DEFORMATION OF OKAMOTO–PAINLEVÉ PAIRS AND PAINLEVÉ EQUATIONS

MASA-HIKO SAITO, TARO TAKEBE & HITOMI TERAJIMA

Abstract. 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 Ẽ_{7}(= P_{II}) and D̃_{8}(= P D̃_{8} ; 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^{2} can be compactified and one can obtain a pair (S, Y) of complex projective surface S and an anti-canonical divisor Y ∈ | − K_{S}| such that S − Y_{red} 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 [Kod2]. This condition can be translated into the following conditions. Let Y = ∑_{i=1}^{r} m_{i}Y_{i} ∈ | − K_{S}| be the irreducible decomposition. Then Y is called of canonical type if and only if

deg(−K_{S})_{|Y_{i}} = deg Y_{|Y_{i}} = Y · Y_{i} = 0 for all i.

In [Sa-Tak], we call such a pair (S, Y) an Okamoto–Painlevé pair if S − Y_{red} contains C^{2} as a Zariski open set and F = S − C^{2} 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 establish the route in the direction:

Date: revised March, 5, 2001.

1991 Mathematics Subject Classification. 14D15, 34M55, 32G10, 14J26.

Key words and phrases. Deformation of Okamoto–Painlevé pairs, Painlevé equations, Kodaira–Spencer class, Local cohomology.

Partly supported by Grant-in Aid for Scientific Research (B-09440015), (B-12440008) and (C-11874008), the Ministry of Education, Science and Culture, Japan .

Painlevé equations ⟹ Okamoto–Painlevé pairs (S, Y) .  (1)

The main purpose of this paper is to establish the route backward, that is, the route in the following direction

Okamoto–Painlevé pairs (S, Y) ⟹ Painlevé equations .  (2)

Our main tool here is the deformation theory of pairs (S, Y) (cf. [KS], [Kaw]) and local cohomology exact sequence (cf. [B-W]). The deformation theory was established by Kodaira–Spencer in [KS] in late 1950’s. Kawamata [Kaw] generalized the deformation theory to compactified complex manifolds, or pairs of smooth compact complex manifolds and simple normal crossing subvarieties. Applying the deformation theory, we can see that the space of infinitesimal deformations of the Okamoto–Painlevé pair (S, Y) is isomorphic to the cohomology group H^{1}(S, Θ_{S}(− log D)) where D = Y_{red}. Looking at the restriction homomorphism

res : H^{1}(S, Θ_{S}(− log D)) ⟶ H^{1}(S − D, Θ_{S−D}),

we see that the kernel of the restriction map res has the important meaning, that is, the direction corresponding to the kernel of res is the infinitesimal deformation of (S, Y) which induces the trivial deformation on S − D. Roughly speaking, one can say that the Painlevé differential equations describe the deformations corresponding to the direction of the kernel of the restriction map.

To be precise, let us consider a family of Okamoto–Painlevé pairs D ↪ S ⟶ B_{R} with onedimensional base space B_{R} with a coordinate t such that the Kodaira–Spencer class ρ(\frac{∂}{∂t}) lies in the kernel of the restriction map res. Then by using the affine covering of S − D and Čech cocycles, we can derive a system of ordinary differential equation. Note that this observation will be applicable to the higher dimensional cases.

From these observation, we see that the kernel of res corresponds to the directions of time variables in the Painlevé differential equation.

Furthermore, we can apply the local cohomology exact sequence to our settings and obtain the exact sequence (cf. [B-W], [Gr])

H_{D}^{1}(S, Θ_{S}(− log D)) ⟶^{μ} H^{1}(S, Θ_{S}(− log D)) \underrightarrow{res} H^{1}(S − D, Θ_{S−D}).

Under the condition that (S, Y) is of non-fibered type, the map μ is injective, and hence, the local cohomology group H_{D}^{1}(S, Θ_{S}(− log D)) coincides with the kernel of res. Therefore, non-zero element of the local cohomology group H_{D}^{1}(S, Θ_{S}(− log D)) corresponds to a time variable of the Painlevé equation. For a generalized rational Okamoto–Painlevé pair (S, Y) of additive type, if Y_{red} is normal crossing divisor, Terajima [T] proved that the dimension of the local cohomology group is positive, hence, we can always obtain a differential equation.

In order to obtain an explicit differential equation for each type R from our setting, we need to construct a global family of generalized rational Okamoto–Painlevé pairs of type R over a parameter space M_{R} × B_{R} which is semiuniversal at each point. Moreover we need to introduce a good affine open covering of the total space such that the rational two form ω_{S} restricted to S − Y_{red} has a canonical form. (See §5 and §6.)

