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Classification of Okamoto-Painlevé pairs. (English) Zbl 1055.34144

There are six types of second-order nonlinear ordinary differential equations without moving singular points named after Painlevé. The present article is closely related to the work of K. Okamoto [Jap. J. Math., New Ser. 5, 1–79 (1979; Zbl 0426.58017)] on these types of equations, in particular to the spaces of initial values introduced by him. Such a space is a noncompact complex surface which can be compactified by a special divisor \(Y\) to a smooth compact complex surface \(S\), and \((S,Y)\) is called an Okamoto-Painlevé pair. Conversely, to each such pair one can associate a Hamiltonian system which is equivalent to a differential equation of Painlevé type. The pair \((S,Y)\) can be described more precisely: \(Y\) is the pole divisor of a meromorphic \(2\)-form \(\omega\) without zeros on \(S\). In particular, \(Y\) represents the anticanonical class of \(S\) and \(\omega\) induces a symplectic structure on \(U:=S-| Y| \). The irreducible components \(Y_i\) of \(Y\) are smooth rational curves with \(Y\cdot Y_i=0\). Moreover \(U\) contains \(\mathbb C^2\) as a Zariski open subset and the divisor \(S\backslash\mathbb C^2\) has normal crossings.
The main result of the paper is the complete classification of those pairs \((S,Y)\). The surface \(S\) is projective rational, and the pair \((S,Y)\) can be obtained by blowings-up and blowings-down of the pair \(\mathbb (P^2,3H)\), \(H\) a line in \(\mathbb P^2\). The possible configurations of \(Y\) including multiplicities are exactly the seven types of singular fibers of elliptic surfaces in the list of K. Kodaira [Ann. Math. (2) 77, 563–626 (1963; Zbl 0118.15802)]. The correspondence between the Kodaira types and the types of Painlevé equations is as follows: \((II^*, P_I)\), \((III^*,P_{II})\), \((I_{3}^*,P_{III}^*)\), \((I_{2}^*,P_{III})\), \((IV^*,P_{IV})\), \((I_{1}^*,P_V),\) \((I_{0}^*,P_{VI})\). The equations of Painlevé type \(P_{III}^*\) are special cases of type \(P_{III}\).

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
32J15 Compact complex surfaces
14J26 Rational and ruled surfaces
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