1977.
[19] A. Nakayashiki, “Commuting partial differential operators and vector bundles over Abelian varieties”, Amer. J. Math., 116 (1994), 65–100.
[20] A. Nakayashiki, “Structure of Baker – Akhiezer Modules of Principally Polarized Abelian varieties, Commuting Partial Differential Operators and Associated Integrable Systems”, Duke Math. J., 62:2 (1991), 315–358.
[21] A. B. Zheglov, “On rings of commuting partial differential operators”, arXiv: 1106.0765.
[22] A. N. Parshin, “Integrable systems and local fields”, Commun. Algebra, 29:9 (2001), 4157–4181.
[23] H. Kurke, D. Osipov, A. Zheglov, “Formal punctured ribbons and two-dimensional local fields”, Journal fu?r die reine und angewandte Mathematik (Crelles Journal), 2009:629, 133–170.
[24] H. Kurke, D. Osipov, A. Zheglov, “Formal groups arising from formal punctured ribbons”, Int. J. of Math., 06 (2010), 755–797.
[25] D.V. Osipov, “The Krichever correspondence for algebraic varieties”, Izv. Ross. Akad. Nauk Ser. Mat., 65:5 (2001), 91–128.
[26] A.B. Zheglov, D.V.Osipov, “O nekotorykh voprosakh, sviazannykh s sootvetstviem Krichevera”, Mat. zametki, 81:4 (2007), 528–539.
[27] M. Atiyah, I. Macdonald, Introduction to Commutative algebra, Addison-Wesley, Reading, Mass, 1969.
[28] N. Bourbaki, Algebre Commutative, Elements de Math., v. 27,28,30,31, Hermann, Paris, 1961–1965.
[29] A. Grothendieck, J. A. Dieudonne?, “E?le?ments de ge?ome?trie alge?brique II”, Publ. Math. I.H.E.S., 8 (1961).
[30] D. Mumford, The red book of varieties and schemes, Springer – Verlag, Berlin, Heidelberg, 1999.
[31] A. E. Mironov, “Kommutativnye kol'tsa differentsial'nykh operatorov, sviazannye s dvumernymi abelevymi mnogoobraziiami”, Sib. mat. zhurnal, 41:6 (2000), 1389–1403.
[32] A. E. Mironov, “Veshchestvennye kommutiruiushchie differentsial'nye operatory, sviazannye s dvumernymi abelevymi mnogoobraziiami”, Sib. mat. zhurnal, 43:1 (2002), 126–143.
[33] A. E. Mironov, “Kommutativnye kol'tsa differentsial'nykh operatorov, otvechaiushchie mnogomernym algebraicheskim mnogoobraziiam”, Sib. matem. zhurnal, 43:5 (2002), 1102–1114.
[34] K. Cho, A. E. Mironov, A. Nakayashiki, “Baker – Akhiezer Modules on the Intersections of Shifted Theta Divisors”, Publications of the Research Institute for Mathematical Sciences, 47:2 (2011), 353–567.
[35] I. A. Melnik, A. E. Mironov, “Baker – Akhiezer modules on rational varieties”, SIGMA, 6 (2010), 030, 15 pp.

To content of the issue