Geodesic
Baker~-- Akhiezer modules, Krichever sheaves, and commuting rings of partial differential operators
Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 1, pp. 20-34.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{DVMG_2012_12_1_a2,
     author = {A. B. Zheglov and A. E. Mironov},
     title = {Baker~-- {Akhiezer} modules, {Krichever} sheaves, and commuting rings of partial differential operators},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {20--34},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {2012},
     language = {ru},
     url = {https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a2/}
}
TY  - JOUR
AU  - A. B. Zheglov
AU  - A. E. Mironov
TI  - Baker~-- Akhiezer modules, Krichever sheaves, and commuting rings of partial differential operators
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2012
SP  - 20
EP  - 34
VL  - 12
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a2/
LA  - ru
ID  - DVMG_2012_12_1_a2
ER  - 
%0 Journal Article
%A A. B. Zheglov
%A A. E. Mironov
%T Baker~-- Akhiezer modules, Krichever sheaves, and commuting rings of partial differential operators
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2012
%P 20-34
%V 12
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a2/
%G ru
%F DVMG_2012_12_1_a2
A. B. Zheglov; A. E. Mironov. Baker~-- Akhiezer modules, Krichever sheaves, and commuting rings of partial differential operators. Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 1, pp. 20-34. https://geodesic-test.mathdoc.fr/item/DVMG_2012_12_1_a2/

[1] I. M. Krichever, “Integrirovanie nelineinykh uravnenii metodami algebraicheskoi geometrii”, Funkts. analiz i ego prilozh., 11:1 (1977), 15–31 | MR | Zbl

[2] I. M. Krichever, “Kommutativnye koltsa lineinykh obyknovennykh differentsialnykh operatorov”, Funkts. analiz. i ego prilozh., 12:3 (1978), 20–31 | MR | Zbl

[3] I. M. Krichever, S. P. Novikov, “Golomorfnye rassloeniya nad algebraicheskimi krivymi i nelineinye uravneniya”, Uspekhi matem. nauk, 35:6 (1980), 47–68 | MR | Zbl

[4] O. I. Mokhov, “Kommutiruyuschie differentsialnye operatory ranga 3 i nelineinye uravneniya”, Izvestiya AN SSSR. Seriya matem., 53:6 (1989), 1291–1314

[5] A. E. Mironov, “Ob odnom koltse kommutiruyuschikh differentsialnykh operatorov ranga dva, otvechayuschem krivoi roda dva”, Matem. sb., 195:5 (2004), 103–-114 | DOI | MR | Zbl

[6] A. E. Mironov, “Kommutiruyuschie differentsialnye operatory ranga 2, otvechayuschie krivoi roda 2”, Funkts. analiz i ego pril., 39:3 (2005), 91–-94 | DOI | MR | Zbl

[7] D. Zuo, “Commuting differential operators of rank 3 associated to a curve of genus 2”, arXiv: 1105.5774 | MR

[8] I. M. Krichever, “Algebraicheskie krivye i kommutiruyuschie matrichnye differentsialnye operatory”, Funkts. analiz i ego prilozh., 10:2 (1976), 75–77 | MR

[9] B. A. Dubrovin, “Matrichnye konechnozonnye operatory.”, Sovremennye problemy matematiki., 23, Itogi nauki i tekhniki (1983), 33–78 | MR

[10] P. G. Grinevich, “Vektornyi rang kommutiruyuschikh matrichnykh differentsialnykh operatorov. Dokazatelstvo kriteriya S. P. Novikova”, Izv. AN SSSR. Ser. matem., 50:3 (1986), 458–478 | MR | Zbl

[11] I. M. Krichever, “Metody algebraicheskoi geometrii v teorii nelineinykh uravnenii”, Uspekhi matem. nauk, 32:6 (1977), 183–208 | MR | Zbl

[12] A. Kasman, E. Previato, “Commutative partial differential operators”, Physica D., 152–153 (2001), 66–77 | DOI | MR | Zbl

[13] O. Chalykh, A. Veselov, “Commutative rings of partial differential operators and Lie algebras”, Comm. Math. Phys., 126:3 (1990), 597–611 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[15] V. M. Bukhshtaber, S. Yu. Shorina, “Kommutiruyuschie differentsialnye mnogomernye operatory tretego poryadka, zadayuschie KdF-ierarkhiyu”, Uspekhi matem. nauk, 58:3 (2003), 187–188 | DOI | MR | Zbl

[16] S. L. Kleiman, “Toward a numerical theory of ampleness”, Annals of Math., 84 (1966), 293–344 | DOI | MR | Zbl

[17] U. Fulton, Teoriya peresechenii, Mir, Moskva, 1989 | MR

[18] R. Hartshorne, Algebraic geometry, Springer, New York – Berlin – Heilderberg, 1977 | MR | Zbl

[19] A. Nakayashiki, “Commuting partial differential operators and vector bundles over Abelian varieties”, Amer. J. Math., 116 (1994), 65–100 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[23] H. Kurke, D. Osipov, A. Zheglov, “Formal punctured ribbons and two-dimensional local fields”, Journal für die reine und angewandte Mathematik (Crelles Journal), 2009:629, 133–170 | DOI | MR | Zbl

[24] H. Kurke, D. Osipov, A. Zheglov, “Formal groups arising from formal punctured ribbons”, Int. J. of Math., 06 (2010), 755–797 | DOI | MR

[25] D.V. Osipov, “The Krichever correspondence for algebraic varieties”, Izv. Ross. Akad. Nauk Ser. Mat., 65:5 (2001), 91–128 | DOI | MR | Zbl

[26] A.B. Zheglov, D.V.Osipov, “O nekotorykh voprosakh, svyazannykh s sootvetstviem Krichevera”, Mat. zametki, 81:4 (2007), 528–539 | DOI | MR | Zbl

[27] M. Atiyah, I. Macdonald, Introduction to Commutative algebra, Addison-Wesley, Reading, Mass, 1969 | MR | Zbl

[28] N. Bourbaki, Algebre Commutative, Elements de Math., v. 27,28,30,31, Hermann, Paris, 1961–1965

[29] A. Grothendieck, J. A. Dieudonné, “Éléments de géométrie algébrique II”, Publ. Math. I.H.E.S., 8 (1961)

[30] D. Mumford, The red book of varieties and schemes, Springer – Verlag, Berlin, Heidelberg, 1999 | MR | Zbl

[31] A. E. Mironov, “Kommutativnye koltsa differentsialnykh operatorov, svyazannye s dvumernymi abelevymi mnogoobraziyami”, Sib. mat. zhurnal, 41:6 (2000), 1389–1403 | MR | Zbl

[32] A. E. Mironov, “Veschestvennye kommutiruyuschie differentsialnye operatory, svyazannye s dvumernymi abelevymi mnogoobraziyami”, Sib. mat. zhurnal, 43:1 (2002), 126–143 | MR | Zbl

[33] A. E. Mironov, “Kommutativnye koltsa differentsialnykh operatorov, otvechayuschie mnogomernym algebraicheskim mnogoobraziyam”, Sib. matem. zhurnal, 43:5 (2002), 1102–1114 | MR | Zbl

[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 | MR

[35] I. A. Melnik, A. E. Mironov, “Baker – Akhiezer modules on rational varieties”, SIGMA, 6 (2010), 030, 15 pp. | MR | Zbl