Deformation of Okamoto–Painlevé pairs and Painlevé equations

Authors:
Masa-Hiko Saito, Taro Takebe and Hitomi Terajima

Journal:
J. Algebraic Geom. **11** (2002), 311-362

DOI:
https://doi.org/10.1090/S1056-3911-01-00316-2

Published electronically:
December 19, 2001

MathSciNet review:
1874117

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Abstract |
References |
Additional Information

Abstract: In this paper, we introduce the notion of a generalized rational Okamoto–Painlevé pair $(S, Y)$ by generalizing the notion of the spaces of initial conditions of Painlevé equations. After classifying those pairs, we will establish an algebro-geometric approach to derive the Painlevé differential equations from the deformation of Okamoto–Painlevé pairs by using the local cohomology groups. Moreover the reason why the Painlevé equations can be written in Hamiltonian systems is clarified by means of the holomorphic symplectic structure on $S - Y$. Hamiltonian structures for Okamoto–Painlevé pairs of type $\tilde {E}_7 (= P_{II})$ and $\tilde {D}_8 (= P_{III}^{\tilde {D}_8})$ are calculated explicitly as examples of our theory.

[AL]AL D. Arinkin and S. Lysenko, Isomorphisms between moduli spaces of $SL(2)$-bundles with connections on $\mathbf {P}^1 \backslash \{x_1, \cdots . x_4 \}$, Math. Res. Letters **4**, (1997), 181–190.
[B-W]B:W D. M. Burns, Jr. and J. M. Wahl, Local contributions to global deformations of surfaces, Invent. Math. **26** (1974), 67-88.
[D]D P. Deligne, Theórie de Hodge, II, Publ. Math. IHES, **40**, (1971), 5–57.
[Gr]Gr A. Grothendieck, *Local cohomology*, (noted by R. Hartshorne), Lecture Notes in Math. 41, Springer-Verlag, Berlin, Heidelberg, New York (1967), 106 pp.
[IKSY]IKSY K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991.
[Kaw]Kaw Y. Kawamata, On deformations of compactifiable manifolds, Math. Ann., **235**, (1978), 247–265.
[Kod1]Kod1 K. Kodaira, On deformations of some complex psuedo-group structures, Ann. of Math., **71**, (1960), 224–302.
[Kod2]Kod2 K. Kodaira, On compact analytic surfaces, II, Ann. of Math., **77**, (1963), pp. 563–626.
[KodT]KodT K. Kodaira, Complex manifolds and deformations of complex structures, Springer–Verlag, 1985.
[KS]KS K. Kodaira and D.C. Spencer, On deformations of complex analytic structures, I, II, Ann. of Math., **67**, (1958), pp. 328–466.
[NY]NY M. Noumi and Y. Yamada, Affine Weyl Groups, Discrete Dynamical Systems and Painlevè Equations, Comm Math Phys **199**, (1998), 2, pp281-295
[MMT]MMT T. Matano, A. Matumiya and K. Takano, On some Hamiltonian structures of Painlevé systems, II, J. Math. Soc. Japan, **51**, No.4, 1999, 843–866.
[O1]O1 K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Espaces des conditions initiales, Japan. J. Math., **5**, 1979, 1–79.
[O2]O2 K. Okamoto, Polynomial Hamiltonians associated with Painlevé equations, I, II, Proc. Japan Acad., **56**, (1980), 264–268; ibid, 367–371.
[O3]O3 K. Okamoto, Studies on the Painlevé equations I. Annali di Mathematica pura ed applicata **CXLVI** 1987, 337–381; II. Japan. J. Math., **13**, (1987), 47–76; III. Math. Ann. **275** (1986), 221–255; IV. Funkcial. Ekvac. Ser. Int. **30** (1987), 305–332.
[SSU]SSU M.-H. Saito, Y. Shimizu and S. Usui, Variation of Hodge Structure and the Torelli Problem, Advanced Studies in Pure Math. **10**, 1987, 649–693.
[Sa-Tak]Sa:Tak M.-H. Saito and T. Takebe, Classification of Okamoto–Painlevé pairs. preprint, Kobe 2000. math.AG 0006028
[Sa-Te]Sa:Te M.-H. Saito and H. Terajima, Semiuniversal families of generalized Okamoto–Painlevé pairs and explicit descriptions of Painlevé equations. in preparation.
[SU]SU M.-H. Saito and H. Umemura, Painlevé equations and deformations of rational surfaces with rational double points. preprint, Nagoya, (2000).
[Sakai]Sakai H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Commun. Math. Phys. 220, 165–229 (2001).
[ST]ST T. Shioda and K. Takano, On some Hamiltonian structures of Painlevé systems I, Funkcial. Ekvac., **40**, 1997, 271–291.
[T]T Hitomi Terajima, Local cohomology of generalized Okamoto–Painlevé pairs and Painlevé equations. Preprints, Kobe, May, 2000, math.AG 0006027.