The organization of this paper is as follows. In §1, we define the notion of generalized Okamoto– Painlevé pairs and recall the relation to generalized Halphen surfaces, which are studied by Sakai [Sakai]. We also classify generalized rational Okamoto–Painlevé pairs (S, Y) such that Y_{red} are normal crossing divisors. A generalized rational Okamoto–Painlevé pair (S, Y) is called of fibered type if there exists a structure of elliptic fibration f : S ⟶ P^{1} such that f^{∗}(∞) = nY for some n ≥ 1. We show that (S, Y) is not of fibered type if and only if regular algebraic functions on S−Y_{red} are just constant functions. This fact is also important for later purpose. After recalling the theory of deformation of pairs in §2, we investigate the cohomology groups for generalized rational Okamoto–Painlevé pairs. In §3, we will apply the theory of local cohomology to our situation, and have the fundamental exact sequence (Proposition 3.1). Moreover, we state an important result, Theorem 3.1, which is proved in [T]. After reviewing the Kodaira–Spencer theory in §4, in §5, we will explain how one can construct the family of generalized rational Okamoto–Painlevé pairs and their open coverings. In §6, we will state our main theorem (Theorem 6.1), which states that from the special global deformation of generalized Okamoto–Painlevé pairs one can obtain the differential equations. Moreover the reason for the equations to be in Hamiltonian systems will be explained geometrically. In §8, we will derive the Painlevé equations from the deformations of Okamoto–Painlevé pairs of type Ẽ_{7} and D̃_{8}.

Prior to our work here, in [SU], M.-H. Saito and H. Umemura essentially pointed out that the deformation of spaces of initial conditions describes Painlevé equation completely. In this sense, this paper is a continuation of [SU], though we have clarified the meaning of time variables by means of local cohomology groups in this paper.

The recent work due to Sakai [Sakai] introduce the following beautiful viewpoint: The geometry of certain rational surfaces with the symmetries induced by Cremona transformations describe the discrete Painlevé equations and the Painlevé equation can be obtained as a limit of the discrete Painlevé equations. We owe much to his beautiful paper [Sakai]. In particular, some of the explicit calculations are done by using his descriptions of the family of Okamoto–Painlevé pairs in [Sakai].

The works of Takano et al [MMT], [ST] is also essential to our work. In §8, we use the coordinate systems introduced by them.

The series of the work is started by [Sa-Tak], where we introduce the notion of Okamoto-Painlevé pair and classify Okamoto–Painlevé pairs (S, Y).

One of missing points in our work here is the theory of Bäcklund transformation of Okamoto– Painlevé pairs. In this direction, one should refer to a series of works of M. Noumi and Y. Yamada (e.g. [NY]), also Sakai’s work [Sakai]. In [SU], the authors tried to understand the Bäcklund transformation by using the notion of “flip ” or “flop” in the theory of the minimal models of higher dimensional varieties. We will investigate this point in future.

1. Generalized Okamoto–Painlevé Pairs

Let S be a complex projective surface. We denote by K_{S} the canonical line bundle or the canonical divisor class of S. Assume that the anti-canonical divisor class −K_{S} is effective, that is, there exists an effective divisor Y ∈ | − K_{S}|. Geometrically, this is equivalent to the existence of a rational 2-form ω_{Y} on S whose corresponding divisor (ω_{Y}) = −(ω_{Y})_{∞} = −Y. Such a divisor Y is called an anti-canonical divisor of S as usual. Since ω_{Y} dose not vanish on S − Y, it induces a holomorphic symplectic structure on S − Y.

In [Sa-Tak], we introduce a notion of Okamoto–Painlevé pair (S, Y) which is a pair of complex projective surface S and an anti-canonical divisor Y satisfying certain conditions ([Definition 2.1 [Sa-Tak]]). Generalizing the notion, we will start this section with the following definition. Definition 1.1. Let (S, Y) be a pair of a complex projective surface S and an anti-canonical divisor Y ∈ | − K_{S}| of S. Let Y = ∑_{i=1}^{r} m_{i}Y_{i} be the irreducible decomposition of Y. We call a pair (S, Y) a generalized Okamoto–Painlevé Pair if for all i, 1 ≤ i ≤ r,

Y · Y_{i} = deg Y_{|Y_{i}} = 0.  (3)

According to [Sa-Tak], we listed the additional conditions for Okamoto–Painlevé pairs besides the condition (3) as follows.

(1) Let us set D := Y_{red} = ∑_{i=1}^{r} Y_{i}. Then S − D contains the complex affine plane C^{2} as a Zariski open set.

(2) Set F = S − C^{2} where C^{2} is the same Zariski open set as in (1). Then F is a (reduced) divisor with normal crossings. In particular, D = Y_{red} is also a reduced divisor with normal crossings.

Under this definition, we proved the following classification of Okamoto–Painlevé pairs in [Sa-Tak]. We remark that an Okamoto–Painlevé pair of type D̃_{7} did not appear in the classification of classical Painlevé equations [O1]. Theorem 1.1. ([Sa-Tak].) Let (S, Y) be a generalized Okamoto-Painlevé pair and assume that S − Y_{red} contains C^{2} and F = S − C^{2} is a reduced divisor with normal crossings. (That is, (S, Y) is an Okamoto–Painlevé pair in original sense. ) Then we have the following assertions.

(1) The surface S is a projective rational surface.

(2) The configuration of Y counting with multiplicity is in the list of Kodaira’s classification of singular fibers of elliptic surfaces (cf. [Kod2]). More precisely, it coincides with one in the following Table 1. (In Figure 1, each line denotes a smooth rational curve C with C^{2} = −2 and the configuration of lines show how they intersect to each other. The number next to each line denotes the multiplicity of each curve in Y = −K_{S}.)

