×

Surprising examples of nonrational smooth spectral surfaces. (English. Russian original) Zbl 1408.13069

Sb. Math. 209, No. 8, 1131-1154 (2018); translation from Mat. Sb. 209, No. 8, 29-55 (2018).
The aim of this paper is to study necessary and sufficient algebro-geometric conditions for the existence of a nontrivial commutative subalgebra of rank \(1\) in \(\widehat{D}\), a completion of the algebra of partial differential operators in two variables, which can be thought of as a simple algebraic analogue of the algebra of analytic pseudo-differential operators on a manifold. These are conditions on a projective (spectral) surface; they are encoded in a new notion of pre-spectral data. For smooth surfaces the sufficient conditions look especially simple. On a smooth projective surface there should exist an ample integral curve \(C\) with \(C^2=1\) and \(h^0(X, \mathcal{O}_X(C))=1\), and a divisor \(D\) with \((D, C)_X=g(C)-1\), \(h^i(X, \mathcal{O}_X(D))=0\), \(i=0, 1, 2\), and \(h^0(X, \mathcal{O}_X(D+C))=1\). What is amazing is that there are examples of such surfaces for which the corresponding commutative subalgebras do not admit isospectral deformations. This paper is organized as follows: the first section is an introduction to the subject. In the second section the author gives a review of the classification theory of commuting operators. At the end of the section he proves a refined version of the classification theorem for commutative algebras of rank \(1\). In the third section, the author introduces the notion of pre-spectral data of rank \(1\) and shows that they can be extended to spectral data, thus reducing the problem of finding examples of commutative subalgebras in the algebra \(\widehat{D}\) to a purely algebro-geometric problem. In particular, he shows that the divisor \(C\) (which is a priori \(\mathbb{Q}\)-Cartier) is Cartier. Section four is devoted to the question of the existence of smooth spectral surfaces. He recalls the notion of trivial commutative subalgebras and shows that smooth spectral surfaces with ample divisors of arithmetical genus less than or equal to \(1\) lead to trivial commutative subalgebras. Then he shows that the properties of the smooth surfaces mentioned in the abstract are sufficient for the existence of pre-spectral data with smooth spectral surface and a locally free spectral sheaf. There are examples of such surfaces, and the moduli space of spectral sheaves is finite in these examples, that is, the corresponding commutative subalgebras do not have any isospectral deformations.

MSC:

13N15 Derivations and commutative rings
14H81 Relationships between algebraic curves and physics
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
Full Text: DOI

References:

