Communications in Mathematical Physics

, Volume 220, Issue 1, pp 165–229 | Cite as

Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

  • Hidetaka Sakai


We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsystem of E (1) 8 inside the Picard lattice of X. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlevé equations, and that the above action constitutes their group of symmetries by Bäcklund transformations. The six Painlevé differential equations appear as degenerate cases of this construction. In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure.


Differential Equation Root System Weyl Group Symplectic Form Degenerate Case 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Hidetaka Sakai
    • 1
  1. 1.Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606, Japan.JP

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