Generalized Halphen surfaces

According to Sakai [§4, [Sakai]], we recall the following definition. Definition 1.2. (1) Let S be a rational surface with an effective anti-canonical divisor Y ∈

| − K_{S}|. Let Y = ∑_{i=1}^{r} m_{i}Y_{i} be the irreducible decomposition. The divisor Y is called of canonical type if

K_{S} · Y_{i} = −Y · Y_{i} = 0 for all i

(2) A rational surface S is called a generalized Halphen surface if S has an effective anticanonical divisor Y of canonical type. A generalized Halphen surface S is called of index

1  2 3  4 5  6 3  4 2 Ẽ_{8}  1  2 3  4 2  3 2 1 Ẽ_{7}

[figure: 1; 1; 2; 3; 2; 2; 1; Ẽ_{6}; 1; 1; 2; 2; 2; 2; 1; 1; D̃_{7}]

[figure: 1; 1; 2; 2; 2; 1; 1; D̃_{6}; 1; 1; 2; 2; 1; 1; D̃_{5}]

2 1 1  1 1 D̃_{4}

Y  Ẽ_{8}  Ẽ_{7}  D̃_{7}  D̃_{6}  Ẽ_{6} D̃_{5} D̃_{4}

Kodaira’s notation II^{∗} III^{∗} I_{3}^{∗}  I_{2}^{∗}  IV^{∗} I_{1}^{∗} I_{0}^{∗}

Painlevé equation P_{I} P_{II} P D̃_{7} ; III P_{III} = P D̃_{6} ; III P_{IV} P_{V} P_{V I}

Table 1

one if

dim | − K_{S}| = 1.

Remark 1.1. By Riemann-Roch theorem, it is easy to see that for a generalized Halphen surface S dim | − K_{S}| ≤ 1.

The following Proposition ensures that one can obtain a generalized Halphen surfaces from blowing-up of 9-points of P^{2}. Proposition 1.1. ( Proposition 2, §2, [Sakai] ). Let S be a generalized Halphen surface, then there exists a birational morphism ρ : S ⟶ P^{2}.

Let (S, Y) be a generalized Okamoto-Painlevé pair such that S is a rational surface. Then S is a generalized Halphen surface with a specified anti-canonical divisor Y. As a corollary of Proposition 1.1, we obtain the following corollary. Corollary 1.1. Let (S, Y) be a generalized rational Okamoto–Painlevé pair. Then S can be obtained as 9 points blowing-up of P^{2}.

One can show that Y has a same configuration as one of Kodaira’s degenerate elliptic curves for a generalized rational Okamoto–Painlevé pair (S, Y) (cf. Proof of Theorem 2.1 in [Sa-Tak]).

If S is a generalized Halphen surface of index one, the morphism associated to the linear system | − K_{S}| induces an elliptic fibration φ : S ⟶ P^{1} with φ^{∗}(∞) = Y. (Here φ^{∗}(∞) denotes the scheme theoretic fibers at ∞ ∈ P^{1}. ) This leads us the following Definition 1.3. A generalized Okamoto-Painlevé pair (S, Y) is called “of fibered type” if there exists an elliptic fibration φ : S ⟶ P^{1} such that φ^{∗}(∞) = nY for some n ≥ 1. If (S, Y) is not of fibered type, we call (S, Y) “of non-fibered type ”.

Note that if (S, Y) is of fibered type and φ : S ⟶ P^{1} is elliptic surface with φ^{∗}(∞) = nY with n > 1, φ^{∗}(∞) is called a multiple fiber. This happens only when Y is of elliptic type or multiplicative type in the notation below.

In [Sakai], Sakai classified generalized Halphen surface S with dim |−K_{S}| = 0. In the case when dim | − K_{S}| = 0, the associated Okamoto-Painlevé pair (S, Y) with a unique member Y ∈ | − K_{S}| is of non-fibered type and they can be classified by means of the configuration of Y.

Let Y = ∑_{i=1}^{r} m_{i}Y_{i} be the irreducible decomposition of Y. Denote by M(Y) the sublattice of Pic(S) ≃ H^{2}(S, Z) generated by the irreducible components {Y_{i}}_{i=1}^{r}. Here the bilinear form on Pic(S) is (−1) times the intersection form on Pic(S). Then {Y_{i}}_{i=1}^{r} forms a root basis of M(Y) and we denote by R(Y) the type of the root system.

One can easily classify R(Y) as in Table 3. Note that according to the type of Y, R(Y) can be classified into three classes: elliptic type when Y is a smooth elliptic curve, multiplicative type when Y is a cycle of rational curves, additive type when the configuration of Y is tree. These types also correspond to the type of generalized Jacobean Pic^{0}(Y) of Y. If we denote by (Pic^{0}(Y))^{0} the component of identity of Pic^{0}(Y), we have the following correspondence (cf. Table 2).