[1] Bădescu, L., IMPAN Monogr. Mat. (N. S.), 65, (2004), Birkhäuser Verlag: Birkhäuser Verlag, Basel · Zbl 1077.14001 · doi:10.1007/978-3-0348-7936-1
[2] Berest, Yu.; Etingof, P.; Ginzburg, V., Cherednik algebras and differential operators on quasi-invariants, Duke Math. J., 118, 2, 279-337, (2003) · Zbl 1067.16047 · doi:10.1215/S0012-7094-03-11824-4
[3] Berest, Yu.; Kasman, A., Lett. Math. Phys., 43, 3, 279-294, (1998) · Zbl 0979.13025 · doi:10.1023/A:1007436917801
[4] Burban, I.; Zheglov, A., Cohen-Macaulay modules over the algebra of planar quasi- invariants and Calogero-Moser systems · Zbl 1453.14011
[5] Bourbaki, N., Actualités Sci. Indust., 1290, 1293, 1308, 1314, (19611965), Hermann: Hermann, Paris · Zbl 0108.04002
[6] Chalykh, O., Algebro-geometric Schrödinger operators in many dimensions, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366, 1867, 947-971, (2008) · Zbl 1153.14303 · doi:10.1098/rsta.2007.2057
[7] Chalykh, O. A.; Veselov, A. P., Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys., 126, 3, 597-611, (1990) · Zbl 0746.47025 · doi:10.1007/BF02125702
[8] Veselov, A. P.; Styrkas, K. L.; Chalykh, O. A., Algebraic integrability for the Schrödinger equation and finite reflection groups, Teor. Mat. Fiz., 94, 2, 253-275, (1993) · Zbl 0805.47070 · doi:10.1007/BF01019330
[9] Chalykh, O.; Feigin, M.; Veselov, A., New integrable generalizations of Calogero-Moser quantum problem, J. Math. Phys., 39, 2, 695-703, (1998) · Zbl 0906.34061 · doi:10.1063/1.532347
[10] Dubrovin, B. A., Matrix finite-zone operators, J. Soviet Math., 23, 1, 33-78, (1983) · Zbl 0561.58043 · doi:10.1007/BF02104895
[11] Dubrovin, B. A.; Matveev, V. B.; Novikov, S. P., Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Uspekhi Mat. Nauk, 31, 1-187, 55-136, (1976) · Zbl 0326.35011 · doi:10.1070/RM1976v031n01ABEH001446
[12] Dubrovin, B. A.; Krichever, I. M.; Novikov, S. P., Integrable systems. I, Dynamical systems – 4, 4, 179-277, (1985) · Zbl 0591.58013 · doi:10.1007/978-3-662-06793-2_3
[13] Feigin, M.; Johnston, D., A class of Baker-Akhiezer arrangements, Comm. Math. Phys., 328, 3, 1117-1157, (2014) · Zbl 1301.52039 · doi:10.1007/s00220-014-1921-4
[14] Grinevich, P. G., Vector rank of commuting matrix differential operators. Proof of S. P. Novikov’s criterion, Izv. Akad. Nauk SSSR Ser. Mat., 50, 3, 458-478, (1986) · Zbl 0623.47049 · doi:10.1070/IM1987v028n03ABEH000892
[15] Grothendieck, A., Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math., 8, 5-222, (1961)
[16] Grothendieck, A., Éléments de géométrie algèbrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math., 28, 5-255, (1966) · Zbl 0144.19904
[17] Hartshorne, R., Grad. Texts in Math., 52, (1977), Springer-Verlag: Springer-Verlag, New York-Heidelberg · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0
[18] Heckman, G. J., A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups, 101, 181-191, (1991) · Zbl 0749.33005
[19] Heckman, G. J.; Opdam, E. M., Root systems and hypergeometric functions. I, Compositio Math., 64, 3, 329-352, (1987) · Zbl 0656.17006
[20] Krichever, I. M., Methods of algebraic geometry in the theory of non-linear equations, Uspekhi Mat. Nauk, 32, 6-198, 183-208, (1977) · Zbl 0372.35002 · doi:10.1070/RM1977v032n06ABEH003862
[21] Krichever, I. M., Commutative rings of ordinary linear differential operators, Funktsional. Anal. i Prilozhen., 12, 3, 20-31, (1978) · Zbl 0408.34008 · doi:10.1007/BF01681429
[22] Krichever, I. M.; Novikov, S. P., Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles, Uspekhi Mat. Nauk, 58, 3-351, 51-88, (2003) · Zbl 1060.37068 · doi:10.4213/rm628
[23] Kulikov, Vik. S., On divisors of small canonical degree on Godeaux surfaces, Mat. Sb., 209, 8, 56-65, (2018) · Zbl 1428.14068 · doi:10.1070/SM9032
[24] Kurke, H.; Osipov, D.; Zheglov, A., Commuting differential operators and higher- dimensional algebraic varieties, Selecta Math. (N. S.), 20, 4, 1159-1195, (2014) · Zbl 1306.37077 · doi:10.1007/s00029-014-0155-9