[AL]AL D. Arinkin and S. Lysenko, Isomorphisms between moduli spaces of $SL(2)$-bundles with connections on $\mathbf {P}^1 \backslash \{x_1, \cdots . x_4 \}$, Math. Res. Letters **4**, (1997), 181–190.
[B-W]B:W D. M. Burns, Jr. and J. M. Wahl, Local contributions to global deformations of surfaces, Invent. Math. **26** (1974), 67-88.
[D]D P. Deligne, Theórie de Hodge, II, Publ. Math. IHES, **40**, (1971), 5–57.
[Gr]Gr A. Grothendieck, *Local cohomology*, (noted by R. Hartshorne), Lecture Notes in Math. 41, Springer-Verlag, Berlin, Heidelberg, New York (1967), 106 pp.
[IKSY]IKSY K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991.
[Kaw]Kaw Y. Kawamata, On deformations of compactifiable manifolds, Math. Ann., **235**, (1978), 247–265.
[Kod1]Kod1 K. Kodaira, On deformations of some complex psuedo-group structures, Ann. of Math., **71**, (1960), 224–302.
[Kod2]Kod2 K. Kodaira, On compact analytic surfaces, II, Ann. of Math., **77**, (1963), pp. 563–626.
[KodT]KodT K. Kodaira, Complex manifolds and deformations of complex structures, Springer–Verlag, 1985.
[KS]KS K. Kodaira and D.C. Spencer, On deformations of complex analytic structures, I, II, Ann. of Math., **67**, (1958), pp. 328–466.
[NY]NY M. Noumi and Y. Yamada, Affine Weyl Groups, Discrete Dynamical Systems and Painlevè Equations, Comm Math Phys **199**, (1998), 2, pp281-295
[MMT]MMT T. Matano, A. Matumiya and K. Takano, On some Hamiltonian structures of Painlevé systems, II, J. Math. Soc. Japan, **51**, No.4, 1999, 843–866.
[O1]O1 K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Espaces des conditions initiales, Japan. J. Math., **5**, 1979, 1–79.
[O2]O2 K. Okamoto, Polynomial Hamiltonians associated with Painlevé equations, I, II, Proc. Japan Acad., **56**, (1980), 264–268; ibid, 367–371.
[O3]O3 K. Okamoto, Studies on the Painlevé equations I. Annali di Mathematica pura ed applicata **CXLVI** 1987, 337–381; II. Japan. J. Math., **13**, (1987), 47–76; III. Math. Ann. **275** (1986), 221–255; IV. Funkcial. Ekvac. Ser. Int. **30** (1987), 305–332.
[SSU]SSU M.-H. Saito, Y. Shimizu and S. Usui, Variation of Hodge Structure and the Torelli Problem, Advanced Studies in Pure Math. **10**, 1987, 649–693.
[Sa-Tak]Sa:Tak M.-H. Saito and T. Takebe, Classification of Okamoto–Painlevé pairs. preprint, Kobe 2000. math.AG 0006028
[Sa-Te]Sa:Te M.-H. Saito and H. Terajima, Semiuniversal families of generalized Okamoto–Painlevé pairs and explicit descriptions of Painlevé equations. in preparation.
[SU]SU M.-H. Saito and H. Umemura, Painlevé equations and deformations of rational surfaces with rational double points. preprint, Nagoya, (2000).
[Sakai]Sakai H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Commun. Math. Phys. 220, 165–229 (2001).
[ST]ST T. Shioda and K. Takano, On some Hamiltonian structures of Painlevé systems I, Funkcial. Ekvac., **40**, 1997, 271–291.
[T]T Hitomi Terajima, Local cohomology of generalized Okamoto–Painlevé pairs and Painlevé equations. Preprints, Kobe, May, 2000, math.AG 0006027.

Additional Information

**Masa-Hiko Saito**

Affiliation:
Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 657-8501, Japan

Email:
mhsaito@math.kobe-u.ac.jp

**Taro Takebe**

Affiliation:
Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 657-8501, Japan

Email:
takebe@math.kobe-u.ac.jp

**Hitomi Terajima**

Affiliation:
Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 657-8501, Japan

Email:
terajima@math.kobe-u.ac.jp

Received by editor(s):
July 7, 2000

Published electronically:
December 19, 2001

Additional Notes:
Partly supported by Grant-in-Aid for Scientific Research (B-09440015), (B-12440008) and (C-11874008), the Ministry of Education, Science and Culture, Japan