R(Y)  (Pic^{0}(Y))^{0}
elliptic type  smooth elliptic curve Y
multiplicative type G_{m} ≃ C^{×}
additive type  G_{a} ≃ C

Table 2

R(Y) (Kodaira type)

elliptic type  Ã_{0}(= I^{0})

multiplicative type Ã_{0}^{∗}(= I_{1}), Ã_{1}(= I_{2}), ⋯, Ã_{7}(= I_{8}), Ã_{8}(= I_{9})

additive type  Ã_{0}^{∗∗}(= II), Ã_{1}^{∗}(= III), Ã_{2}^{∗}(= IV)

D̃_{4}(= I_{0}^{∗}), ⋯, D̃_{8}(= I_{4})

Ẽ_{6}(= IV^{∗}), Ẽ_{7}(= III^{∗}), Ẽ_{8}(= II^{∗})

Table 3

Proposition 1.2. Let (S, Y) be a generalized rational Okamoto–Painlevé pair such that Y_{red} is a divisor with only normal crossings. Then besides the list of Okamoto–Painlevé pairs in Table 1, we have a pair (S, Y) of type D̃_{8} and also Ã_{r} for 0 ≤ r ≤ 8 and Ã_{0}^{∗}. Here for Ã_{0}, Y is a smooth elliptic curve (Kodaira I_{0}-type) and for Ã_{0}^{∗}, Y is a rational curve with a node (Kodaira I_{1}-type).

We list up generalized rational Okamoto–Painlevé pairs with normal crossing divisor Y_{red} in Table 4.

Y  Ẽ_{8} D̃_{8}  Ẽ_{7}  D̃_{7} D̃_{6} Ẽ_{6} D̃_{5} D̃_{4}  Ã_{r−1}  Ã_{0}  Ã_{0}^{∗}
2 ≤ r ≤ 9 r = 1 r = 1
Kodaira’s notation II^{∗} I_{4}^{∗} III^{∗} I_{3}^{∗} I_{2}^{∗} IV^{∗} I_{1}^{∗} I_{0}^{∗}  I_{r}  I_{0}  I_{1}

r = ♯ of comps. of Y 9 9  8  8 7  7  6 5  r  1  1

Table 4

Regular algebraic functions on S − Y.

Let (S, Y) be a generalized rational Okamoto-Painlevé pair. If (S, Y) is of fibered type, that is, if there exists an elliptic fibration φ : S ⟶ P^{1} with φ(∞)^{∗} = nY, pulling back non-constant regular algebraic functions on P^{1} − {∞} ≃ C, we have many non-constant regular functions on S − Y. We can prove the converse of this fact. Proposition 1.3. Let (S, Y) be a generalized rational Okamoto–Painlevé pair. The following conditions are equivalent to each other.

(1) (S, Y) is of non-fibered type.

(2) H^{0}(S−Y, O^{alg}) ≃ C, that is, all regular algebraic functions of S−Y are constant functions. Proof. As we remarked as above, the implication (2) ⇒ (1) is obvious. Assume that there exists

a non-constant regular function f on S − Y. Then the morphism f : S − Y ⟶ C extends to a morphism

\overline{f} : S ⟶ P^{1}.

Set Y′ = \overline{f}^{∗}(∞). Since \overline{f} is regular on S − Y, recalling the irreducible decomposition of Y = ∑_{i=1}^{r} m_{i}Y_{i}, we can write

Y′ = ∑_{i=1}^{r} a_{i}Y_{i}

with a_{i} ≥ 0. First we show that a_{i} > 0 for every 1 ≤ i ≤ r. If a_{i} = 0 for some i, the configuration of Y′_{red} becomes a proper sub-graph of the configuration of Y_{red}. Since the graph of Y_{red} corresponds to an affine Dynkin diagram, one can easily see that Y′ can be contracted to rational double points {p_{1}, ⋯, p_{s}} and obtain a normal surface S′ with normal singular points {p_{1}, ⋯, p_{s}}. Since S −Y′ and S′ − \{p_{1}, ⋯, p_{s}\} are isomorphic and \overline{f} is regular on S − Y′ ≃ S′ − \{p_{1}, ⋯, p_{s}\}, \overline{f} extends to a regular function on S′. Since S′ is proper, \overline{f} must be constant and hence f is also constant. This contradicts to the original assumption. Hence we see that a_{i} > 0 for all 1 ≤ i ≤ r.

[figure: 1; 1; 2; 2; 2; 2; 2; 1; 1; D̃_{8}]

[figure: 1; 1; 1; 1; 1; 1; 1; 1; Ã_{8}; 1]

Figure 2

This implies that Y′_{red} = Y_{red} and since Y_{red} is connected, so is Y′_{red}. Taking the Stein factorization if necessary, we may assume that all of the fiber of \overline{f} : S ⟶ P^{1} is connected and \overline{f}^{∗}(∞)_{red} = Y′_{red} = Y_{red}. (Note that S is a rational surface, hence the irregularity of S is zero.) We will show that general fiber of \overline{f} is an elliptic curve. Since S is smooth and \overline{f} has connected fibers, we only have to show that the virtual genus of Y′ is one. Since −K_{S} = Y and Y′ = ∑_{i=1}^{r} a_{i}Y_{i}, we see that

K_{S} · Y′ = ∑_{i=1}^{r} a_{i}(−Y) · Y_{i} = 0

by definition of Okamoto–Painlevé pair. Moreover since Y′ is linear equivalent to a general fiber of \overline{f}, we see that Y′ · Y_{i} = 0 for all 1 ≤ i ≤ r. Hence we see that (Y′)^{2} = 0. Then the virtual genus of Y′ is given by

π(Y′) = \frac{K_{S} · Y′ + (Y′)^{2}}{2} + 1 = 1,

and this completes the proof of proposition. □

2. Deformation of generalized rational Okamoto–Painlevé pairs

In this section, we will recall necessary background of theory of infinitesimal deformation of Okamoto–Painlevé pairs. First, we will recall the general theory of deformation of pairs.

General Theory of Deformation of Pairs

Let (X, H) be a pair of a complex manifold X and a (reduced) normal crossing divisor and let H = ∑_{i=1}^{r} H_{i} be the irreducible decomposition of H. By a technical reason we will assume that H is a simple normal crossing divisor, that is, each irreducible component H_{i} is a smooth divisor. We call such a pair (S, H) is a non-singular pair .

For such a non-singular pair (S, H), the normalization H̃ of H is given by the disjoint union ∐_{i=1}^{r} H_{i} of H_{i}’s, and we denote by ν : H̃ ⟶ H the normalization map.

First, we recall the general theory of deformation of a non-singular pair (X, H) due to Kawamata [Kaw]. (See also [SSU]).

Let Ω_{X}^{1}(log H) denote the sheaf of germs of meromorphic one forms on X which have logarithmic poles along H. Moreover, we set

Θ_{X}(− log H) := \underline{Hom}(Ω_{X}^{1}(log H), O_{X}).

This is the sheaf of germs of regular vector fields which have logarithmic zero along H. Definition 2.1. (Cf. [Definition 3, [Kaw]]) A deformation of a non-singular pair (X, H) is a 5-tuple (X , H, π, B, ι)

(1) π : X ⟶ B is a proper smooth holomorphic map from a complex manifold X to a connected complex manifold B

(2) H = ∑_{i=1}^{r} H_{i} is a simple normal crossing divisor of X

(3) For a point 0 ∈ B, we have an isomorphism ι : (π^{−1}(0), π^{−1}(0) ∩ H) = (X_{0}, H_{0}) ≃ (X, H).

(4) π is locally a projection of a product space as well as the restriction of it to H, that is, for each p ∈ X there exists an open neighborhood U of p and an isomorphism φ : U ⟶ V ×W, where V = π(U) and W = π^{−1}(π(p)), such that the following diagram

U ⟶^{φ} V × W
π ↘  ↙ pr_{1}
V

is commutative and φ(U ∩ H) = V × (W ∩ H).

The deformation of a pair is often denoted by the following diagram:

\begin{matrix}X & ↩ & H \\ π ↓ & ↙ \\ B \end{matrix}

For a deformation π : X ⟶ B of complex manifold X = X_{0}, we can define the Kodaira–Spencer class

ρ : T_{0}(B) ⟶ H^{1}(X, Θ_{X}).

Similarly, for a deformation of a pair (X, H) as above, we can define the Kodaira–Spencer map

ρ : T_{0}(B) ⟶ H^{1}(X, Θ_{X}(− log H)).

As for the existence of Kuranishi space of local semiuniversal deformation of a pair, we have the following theorem due to Kawamata [Cor. 4, [Kaw]]. Theorem 2.1. For each pair (X, H) of a compact complex manifold X and a normal crossing divisor H, there exist a germ of a complex variety (B, o) and the semiuniversal deformation of (X, H)

\begin{matrix}X & ↩ & H = ∑_{i=1}^{r} H_{i} \\ π ↓ & ↙ φ \\ B \end{matrix} .

Moreover if

H^{2}(X, Θ_{X}(− log H)) = {0},

the germ (B, 0) is smooth and the Kodaira–Spencer map induces an isomorphism

T_{0}(B) ⟶^{≃} H^{1}(X, Θ_{X}(− log H)).

The following Lemma is well known and easy to verify. Lemma 2.1. Let (X, H = ∑_{i=1}^{l} H_{i}) be as above, and let ν : H̃ = ∐_{i=1}^{r} H_{i} ⟶ H be the normalization map. Then we have exact sequences of sheaves:

0 ⟶ Ω_{X}^{1} ⟶ Ω_{X}^{1}(log H) ⟶^{P.R.} ν_{∗}(⊕_{i=1}^{l}O_{H_{i}}) ⟶ 0  (4)
0 ⟶ Θ_{X}(− log H) ⟶ Θ_{X} ⟶ ν_{∗}(⊕_{i=1}^{l}N_{H_{i}/X}) ⟶ 0  (5)

Here the map P.R. : Ω_{X}^{1}(log H) ⟶ ν_{∗}(⊕_{i=1}^{l}O_{H_{i}}) is induced by the Poincaré residue and N_{H_{i}/X} = O_{X}(H_{i})/O_{X} denotes the normal bundle of the divisor H_{i} ⊂ X.

Deformation of generalized rational Okamoto–Painlevé pairs

Let (S, Y) be a generalized rational Okamoto–Painlevé pair. Recall that Y = ∑_{i=1}^{r} m_{i}Y_{i} is the anti-canonical divisor −K_{S}. Moreover we set D_{i} = Y_{red} = ∑_{i=1}^{r} Y_{i}. From now on, we will calculate some cohomology groups of the pair (S, D) for Okamoto–Painlevé pair (S, Y) which we will use later.

