Autonomous limit of the 4-dimensional Painlevé-type equations and degeneration of curves of genus two
Annales de l'Institut Fourier, to appear, 49 p.

In recent studies, 4-dimensional analogs of the Painlevé equations were listed and there are 40 types. The aim of the present paper is to geometrically characterize these 40 Painlevé-type equations. For this purpose, we study the autonomous limit of these equations and degeneration of their spectral curves. The spectral curves are 2-parameter families of genus two curves and their generic degeneration are one of the types classified by Namikawa and Ueno. Liu’s algorithm enables us to find the degeneration types of the spectral curves for our 40 types of integrable systems. This result is analogous to the following fact; the families of the spectral curves of the autonomous 2-dimensional Painlevé equations P I , P II , P IV , P III D 8 , P III D 7 , P III D 6 , P V and P VI define elliptic surfaces with the singular fiber at H= of the Dynkin types E 8 (1) , E 7 (1) , E 6 (1) , D 8 (1) , D 7 (1) , D 6 (1) , D 5 (1) and D 4 (1) , respectively.

Dans des études récentes, des analogues en 4 dimensions des équations de Painlevé ont été répertoriés et il existe 40 types. Le but du présent article est de caractériser géométriquement ces 40 équations de type Painlevé. A cet effet, nous étudions la limite autonome de ces équations et la dégénérescence de leurs courbes spectrales. Les courbes spectrales sont des familles à 2 paramètres de courbes de genre deux et leur dégénérescences génériques sont d’un des types classés par Namikawa et Ueno. L’algorithme de Liu nous permet de trouver le types de dégénérescence de courbes spectrales pour nos 40 types de systémes intégrables. Ce résultat est analogue au fait suivant ; les familles des courbes spectrales des équations de Painlevé autonomes bidimensionnelles P I , P II , P IV , P III D 8 , P III D 7 , P III D 6 , P V et P VI définissent des surfaces elliptiques avec une fibre singuliére à H= des types Dynkin E 8 (1) , E 7 (1) , E 6 (1) , D 8 (1) , D 7 (1) , D 6 (1) , D 5 (1) et D 4 (1) , respectivement.

Received : 2016-11-14
Revised : 2017-08-12
Accepted : 2017-11-14
Classification:  34M55,  33E17,  14H70
Keywords: integrable system, Painlevé-type equations, isospectral limit, spectral curve, hyperelliptic curve, degeneration of curves
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     author = {Nakamura, Akane},
     title = {Autonomous limit of the 4-dimensional Painlev\'e-type equations and degeneration of curves of genus two},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Nakamura, Akane. Autonomous limit of the 4-dimensional Painlevé-type equations and degeneration of curves of genus two. Annales de l'Institut Fourier, to appear, 49 p.

[1] Adams, Malcolm; Harnad, John; Hurtubise, Jacques Dual moment maps into loop algebras, Lett. Math. Phys., Tome 20 (1990) no. 4, pp. 299-308 | Article | MR 1077962

[2] Adams, Malcolm; Harnad, John; Previato, Emma Isospectral Hamiltonian flows in finite and infinite dimensions. I. Generalized Moser systems and moment maps into loop algebras, Commun. Math. Phys., Tome 117 (1988) no. 3, pp. 451-500 | MR 953833

[3] Adler, Mark; Van Moerbeke, Pierre The complex geometry of the Kowalewski-Painlevé analysis, Invent. Math., Tome 97 (1989) no. 1, pp. 3-51 | Article | MR 999312

[4] Adler, Mark; Van Moerbeke, Pierre; Vanhaecke, Pol Algebraic integrability, Painlevé geometry and Lie algebras, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 47 (2004), xii+483 pages | Article | MR 2095251

[5] Audin, Michèle Spinning tops. A course on integrable systems, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 51 (1996), viii+139 pages | MR 1409362 | Zbl 0867.58034

[6] Bolsinov, Alekseí V.; Fomenko, Anatolii T. Integrable Hamiltonian systems. Geometry, topology, classification, Chapman & Hall/CRC (2004), xvi+730 pages (Translated from the 1999 Russian original) | Article | MR 2036760 | Zbl 1056.37075

[7] Chazy, Jean Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, Acta Math., Tome 34 (1911) no. 1, pp. 317-385 | Article | MR 1555070

[8] Cosgrove, Christopher M. Higher-order Painlevé equations in the polynomial class. I. Bureau symbol P2, Stud. Appl. Math., Tome 104 (2000) no. 1, pp. 1-65 | Article | MR 1738749

[9] Cosgrove, Christopher M. Higher-order Painlevé equations in the polynomial class. II. Bureau symbol P1, Stud. Appl. Math., Tome 116 (2006) no. 4, pp. 321-413 | Article | MR 2220477

[10] Dettweiler, Michael; Reiter, Stefan Middle convolution of Fuchsian systems and the construction of rigid differential systems, J. Algebra, Tome 318 (2007) no. 1, pp. 1-24 | Article | MR 2363121

