パンルヴェ型微分方程式と代数幾何
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| Published in | 数学 Vol. 62; no. 4; pp. 524 - 544 |
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一般社団法人 日本数学会
2010
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| ISSN | 0039-470X 1883-6127 |
| DOI | 10.11429/sugaku.0624524 |
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| Author | 齋藤, 政彦 |
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| Author_xml | – sequence: 1 fullname: 齋藤, 政彦 organization: 神戸大学大学院理学研究科 |
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| References | [JM] M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D, 2 (1981), 407-448. [IIS2] M. Inaba, K. Iwasaki and M.-H. Saito, Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci., 42 (2006), 987-1089. [OT] K. Okamoto and K. Takano, The proof of the Painlevé property by Masuo Hukuhara, Funkcial. Ekvac., 44 (2001), 201-217. [Yam] D. Yamakawa, Geometry of multiplicative preprojective algebra, Int. Math. Res. Papers, 2008, article ID rpn008, 77pp. [MY] M. Maruyama and K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann., 293 (1992), 77-99. [ShT] T. Shioda and K. Takano, On some Hamiltonian structures of Painlevé systems. I, Funkcial. Ekvac., 40 (1997), 271-291. [MMT] T. Matano, A. Matumiya and K. Takano, On some Hamiltonian structures of Painlevé systems. II, J. Math. Soc. Japan, 51 (1999), 843-866. [Sak] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys., 220 (2001), 165-229. [OI] 岡本和夫, パンルヴェ方程式, 岩波書店, 2009, p 286. [STe] M.-H. Saito and H. Terajima, Nodal curves and Riccati solutions of Painlevé equations, J. Math. Kyoto Univ., 44 (2004), 529-568. [G] B. Gambier, Sur les équations différentiells du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes, Acta Math., 33 (1910), 1-55. [Mar] M. Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ., 18 (1978), 557-614. [Mat] A. Matumiya, On some Hamiltonian structures of Painlevé systems. III, Kumamoto J. Math., 10 (1997), 45-73. [O2] K. Okamoto, Polynomial Hamiltonians associated with Painlevé equations, I, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 264-268; II, ibid. 367-371. [OJ1] 岡本和夫, Painlevéの方程式, 数学, 32 (1980), 30-43. [IIS3] M. Inaba, K. Iwasaki and M.-H. Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II, In: Moduli spaces and arithmetic geometry, Kyoto, 2004, (eds. S. Mukai, Y. Miyaoka, S. Mori, A. Moriwaki and I. Nakamura), Adv. Stud. Pure Math., 45 (2006), 387-432. [STT] M.-H. Saito, T. Takebe and H. Terajima, Deformation of Okamoto-Painlevé pairs and Painlevé equations, J. Algebraic Geom., 11 (2002), 311-362. [UJ2] 梅村浩, Painlevé方程式と古典函数, 数学, 47 (1995), 341-359. [Mal1] B. Malgrange, Sur les déformations isomonodromiques. I. Singularités régulières; Sur les déformation isomonodromiques. II. Singularités irrégulières, In: Mathématique et Physique, Paris, 1979/1982, Progr. Math., 37, Birkhäuser, Boston, 1983, pp. 401-426; pp. 427-438. [In2] M. Inaba, Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence, preprint, arXiv:math.AG/0602004. [PaO] P. Painlevé, Oeuvres de Paul Painle\'{v}e. Tome I, Éditions du Centre National de la Recherche Scientifique, Paris, 1973. [STa] M.-H. Saito and T. Takebe, Classification of Okamoto-Painlevé pairs, Kobe J. Math., 19 (2002), 21-50. [Iw1] K. Iwasaki, Moduli and deformation for Fuchsian projective connections on a Riemann surface. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 38 (1991), 431-531. [IKSY] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, Wiesbaden, 1991. [IU2] K. Iwasaki and T. Uehara, Chaos in the sixth Painlevé equation, In: Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, Proceedings of the Conference, Kyoto Univ., 2006, (ed. Y. Takei), RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci., Kyoto, 2007, pp. 73-88. [IU1] K. Iwasaki and T. Uehara, An ergodic study of Painlevé VI, Math. Ann., 338 (2007), 295-345. [AB] D. Arinkin and A. Borodin, Moduli spaces of d-connections and difference Painlevé equations, Duke Math. J., 134 (2006), 515-556. [Iw2] K. Iwasaki, Fuchsian moduli on a Riemann surface-its Poisson structure and Poincaré-Lefschetz duality, Pacific J. Math., 155 (1992), 319-340. [N] H. Nakajima, Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces, In: Moduli of vector bundles, Sanda, 1994; Kyoto, 1994, (ed. M. Maruyama), Lecture Notes in Pure and Appl. Math., 179, Dekker, New York, 1996, 199-208. [D] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math., 163, Springer, Berlin, 1970. [Mu] D. Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band, 34, Springer, Berlin-New York, 1965. [NY] 野海正俊・山田泰彦, Painlevé方程式の対称性, 数学, 53 (2001), 62-75. [Te1] H. Terajima, Local cohomology of generalized Okamoto-Painlevé pairs and Painlevé equations, Japan. J. Math. (N.S.), 28 (2002), 137-162. [Iw3] K. Iwasaki, A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. Ser. A Math. Sci., 78 (2002), 131-135. [AL2] D. Arinkin and S. Lysenko, Isomorphisms between moduli spaces of $SL(2)$-bundles with connections on $\boldsymbol{P}^1 \backslash \{x_1, \dots, x_4 \}$, Math. Res. Lett., 4 (1997), 181-190. [Mal2] B. Malgrange, Déformations isomonodromiques, forme de Liouville, fonction $\tau$, Ann. Inst. Fourier (Grenoble), 54 (2004), 1371-1392. [Sas] Y. Sasano, Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $D_6^{(1)}$. II, In: Algebraic analysis and the exact WKB analysis for systems of differential equations, RIMS Kôkyûroku Bessatsu, B5, Res. Inst. Math. Sci., Kyoto, 2008, pp. 137-152. [Po] H. Poincaré, Sur un théorème de M. Fuchs, Acta Math., 7 (1885), 1-32. [Si] C. Simpson, The Hodge filtration on nonabelian cohomology, In: Algebraic geometry-Santa Cruz 1995, Proc. Sympos. Pure Math., Amer. Math. Soc., 62 (1997), 217-281. [SU] M.-H. Saito and H. Umemura, Painlevé equations and deformations of rational surfaces with rational double points, In: Physics and combinatorics 1999, Proceedings of the International Workshop, Nagoya Univ., 1999, (eds. A. N. Kirillov, A. Tsuchiya and H. Umemura), World Sci. Publ., River Edge, NJ, 2001, pp. 320-365. [IISA] M. Inaba, K. Iwasaki and M.-H. Saito, Dynamics of the sixth Painlevé Equation, In: Théories asymptotiques et équations de Painlevé, Angers, 2004, Semin. Congr., 14, Soc. Math. France, 2006, 103-167. [J] M. Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci., 18 (1982), 1137-1161. [AL1] D. Arinkin and S. Lysenko, On the moduli of ${\rm SL}(2)$-bundles with connections on $\boldsymbol{P}^1 \backslash \{t_1, \cdots, t_4 \}$, Internat. Math. Res. Notices, 1997, no. 19, 983-999. [F] R. Fuchs, Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann., 63 (1907), 301-321. [O3] K. Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{VI}$, Ann. Math. Pura Appl. (4), 146 (1987), 337-381. [Sh] S. Shimomura, Proofs of the Painlevé property for all Painlevé equations, Japan. J. Math. (N.S.), 29 (2003), 159-180. [UJ3] 梅村浩, Painlevé方程式の100年, 数学, 51 (1999), 59-84. [Pa1] P. Painlevé, Mémoire sur les équations différentielles dont l'intégrale générale est uniforme, Bull. Soc. Math. France, 28 (1900), 201-261. [Bo] P. Boalch, From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. (3), 90 (2005), 167-208. [PuSa] M. van der Put and M.-H. Saito, Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier, 59 (2009), no. 7, 2611-2667. preprint, 2009, arXiv:0902.1702, to appear in Ann. Inst. Fourier. [IIS1] M. Inaba, K. Iwasaki and M.