Let i : D ↪ S be the natural inclusion and ν : D̃ = ∐_{i=1}^{r} Y_{i} → D the normalization map. Set j = i · ν. First, let us consider the following Gysin exact sequence

H^{1}(S, C) ⟶ H^{1}(S − D, C) ⟶ H^{0}(D̃, C) ⟶^{j!} H^{2}(S, C) ⟶ · · · .  (6)

The following lemma is important. Lemma 2.2. Under the same notation as above, the Gysin map gives an injective homomorphism

H^{0}(D̃, C) ↪ H^{2}(S, C).

Proof. Since

H^{0}(D̃, C) = ⊕_{i=1}^{r}H^{0}(Y_{i}, C) = ⊕_{i=1}^{r}C · 1_{Y_{i}},

and the image of the Gysin map of 1_{Y_{i}} is just the divisor class Y_{i} ∈ H^{2}(S, C) ≃ Pic(S) ⊗ C, we only have to show that the classes {Y_{i}}_{i=1}^{r} is lineally independent in H^{2}(S, C). Looking at the intersection matrix of {Y_{i}}_{i=1}^{r}, we easily see that only possible linear relation is

Y = ∑_{i=1}^{r} m_{i}Y_{i} = 0.

On the other hand, K_{S} = −Y and S has at least one (−1)-smooth rational curve E. By adjunction formula, we see that Y · E = −(K_{S}) · E = 1, hence we see that {Y_{i}}_{i=1}^{r} is linearly independent. □

As a corollary to Lemma 2.2, we obtain: Corollary 2.1. For a generalized rational Okamoto–Painlevé pair (S, Y), we have the following.

(1) H^{1}(S − D, C) = 0.

(2) H^{0}(S, Ω_{S}^{1}(log D)) = 0.
(3) H^{2}(S, Θ_{S}(− log D)) = 0.
(4) H^{2}(S, Θ_{S}) = 0.

Proof. Since S is a rational surface, we have H^{1}(S, C) = 0. From the exact sequence (6) and Lemma 2.2, we have the first assertion. Then from the mixed Hodge theory, we have an inclusion

H^{0}(S, Ω_{S}^{1}(log D)) ↪ H^{1}(S − D, C).

This proves the second assertion. For the third assertion, let us consider the Serre duality

H^{2}(S, Θ_{S}(− log D))^{∨} ≃ H^{0}(S, Ω_{S}^{1}(log D) ⊗ K_{S}).

Since K_{S} = O_{S}(−Y), we have an inclusion

H^{0}(S, Ω_{S}^{1}(log D) ⊗ K_{S}) ↪ H^{0}(S, Ω_{S}^{1}(log D)) = {0},

This shows the third assertion. From the exact sequence (5), we obtain the exact sequence

⟶ H^{2}(S, Θ_{S}(− log D)) ⟶ H^{2}(S, Θ_{S}) ⟶ H^{2}(D, N_{D_{i}/S}) ⟶ 0.

Since dim D = 1, H^{2}(D, N_{D_{i}/S}) = 0, hence, from the third assertion we obtain the fourth assertion.

The following geometric facts are very important for our purpose. (cf. [Lemma 3, [AL]], [SU]). Proposition 2.1. Let (S, Y) be a generalized rational Okamoto–Painlevé pair such that Y is a divisor with normal crossing and (S, Y) is not of fibered type.

(1) H^{0}(S − D, O^{alg}) ≃ C

(2) H^{0}(S − D, Θ_{S−D}^{alg}) ≃ 0. Here Θ_{S−D}^{alg} denotes the sheaf of germs of algebraic regular infinitesimal automorphisms.

(3) H^{0}(S, Θ_{S}(H)) = 0 for any effective divisor H supported on D.

(4) H^{0}(S, Θ_{S}(− log D)(H)) = 0 for any effective divisor H supported on D.

Proof. The first assertion follows from Proposition 1.3. Since on S there exists a non-zero rational 2-forms ω_{S} which is non-degenerate on S −D, ω_{S} induces an isomorphism Θ_{S−D} ≃ Ω_{S−D}^{1}. Hence it suffices to show that H^{0}(S − D, Ω_{S−D}^{1,alg}) = (0). Taking a section η ∈ H^{0}(S − D, Ω_{S−D}^{1,alg}), we see that dη/ω_{S} is a regular holomorphic function on S − D, hence constant c (cf. Proposition 1.3). So this implies that dη = c · ω_{S}. On the other hand, we can easily see that ω_{S} is non-zero element in H_{DR}^{2}(S − D, C), hence dη = 0. Hence it lies in H^{0}(S − D, dO_{S−D}^{alg}). Since we have an isomorphism (cf. [3.1.7.1, [D]])

H^{1}(S − D, C_{S−D}) ≃ H_{DR}^{1}(S − D),

and H^{1}(S − D, C_{S−D}) = (0) from Corollary 2.1, (1), we see that η can be written as df for some f ∈ H^{0}(S − D, O_{S−D}^{alg}). However since f is constant (Proposition 1.3), we see that η = df = 0. The last two assertions easily follow from the second assertion. □

The following proposition shows that the Kuranishi space of a generalized rational Okamoto– Painlevé pair (S, Y) is smooth and has dimension 10 − r where r denotes the number of irreducible components of Y. Proposition 2.2. Let (S, Y) be a generalized rational Okamoto–Painlevé pair such that D = Y_{red} is a simple normal crossing divisor and Y ≠ Ã_{0}-type. Then we have

c_{2}(S) = topological Euler characteristic = 12,  (7)
b_{2}(S) = rank H^{2}(S, Z) = 10,  (8)
dim H^{1}(S, Θ_{S}) = 10,  (9)

and

dim H^{1}(S, Θ_{S}(− log D)) = 10 − r  (10)

where r is the number of irreducible components of Y. Moreover, the Kuranishi space of the local deformation of the pair (S, D) is smooth and of dimension 10 − r.