[11] Fuchs, Reinhard Sur quelques équations différentielles linéaires du second ordre., C. R. Math. Acad. Sci. Paris, Tome 141 (1906), pp. 555-558 | Zbl 36.0397.02

[12] Haraoka, Yoshishige; Filipuk, Galina Middle convolution and deformation for Fuchsian systems, J. Lond. Math. Soc., Tome 76 (2007) no. 2, pp. 438-450 | Article | MR 2363425

[13] Harnad, John Dual isomonodromic deformations and moment maps to loop algebras, Commun. Math. Phys., Tome 166 (1994) no. 2, pp. 337-365 | MR 1309553

[14] Hiroe, Kazuki; Oshima, Toshio A classification of roots of symmetric Kac-Moody root systems and its application, Symmetries, integrable systems and representations, Springer (Springer Proceedings in Mathematics & Statistics) Tome 40 (2013), pp. 195-241 | Article | MR 3077686

[15] Hitchin, Nigel Stable bundles and integrable systems, Duke Math. J., Tome 54 (1987) no. 1, pp. 91-114 | Article | MR 885778

[16] Hukuhara, Masuo Sur les points singuliers des équations différentielles linéaires. II., J. Fac. Sci., Hokkaido Univ., Ser. I, Tome 5 (1937), pp. 123-166 | Zbl 0016.30502

[17] Iitaka, Shigeru On the degenerates of a normally polarized abelian varieties of dimension 2 and algebraic curves of genus 2 (in Japanese), University of Tokyo (1967) (Masters thesis)

[18] Inaba, Michi-Aki; Iwasaki, Katsunori; Saito, Masa-Hiko Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci., Tome 42 (2006) no. 4, pp. 987-1089 | MR 2289083

[19] Jimbo, Michio; Miwa, Tetsuji; Ueno, Kimio Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function, Physica D, Tome 2 (1981) no. 2, pp. 306-352 | Article | MR 630674

[20] Katz, Nicholas M. Rigid local systems, Princeton University Press, Annals of Mathematics Studies, Tome 139 (1996), viii+223 pages | MR 1366651

[21] Kawakami, Hiroshi Four-dimensional Painlevé-type equations associated with ramified linear equations I: Matrix Painlevé systems (2016) (https://arxiv.org/abs/1608.03927, to appear in Funkcial, Ekvac)

[22] Kawakami, Hiroshi Four-dimensional Painlevé-type equations associated with ramified linear equations III: Garnier systems and Fuji-Suzuki systems, SIGMA, Symmetry Integrability Geom. Methods Appl., Tome 13 (2017), 096, 50 pages (Art. ID 096, 50 p.) | Zbl 1402.34095

[23] Kawakami, Hiroshi Four-dimensional Painlevé-type equations associated with ramified linear equations II: Sasano systems, J. Integrable Sys., Tome 3 (2018) no. 1, xyy013, 36 pages (Art. ID xyy013, 36 p.) | Zbl 1402.34096

[24] Kawakami, Hiroshi; Nakamura, Akane; Sakai, Hidetaka Toward a classification of four-dimensional Painlevé-type equations, Algebraic and geometric aspects of integrable systems and random matrices, American Mathematical Society (Contemporary Mathematics) Tome 593 (2013), pp. 143-161 | Article | MR 3087954

[25] Kawakami, Hiroshi; Nakamura, Akane; Sakai, Hidetaka Degeneration scheme of 4-dimensional Painlevé-type equations, 4-dimensional Painlevé-type equations, Mathematical Society of Japan (MSJ Memoirs) Tome 37 (2018) | Zbl 07003998

[26] Kawamuko, Hiroyuki On the Garnier system of half-integer type in two variables, Funkc. Ekvacioj, Tome 52 (2009) no. 2, pp. 181-201 | Article | MR 2547101

[27] Kimura, Hironobu The degeneration of the two-dimensional Garnier system and the polynomial Hamiltonian structure, Ann. Mat. Pura Appl., Tome 155 (1989), pp. 25-74 | Article | MR 1042827

[28] Kimura, Hironobu Uniform foliation associated with the Hamiltonian system H n , Ann. Sc. Norm. Super. Pisa, Cl. Sci., Tome 20 (1993) no. 1, pp. 1-60 | MR 1215998

[29] Kimura, Hironobu Initial value spaces of degenerate Garnier systems, RIMS Kokyuroku, Tome 1133 (2000), pp. 18-27 | MR 1778322 | Zbl 0958.34514

[30] Kodaira, Kunihiko On compact analytic surfaces. II, Ann. Math., Tome 77 (1963), pp. 563-626 | MR 0184257 | Zbl 0118.15802

[31] Kodaira, Kunihiko On compact analytic surfaces. III, Ann. Math., Tome 78 (1963), pp. 1-40 | Zbl 0171.19601