-H. Saito, Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert Correspondence, Int. Math. Res. Not., 2004, no. 1, 1-30. [SY] Y. Sasano and Y. Yamada, Symmetry and holomorphy of Painlevé type systems, In: Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, Proceedings of the Conference, Kyoto Univ., 2006, (ed. Y. Takei), RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci., Kyoto, 2007, pp. 73-88. [O1] K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. (N.S.), 5, (1979), 1-79. [A] D. Arinkin, Orthogonality of natural sheaves on moduli stacks of $SL(2)$-bundles with connections on $\boldsymbol{P}^1$ minus 4 points, Selecta Math. (N.S.), 7 (2001), 213-239. [Iw4] K. Iwasaki, An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation, Comm. Math. Phys., 242 (2003), 185-219. [Te2] H. Terajima, Families of Okamoto-Painlevé pairs and Painlevé equations, Ann. Mat. Pura Appl. (4), 186 (2007), 99-146. [K] K. Kodaira, On compact analytic surfaces. II, Ann. of Math. (2), 77 (1963), 563-626. [JMU] M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D, 2 (1981), 306-352. [Hu] M. Hukuhara, 1. Meromofeco de la solvo de (F) $y"=6y^2+x$, 2. La ekvacio (VI) de Painlevé-Gambier, unpublished lecture notes. [Mi] T. Miwa, Painlevé property of monodromy preserving equations and the analyticity of $\tau$ function, Publ. Res. Inst. Math. Sci., Kyoto Univ., 17 (1981), 703-721. [PuSi] M. van der Put and M. F. Singer, Galois theory of linear differential equations, Grundlehren Math. Wiss., 328, Springer, Berlin, 2003. [Pa2] P. Painlevé, Sur les équations différentielles du second ordre à points critiques fixes, C. R. Acad. Sci. Paris, 143 (1906), 1111-1117. [OS] 岡本和夫, パンルヴェ方程式序説, 上智大学数学講究録, no. 19, 1985. [UJ1] 梅村浩, Painlevé方程式の既約性について, 数学, 40 (1988), 47-60. [In1] M. Inaba, Moduli of parabolic stable sheaves on a projective scheme, J. Math. Kyoto Univ., 40 (2000), 119-136. |
| References_xml | – reference: [Te1] H. Terajima, Local cohomology of generalized Okamoto-Painlevé pairs and Painlevé equations, Japan. J. Math. (N.S.), 28 (2002), 137-162. – reference: [Hu] M. Hukuhara, 1. Meromofeco de la solvo de (F) $y"=6y^2+x$, 2. La ekvacio (VI) de Painlevé-Gambier, unpublished lecture notes. – reference: [F] R. Fuchs, Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann., 63 (1907), 301-321. – reference: [MMT] T. Matano, A. Matumiya and K. Takano, On some Hamiltonian structures of Painlevé systems. II, J. Math. Soc. Japan, 51 (1999), 843-866. – reference: [Mal2] B. Malgrange, Déformations isomonodromiques, forme de Liouville, fonction $\tau$, Ann. Inst. Fourier (Grenoble), 54 (2004), 1371-1392. – reference: [JM] M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D, 2 (1981), 407-448. – reference: [D] P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math., 163, Springer, Berlin, 1970. – reference: [OT] K. Okamoto and K. Takano, The proof of the Painlevé property by Masuo Hukuhara, Funkcial. Ekvac., 44 (2001), 201-217. – reference: [PuSi] M. van der Put and M. F. Singer, Galois theory of linear differential equations, Grundlehren Math. Wiss., 328, Springer, Berlin, 2003. – reference: [Po] H. Poincaré, Sur un théorème de M. Fuchs, Acta Math., 7 (1885), 1-32. – reference: [J] M. Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci., 18 (1982), 1137-1161. – reference: [OI] 岡本和夫, パンルヴェ方程式, 岩波書店, 2009, p 286. – reference: [Mi] T. Miwa, Painlevé property of monodromy preserving equations and the analyticity of $\tau$ function, Publ. Res. Inst. Math. Sci., Kyoto Univ., 17 (1981), 703-721. – reference: [UJ2] 梅村浩, Painlevé方程式と古典函数, 数学, 47 (1995), 341-359. – reference: [PaO] P. Painlevé, Oeuvres de Paul Painle\'{v}e. Tome I, Éditions du Centre National de la Recherche Scientifique, Paris, 1973. – reference: [ShT] T. Shioda and K. Takano, On some Hamiltonian structures of Painlevé systems. I, Funkcial. Ekvac., 40 (1997), 271-291. – reference: [IISA] M. Inaba, K. Iwasaki and M.