Proof. First, from Noether’s formula, we obtain

χ(S, O_{S}) = \frac{1}{12}((K_{S})^{2} + c_{2}(S)).  (11)

From the definition of a generalized rational Okamoto–Painlevé pair (S, Y), we have K_{S} = −Y and (K_{S})^{2} = Y^{2} = 0. Since S is rational, we have χ(S, O_{S}) = 1. Hence, we have

c_{2}(S) = 12.^{1}

Since B_{1}(S) = 0, the above equality implies that B_{2}(S) = 12 − 2 = 10.

From Riemann-Roch-Hirzeburuch formula, we obtain

χ(S, Θ_{S}) = ∑_{i=1}^{2}(−1)^{i} dim H^{i}(S, Θ_{S}) = \frac{1}{6}(7 · (K_{S})^{2} − 5c_{2}(S)).

1Note that this also follows from the fact that S is a 9-points blown-up of P^{2}.

Table of the deformation of generalized rational Okamoto–Painlevé pairs.

Y  Ẽ_{8} D̃_{8}  Ẽ_{7}  D̃_{7}  D̃_{6}  Ẽ_{6} D̃_{5} D̃_{4} Ã_{r−1}, r ≥ 2

Number of components of Y 9  9  8  8  7  7  6  5  r

dim H^{1}(S, Θ_{S}(− log D))  1  1  2  2  3  3  4  5  10 − r

Painlevé equation  P_{I} P D̃_{8} ; III P_{II} P D̃_{7} ; III P D̃_{6} ; III P_{IV} P_{V} P_{V I}  none

Table 5

Then again from (K_{S})^{2} = 0 we obtain

χ(S, Θ_{S}) = −\frac{5}{6}c_{2}(S) = −\frac{5}{6} × 12 = −10.

Moreover from Corollary 2.1, (4) and Proposition 2.1 we obtain H^{i}(S, Θ_{S}) = 0 for i = 0, 2, and hence

dim H^{1}(S, Θ_{S}) = 10.  (12)

For H^{1}(S, Θ_{S}(− log D)), consider the exact sequence

0 ⟶ Θ_{S}(− log D) ⟶ Θ_{S} ⟶ ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⟶ 0.  (13)

Remember D = Y_{red} = ∑_{i=1}^{r} Y_{i}. Since K_{S} · Y_{i} = 0 and Y_{i} ≃ P^{1} by assumption, the adjunction formula shows that

deg N_{Y_{i}/S} = −2 or N_{Y_{i}/S} ≃ O_{P^{1}}(−2).

Then since H^{0}(P^{1}, O_{P^{1}}(−2)) = 0, H^{1}(P^{1}, O_{P^{1}}(−2)) ≃ C and H^{2}(S, Θ_{S}(− log D)) = 0, we have the following exact sequence

0 ⟶ H^{1}(S, Θ_{S}(− log D)) ⟶ H^{1}(S, Θ_{S}) ⟶ ⊕_{i=1}^{r}C[Y_{i}] ⟶ 0  (14)

This implies that the assertion (10) holds. The last assertion follows from Theorem 2.1 and the fact that H^{2}(S, Θ_{S}(− log D)) = 0.

3. Local cohomology sequences and Time variables

Let (S, Y) be a generalized rational Okamoto–Painlevé pair and set D = Y_{red}. Moreover, in this section, we assume that (1) (S, Y) is of non-fibered type and

(2) Y_{red} is a normal crossing divisor with at least two irreducible components, i.e. (r ≥ 2) so that all irreducible components of Y_{red} are smooth rational curves.

In what follows, O_{S} and O_{S−D} denote the sheaves of germs of algebraic regular functions on S and S − D respectively. Moreover all sheaves of O_{S}-modules are considered in algebraic category unless otherwise stated. Let us consider the following exact sequence of local cohomology groups ([Corollary 1.9, [Gr]])

H^{0}(S, Θ_{S}(− log D)) → H^{0}(S − D, Θ_{S}(− log D)) → H_{D}^{1}(Θ_{S}(− log D)) →  (15)
H^{1}(S, Θ_{S}(− log D)) →^{res} H^{1}(S − D, Θ_{S}(− log D)) .  (16)

Since (S, Y) is of non-fibered type, from (2), Proposition 2.1, we see that

H^{0}(S − D, Θ_{S}(− log D)) = H^{0}(S − D, Θ_{S}) = {0}.

Hence, we have the following Proposition 3.1. For a generalized rational Okamoto–Painlevé pair of non-fibered type, we have the following exact sequence:

0 → H_{D}^{1}(Θ_{S}(− log D)) → H^{1}(S, Θ_{S}(− log D)) →^{res} H^{1}(S − D, Θ_{S}(− log D)) (17)

The following theorem is proved in [T]. Theorem 3.1. Let (S, Y) be a generalized rational Okamoto-Painlevé pair (S, Y) with the condition above. Moreover D = Y_{red} is of additive type. Then we have

dim H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) = 1.  (18)

Here we put N_{D} = O_{S}(D)/O_{S}.