[25] Zheglov, A. B.; Kurke, H., Geometric properties of commutative subalgebras of partial differential operators, Mat. Sb., 206, 5, 61-106, (2015) · Zbl 1329.13043 · doi:10.4213/sm8429
[26] Mauleshova, G. S.; Mironov, A. E., One-point commuting difference operators of rank 1, Dokl. Ross. Akad. Nauk, 466, 4, 399-401, (2016) · Zbl 1361.13012 · doi:10.1134/S106456241601021X
[27] Mironov, A. E., Commutative rings of differential operators corresponding to multidimensional algebraic varieties, Sibirsk. Mat. Zh., 43, 5, 1102-1114, (2002) · Zbl 1006.14016 · doi:10.1023/A:1020158924466
[28] Cho, K.; Mironov, A.; Nakayashiki, A., Baker-Akhiezer modules on the intersections of shifted theta divisors, Publ. Res. Inst. Math. Sci., 47, 2, 553-567, (2011) · Zbl 1225.14034 · doi:10.2977/PRIMS/43
[29] Mulase, M., Category of vector bundles on algebraic curves and infinite dimensional Grassmanians, Internat. J. Math., 1, 3, 293-342, (1990) · Zbl 0723.14010 · doi:10.1142/S0129167X90000174
[30] Mulase, M., Algebraic theory of the KP equations, Perspectives in mathematical physics, III, 151-217, (1994) · Zbl 0837.35132
[31] Nakayashiki, A., Commuting partial differential operators and vector bundles over Abelian varieties, Amer. J. Math., 116, 1, 65-100, (1994) · Zbl 0809.14016 · doi:10.2307/2374982
[32] Nakayashiki, A., Structure of Baker-Akhiezer modules of principally polarized Abelian varieties, commuting partial differential operators and associated integrable systems, Duke Math. J., 62, 2, 315-358, (1991) · Zbl 0732.14008 · doi:10.1215/S0012-7094-91-06213-7
[33] Osipov, D. V., The Krichever correspondence for algebraic varieties, Izv. Ross. Akad. Nauk Ser. Mat., 65, 5, 91-128, (2001) · Zbl 1068.14053 · doi:10.4213/im358
[34] Gorchinskiy, S. O.; Osipov, D. V., Continuous homomorphisms between algebras of iterated Laurent series over a ring, Proc. Steklov Inst. Math., 294, 54-75, (2016) · Zbl 1359.13023 · doi:10.1134/S0371968516030031
[35] Gorchinskiy, S. O.; Osipov, D. V., Higher-dimensional Contou-Carrère symbol and continuous automorphisms, Funktsional. Anal. i Prilozhen., 50, 4, 26-42, (2016) · Zbl 1360.19005 · doi:10.4213/faa3252
[36] Zheglov, A. B.; Osipov, D. V., On some questions related to the Krichever correspondence, Mat. Zametki, 81, 4, 528-539, (2007) · Zbl 1134.14023 · doi:10.4213/mzm3695
[37] Olshanetsky, M. A.; Perelomov, A. M., Quantum integrable systems related to Lie algebras, Phys. Rep., 94, 6, 313-404, (1983) · doi:10.1016/0370-1573(83)90018-2
[38] Parshin, A. N., The Krichever correspondence for algebraic surfaces, Funktsional. Anal. i Prilozhen., 35, 1, 88-90, (2001) · Zbl 1078.14525 · doi:10.4213/faa237
[39] Parshin, A. N., Integrable systems and local fields, Comm. Algebra, 29, 9, 4157-4181, (2001) · Zbl 1014.14015 · doi:10.1081/AGB-100105994
[40] Previato, E., Multivariable Burchnall-Chaundy theory, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366, 1867, 1155-1177, (2008) · Zbl 1153.37420 · doi:10.1098/rsta.2007.2064
[41] Przyjalkowski, V.; Shramov, C., On Hodge numbers of complete intersections and Landau-Ginzburg models, Int. Math. Res. Not., 2015, 21, 11302-11332, (2015) · Zbl 1343.14038 · doi:10.1093/imrn/rnv024
[42] Przyjalkowski, V. V., Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds, Mat. Sb., 208, 7, 84-108, (2017) · Zbl 1386.14055 · doi:10.4213/sm8838
[43] Rothstein, M., Dynamics of the Krichever construction in several variables, J. Reine Angew. Math., 2004, 572, 111-138, (2004) · Zbl 1142.37362 · doi:10.1515/crll.2004.046
[44] Zheglov, A. B., On rings of commuting partial differential operators, Algebra i Analiz, 25, 5, 86-145, (2013) · Zbl 1325.13025 · doi:10.1090/S1061-0022-2014-01316-7
[45] Zheglov, A. B., D. Sc. dissertation, (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.