-H. Saito, Dynamics of the sixth Painlevé Equation, In: Théories asymptotiques et équations de Painlevé, Angers, 2004, Semin. Congr., 14, Soc. Math. France, 2006, 103-167. – reference: [Pa2] P. Painlevé, Sur les équations différentielles du second ordre à points critiques fixes, C. R. Acad. Sci. Paris, 143 (1906), 1111-1117. – reference: [Sas] Y. Sasano, Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type $D_6^{(1)}$. II, In: Algebraic analysis and the exact WKB analysis for systems of differential equations, RIMS Kôkyûroku Bessatsu, B5, Res. Inst. Math. Sci., Kyoto, 2008, pp. 137-152. – reference: [A] D. Arinkin, Orthogonality of natural sheaves on moduli stacks of $SL(2)$-bundles with connections on $\boldsymbol{P}^1$ minus 4 points, Selecta Math. (N.S.), 7 (2001), 213-239. – reference: [AB] D. Arinkin and A. Borodin, Moduli spaces of d-connections and difference Painlevé equations, Duke Math. J., 134 (2006), 515-556. – reference: [Iw4] K. Iwasaki, An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation, Comm. Math. Phys., 242 (2003), 185-219. – reference: [IIS2] M. Inaba, K. Iwasaki and M.-H. Saito, Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI. I, Publ. Res. Inst. Math. Sci., 42 (2006), 987-1089. – reference: [JMU] M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D, 2 (1981), 306-352. – reference: [UJ3] 梅村浩, Painlevé方程式の100年, 数学, 51 (1999), 59-84. – reference: [In2] M. Inaba, Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence, preprint, arXiv:math.AG/0602004. – reference: [Sak] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys., 220 (2001), 165-229. – reference: [PuSa] M. van der Put and M.-H. Saito, Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier, 59 (2009), no. 7, 2611-2667. preprint, 2009, arXiv:0902.1702, to appear in Ann. Inst. Fourier. – reference: [UJ1] 梅村浩, Painlevé方程式の既約性について, 数学, 40 (1988), 47-60. – reference: [IU2] K. Iwasaki and T. Uehara, Chaos in the sixth Painlevé equation, In: Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, Proceedings of the Conference, Kyoto Univ., 2006, (ed. Y. Takei), RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci., Kyoto, 2007, pp. 73-88. – reference: [STa] M.-H. Saito and T. Takebe, Classification of Okamoto-Painlevé pairs, Kobe J. Math., 19 (2002), 21-50. – reference: [MY] M. Maruyama and K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann., 293 (1992), 77-99. – reference: [NY] 野海正俊・山田泰彦, Painlevé方程式の対称性, 数学, 53 (2001), 62-75. – reference: [OS] 岡本和夫, パンルヴェ方程式序説, 上智大学数学講究録, no. 19, 1985. – reference: [Mal1] B. Malgrange, Sur les déformations isomonodromiques. I. Singularités régulières; Sur les déformation isomonodromiques. II. Singularités irrégulières, In: Mathématique et Physique, Paris, 1979/1982, Progr. Math., 37, Birkhäuser, Boston, 1983, pp. 401-426; pp. 427-438. – reference: [SY] Y. Sasano and Y. Yamada, Symmetry and holomorphy of Painlevé type systems, In: Algebraic, analytic and geometric aspects of complex differential equations and their deformations. Painlevé hierarchies, Proceedings of the Conference, Kyoto Univ., 2006, (ed. Y. Takei), RIMS Kôkyûroku Bessatsu, B2, Res. Inst. Math. Sci., Kyoto, 2007, pp. 73-88. – reference: [IIS1] M. Inaba, K. Iwasaki and M.-H. Saito, Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert Correspondence, Int. Math. Res. Not., 2004, no. 1, 1-30. – reference: [Sh] S. Shimomura, Proofs of the Painlevé property for all Painlevé equations, Japan. J. Math. (N.S.), 29 (2003), 159-180. – reference: [Mat] A. Matumiya, On some Hamiltonian structures of Painlevé systems. III, Kumamoto J. Math., 10 (1997), 45-73. – reference: [STT] M.