[32] Kostov, Vladimir Petrov The Deligne-Simpson problem for zero index of rigidity, Perspectives of complex analysis, differential geometry and mathematical physics (St. Konstantin, 2000), World Scientific (2001), pp. 1-35 | MR 1877365

[33] Levelt, Antonius H. M. Jordan decomposition for a class of singular differential operators., Ark. Mat., Tome 13 (1975), pp. 1-27 | Article | Zbl 0305.34008

[34] Levin, Andrey M.; Olshanetsky, Mikhail A.; Zotov, Andrei V. Painlevé VI, rigid tops and reflection equation, Commun. Math. Phys., Tome 268 (2006) no. 1, pp. 67-103 | Article | MR 2249796

[35] Liu, Qing Courbes stables de genre 2 et leur schéma de modules, Math. Ann., Tome 295 (1993) no. 2, pp. 201-222 | Article | MR 1202389

[36] Liu, Qing Modèles minimaux des courbes de genre deux, J. Reine Angew. Math., Tome 453 (1994), pp. 137-164 | Article | MR 1285783

[37] Malmquist, Johannes Sur les équations différentielles du second ordre dont l’intégrale générales a ses points critiques fixes, Arkiv. Mat. Astr. Fys., Tome 17 (1922-23), pp. 1-89 | Zbl 49.0305.02

[38] Nakamura, Akane On the Bäcklund transformations of the matrix Painlevé equations (in Japanese), University of Tokyo (Japan) (2011) (Masters thesis)

[39] Nakamura, Akane The degeneration of the Painlevé divisors associated with the autonomous Garnier system of type 9/2, Josai Math. Mono. (2017)

[40] Namikawa, Yukihiko; Ueno, Kenji The complete classification of fibres in pencils of curves of genus two, Manuscr. Math., Tome 9 (1973), pp. 143-186 | MR 0369362

[41] Ogg, Andrew On pencils of curves of genus two, Topology, Tome 5 (1966), pp. 355-362 | MR 0201437

[42] Okamoto, Kazuo Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Jpn. J. Math., New Ser., Tome 5 (1979) no. 1, pp. 1-79 | MR 614694

[43] Okounkov, Andrei; Rains, Eric Noncommutative geometry and Painlevé equations, Algebra Number Theory, Tome 9 (2015) no. 6, pp. 1363-1400 | Article | MR 3397405

[44] Oshima, Toshio Fractional calculus of Weyl algebra and Fuchsian differential equations, Mathematical Society of Japan, MSJ Memoirs, Tome 28 (2012)

[45] Rains, Eric (private communication)

[46] Rains, Eric Generalized Hitchin systems on rational surfaces (2013) (https://arxiv.org/abs/1307.4033 )

[47] Rains, Eric The (noncommutative) geometry of differential and difference equations (2016) (Presented at NSF/CBMS Regional Research Conference on Discrete Painlevé Equations, University of Texas Rio Grande Valley)

[48] Rains, Eric The noncommutative geometry of elliptic difference equations (2016) (https://arxiv.org/abs/1607.08876 )

[49] Sakai, Hidetaka Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Commun. Math. Phys., Tome 220 (2001) no. 1, pp. 165-229 | Article | MR 1882403

[50] Sakai, Hidetaka Ordinary differential equations on rational elliptic surfaces, Symmetries, integrable systems and representations, Springer (Springer Proceedings in Mathematics & Statistics) Tome 40 (2013), pp. 515-541 | Article | MR 3077699

[51] Sakai, Hidetaka Isomonodromic deformation and 4-dimensional Painlevé type equations, 4-dimensional Painlevé-type equations, Mathematical Society of Japan (MSJ Memoirs) Tome 37 (2018) | Zbl 07003998

[52] Simpson, Carlos The Hodge filtration on nonabelian cohomology, Algebraic geometry—Santa Cruz 1995, American Mathematical Society (Proceedings of Symposia in Pure Mathematics) Tome 62 (1997), pp. 217-281 | Article | MR 1492538

[53] Suzuki, Masaki Spaces of initial conditions of Garnier system and its degenerate systems in two variables, J. Math. Soc. Japan, Tome 58 (2006) no. 4, pp. 1079-1117 | MR 2276182

[54] Tahara, Nobuhiko An augmentation of the phase space of the system of type A 4 (1) , Kyushu J. Math., Tome 58 (2004) no. 2, pp. 393-425 | Article | MR 2117253

[55] Tate, John Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer (Lecture Notes in Mathematics) Tome 476 (1975), pp. 33-52 | MR 0393039

[56] Turrittin, Hugh L. Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math., Tome 93 (1955), pp. 27-66 | Article | Zbl 0064.33603

[57] Yamakawa, Daisuke Fourier-Laplace transform and isomonodromic deformations, Funkc. Ekvacioj, Tome 59 (2016) no. 3, pp. 315-349 | MR 3642539