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Poncelet polygons and the Painlevé equations. (English) Zbl 0893.32018
Geometry and analysis. Papers presented at the Bombay colloquium, India, January 6–14, 1992. Oxford: Oxford University Press. Stud. Math., Tata Inst. Fundam. Res. 13, 151-185 (1995).

This paper is central in the recent renaissance in the study of the Painlevé equations [see also N. J. Hitchin, J. Differ. Geom. 42, No. 1, 30-112 (1995; Zbl 0861.53049); Yu. I. Manin, “Sixth Painlevé equation, universal elliptic curve and mirror of ${ℙ}^{2}$”, Preprint, Max-Planck-Inst. Math., Bonn, (1996); L. Mason and N. Woodhouse, “Integrability, self-duality and twistor theory”, Clarendon Press, Oxford Univ. Press, Oxford (1996; Zbl 0856.58002)]. The author gives explicit solutions to the sixth Painlevé equation. These solutions are related to a flat unitary connection with singularities. In the context of a theorem of Narasimhan and Seshadri, the connection mediates between parabolically stable bundles and representations of the fundamental group.

The damand for explicitness motivates the restriction of the study to bundles on the complex projective line. Indeed, the author looks for a connection written as a $2×2$ matrix-valued 1-form with a simple pole at four points on the projective line. Fixing the holonomy, the 1-form associated to each set of four points comes from a solution to the Schlesinger equation of isomonodromic deformation theory and leads to the sixth Painlevé equation. The author concentrates on algebraic solutions coming from a curve defined by the binary dihedral group ${D}_{k}$ in SU(2) and he obtains explicit solutions for small $k$.

The approach is to consider a smooth projective complex threefold with an action of $\text{SL}\left(2,ℂ\right)$ and a dense open orbit. The Maurer-Cartan form gives a meromorphic connection on $Z$ which is flat on $\text{SL}\left(2,ℂ\right)/{D}_{k}$ and has holonomy ${D}_{k}$. The connection has a logarithmic singularity along an anticanonical divisor $Y$ in $Z$ and there is a four-parameter family of rational curves in $Z$ intersecting $Y$ in four points with varying cross ratio. This gives the isomonodromic deformation and the solution to the Painlevé equation.

The construction of the compactification $Z$ builds on work on Schwarzenberger and leads to a description of $Z$ as a projectivized bundle $P\left({V}_{k}\right)$ where ${V}_{k}$ is a rank two vector bundle on $ℂ{\text{P}}^{2}$. One type of rational curve in $Z$ projects to conics in $ℂ{\text{P}}^{2}$ and leads to the old problem (from around 1746) of the Poncelet polygons (given two conics in the projective plane, is it possible to find polygons inscribed in one conic and circumscribed by the other [P. Griffiths and J. Harris, Enseign. Math., II. Ser. 24, No. 1-2, 31-40 (1978; Zbl 0384.14009)]. Using a modern description by Atiyah of Cayley’s solution from 1853 of the Poncelet problem, the author is able to get the connection explicitly. The work of W. P. Barth and J. Michel [Math. Ann. 295, No. 1, 25-49 (1993; Zbl 0789.14033)] is used to find the modular curve giving the algebraic solution of the Painlevé equation corresponding to the dihedral group.

MSC:
 32Q20 Kähler-Einstein manifolds 14J60 Vector bundles on surfaces and higher-order varieties, and their moduli 32L05 Holomorphic fiber bundles and generalizations 37J35 Completely integrable systems, topological structure of phase space, integration methods 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies