CLASSIFICATION-OF-OKAMOTO–PAINLEVE-PAIRS.pdf

´ CLASSIFICATION OF OKAMOTO–PAINLEVE PAIRS MASA-HIKO SAITO AND TARO TAKEBE Abstract. In this paper, we introduce the notion of an Okamoto–Painlev´e pair (S, Y ) which consists of a compact smooth complex surface S and an effective divisor Y on S satisfying certain conditions. Though spaces of initial values of Painlev´e equations introduced by K. Okamoto give examples of Okamoto–Painleve pairs, we find a new example of Okamoto–Painlev´e pairs not listed in [Oka]. We will give the complete classification of Okamoto–Painlev´e pairs. 0. Introduction In this paper, we will introduce the notion of an Okamoto–Painlev´e pair (S, Y ), which is defined as follows: Definition 0.1. (Cf. Definition 2.1). Let S be a compact smooth complex surface and Y = r a Y an effective divisor on S . We say that a pair (S, Y ) is an Okamoto–Painlev´e pair if it i=1 i i satisfies the following conditions: (i) There exists a meromorphic 2-form ω on S such that (ω ) = −Y , that is, ω has the pole divisor Y (counting multiplicities) and has no zero outside Y . (ii) For all i (1 ≤ i ≤ r), Y Yi = deg[Y ] |Yi = 0. (iii) Let us set D := Y = r Y . Then S − D contains C2 as a Zariski open set. red i=1 i (iv) Set F = S − C2 where C2 is the same Zariski open set as in (ii). Then F is a (reduced) divisor with normal crossings. Historically, Okamoto [Oka] introduced the space MJ (t) of initial values for each Painlev´e equa- tion of type PJ (J = I, . . . , V I ) with the time parameter t, which is