-H. Saito, T. Takebe and H. Terajima, Deformation of Okamoto-Painlevé pairs and Painlevé equations, J. Algebraic Geom., 11 (2002), 311-362. – reference: [AL1] D. Arinkin and S. Lysenko, On the moduli of ${\rm SL}(2)$-bundles with connections on $\boldsymbol{P}^1 \backslash \{t_1, \cdots, t_4 \}$, Internat. Math. Res. Notices, 1997, no. 19, 983-999. – reference: [Pa1] P. Painlevé, Mémoire sur les équations différentielles dont l'intégrale générale est uniforme, Bull. Soc. Math. France, 28 (1900), 201-261. – reference: [Iw2] K. Iwasaki, Fuchsian moduli on a Riemann surface-its Poisson structure and Poincaré-Lefschetz duality, Pacific J. Math., 155 (1992), 319-340. – reference: [K] K. Kodaira, On compact analytic surfaces. II, Ann. of Math. (2), 77 (1963), 563-626. – reference: [IKSY] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, Wiesbaden, 1991. – reference: [IIS3] M. Inaba, K. Iwasaki and M.-H. Saito, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. II, In: Moduli spaces and arithmetic geometry, Kyoto, 2004, (eds. S. Mukai, Y. Miyaoka, S. Mori, A. Moriwaki and I. Nakamura), Adv. Stud. Pure Math., 45 (2006), 387-432. – reference: [Si] C. Simpson, The Hodge filtration on nonabelian cohomology, In: Algebraic geometry-Santa Cruz 1995, Proc. Sympos. Pure Math., Amer. Math. Soc., 62 (1997), 217-281. – reference: [SU] M.-H. Saito and H. Umemura, Painlevé equations and deformations of rational surfaces with rational double points, In: Physics and combinatorics 1999, Proceedings of the International Workshop, Nagoya Univ., 1999, (eds. A. N. Kirillov, A. Tsuchiya and H. Umemura), World Sci. Publ., River Edge, NJ, 2001, pp. 320-365. – reference: [Iw1] K. Iwasaki, Moduli and deformation for Fuchsian projective connections on a Riemann surface. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 38 (1991), 431-531. – reference: [IU1] K. Iwasaki and T. Uehara, An ergodic study of Painlevé VI, Math. Ann., 338 (2007), 295-345. – reference: [OJ1] 岡本和夫, Painlevéの方程式, 数学, 32 (1980), 30-43. – reference: [O1] K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. (N.S.), 5, (1979), 1-79. – reference: [In1] M. Inaba, Moduli of parabolic stable sheaves on a projective scheme, J. Math. Kyoto Univ., 40 (2000), 119-136. – reference: [Iw3] K. Iwasaki, A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. Ser. A Math. Sci., 78 (2002), 131-135. – reference: [G] B. Gambier, Sur les équations différentiells du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes, Acta Math., 33 (1910), 1-55. – reference: [Te2] H. Terajima, Families of Okamoto-Painlevé pairs and Painlevé equations, Ann. Mat. Pura Appl. (4), 186 (2007), 99-146. – reference: [AL2] D. Arinkin and S. Lysenko, Isomorphisms between moduli spaces of $SL(2)$-bundles with connections on $\boldsymbol{P}^1 \backslash \{x_1, \dots, x_4 \}$, Math. Res. Lett., 4 (1997), 181-190. – reference: [O2] K. Okamoto, Polynomial Hamiltonians associated with Painlevé equations, I, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 264-268; II, ibid. 367-371. – reference: [Mu] D. Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band, 34, Springer, Berlin-New York, 1965. – reference: [Mar] M. Maruyama, Moduli of stable sheaves. II, J. Math. Kyoto Univ., 18 (1978), 557-614. – reference: [Bo] P. Boalch, From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. (3), 90 (2005), 167-208. – reference: [Yam] D. Yamakawa, Geometry of multiplicative preprojective algebra, Int. Math. Res. Papers, 2008, article ID rpn008, 77pp. – reference: [STe] M.-H. Saito and H. Terajima, Nodal curves and Riccati solutions of Painlevé equations, J. Math. Kyoto Univ., 44 (2004), 529-568. – reference: [N] H. Nakajima, Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces, In: Moduli of vector bundles, Sanda, 1994; Kyoto, 1994, (ed. M. Maruyama), Lecture Notes in Pure and Appl. Math., 179, Dekker, New York, 1996, 199-208. – reference: [O3] K. Okamoto, Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{VI}$, Ann. Math. Pura Appl. (4), 146 (1987), 337-381. |
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| SubjectTerms | パンルヴェ型微分方程式 リーマン・ヒルベルト対応 初期値空間 安定放物接続のモジュライ空間 岡本・パンルヴェ対 |
| Title | パンルヴェ型微分方程式と代数幾何 |
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