un 2 00 0 DEFORMATION OF OKAMOTO – PAINLEVÉ PAIRS AND PAINLEVÉ EQUATIONS

In this paper, we introduce the notion of 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 Ẽ7(= PII) and D̃8(= P D̃8 III ) are calculated explicitly as examples of our theory. 0. Introduction In the study of Painlevé equations, the spaces of initial conditions introduced by K. Okamoto [O1], [O2], [O3] have been playing essential roles. It is known that each Painlevé differential equation is equivalent to one of Hamiltonian systems whose Hamiltonians are given by the polynomials in two variables (x, y). (See Table 7 and 8 in §7). The space (x, y) ∈ C can be compactified and one can obtain a pair (S, Y ) of complex projective surface S and an anti-canonical divisor Y ∈ | −KS | such that S − Yred becomes a space of initial conditions. In the study of the space of initial conditions as in [O1], [MMT], it became clear that after eliminating the singularities of Painlevé differential equation by blowings-up, the boundary divisor Y should have the same configuration as in the list of degenerate elliptic curves classified by Kodaira [Kod]. This condition can be translated into the following conditions. Let Y = ∑r i=1 miYi ∈ | −KS| be the irreducible decomposition. Then Y is called of canonical type if and only if deg(−KS)|Yi = degY|Yi = Y · Yi = 0 for all i. In [Sa-Tak], we call such a pair (S, Y ) an Okamoto–Painlevé pair if S − Yred contains C 2 as a Zariski open set and F = S − C is a normal crossing divisor. One can verify that all compactifications of the spaces of initial conditions of known Painlevé equations satisfy these conditions (cf. [O1], [MMT]). Therefore, in this notation, the former studies of Painlevé equations Date: June, 4, 2000. 1991 Mathematics Subject Classification. 14D15, 34M55, 32G10, 14J26.