Since we have a natural inclusion

H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) ↪ H_{D}^{1}(Θ_{S}(− log D)),

we obtain

dim H_{D}^{1}(Θ_{S}(− log D)) ≥ 1.  (19)

This theorem plays an important role to understand the Painlevé equation related to (S, Y). We will not investigate the further structure of local cohomology here. Instead, we propose the following Conjecture 3.1. Under the same notation and assumption as in Theorem 3.1,

H_{D}^{1}(Θ_{S}(− log D)) ≃ H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) ≃ C.  (20)

From the exact sequence (17), we see that the subspace H_{D}^{1}(S, Θ_{S}(− log Y)) of H^{1}(S, Θ_{S}(− log Y)) coincides with the kernel of μ. This implies that:

H_{D}^{1}(S, Θ_{S}(− log D)) ≃ \{ \begin{matrix}Infinitesimal deformations of (S, D) whose restriction \\ to S − D induces the trivial deformation \end{matrix} \} .

In §6, we will show that any direction corresponding to a non-zero element of the local cohomology group H_{D}^{1}(S, Θ_{S}(− log D)) induces a differential equation (at least locally) by using Čech coboundaries.

At this moment, we can not prove Conjecture 3.1 in the full generality. However, we see that the one dimensional vector subspace H^{1}(D, Θ_{S}(− log D)⊗N_{D}) of H_{D}^{1}(Θ_{S}(− log D)) ⊂ H^{1}(Θ_{S}(− log D)) really corresponds to the time variable t in the known Painlevé equation. It is unlikely that we will have more time variables, so this gives an evidence of Conjecture 3.1.

Let us explain the strategy of proving Theorem 3.1 in [T]. Recall that

H_{D}^{1}(S, Θ_{S}(− log D)) = \underrightarrow{lim} Ext^{1}(O_{nD}, Θ_{S}(− log D))

where O_{nD} = O_{S}/O_{S}(−nD) (cf. [Theorem 2.8, [Gr]]).

On the other hand, since Θ_{S}(− log D) is a locally free O_{S}-module, we see that

Hom(O_{nD}, Θ_{S}(− log D)) = 0,  (21)

and

Ext^{1}(O_{nD}, Θ_{S}(− log D)) = Θ_{S}(− log D) ⊗ N_{nD},  (22)

where N_{nD} = O_{S}(nD)/O_{S}. By an argument using a spectral sequence, we see that

H_{D}^{1}(S, Θ_{S}(− log D)) = \underrightarrow{lim} H^{0}(Θ_{S}(− log D) ⊗ N_{nD})  (23)

Hence, we have a natural inclusion

H^{0}(Θ_{S}(− log D) ⊗ N_{D}) ↪ H_{D}^{1}(S, Θ_{S}(− log D)).  (24)

Lemma 3.1. Let (S, Y) be a generalized rational Okamoto–Painlevé pair as above and set D = Y_{red}. Then we have the following exact sequences

0 ⟶ Θ_{D} ⊗ N_{D} ⟶ Θ_{S} ⊗ N_{D} ⟶ ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⊗ N_{D} ⟶ 0.  (25)
0 ⟶ ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⟶ Θ_{S}(− log D) ⊗ N_{D} ⟶ Θ_{D} ⊗ N_{D} ⟶ 0.  (26)

Here Θ_{D} denotes the tangent sheaf of D and ν : D̃ ⟶ D the normalization map.

Proof. The first exact sequence (25) follows from [(1.9), [B-W]].

Let us consider the following diagram:

0  0  ker λ
↓  ↓  ↓
0 ⟶ Θ_{S}(− log D) ⟶ Θ_{S}(− log D) ⊗ O_{S}(D) ⟶ Θ_{S}(− log D) ⊗ N_{D} ⟶ 0
↓  ↓  ↓ λ
0 ⟶  Θ_{S}  ⟶  Θ_{S} ⊗ O_{S}(D)  ⟶  Θ_{S} ⊗ N_{D}  ⟶ 0
↓  ↓  ↓
ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⟶^{μ} ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⊗ N_{D} ⟶  coker λ  ⟶ 0
↓  ↓  ↓
0  0  0  .

By the snake lemma, we obtain the exact sequence

0 ⟶ ker λ ⟶ ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⟶^{μ} ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⊗ N_{D} ⟶ coker λ ⟶ 0.

From a local consideration, we see that the map μ is the zero map, hence

ker λ ≃ ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}), ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⊗ N_{D} ≃ coker λ.

Moreover since Im λ ≃ ker[Θ_{S} ⊗ N_{D} ⟶ ν_{∗}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⊗ N_{D}], from the exact sequence (25), we obtain the exact sequence (26).

□.

From the exact sequence (26), one can obtain

H^{0}(⊕_{i=1}^{r}N_{Y_{i}/S}) ⟶ H^{0}(Θ_{S}(− log D) ⊗ N_{D}) ⟶ H^{0}(Θ_{D} ⊗ N_{D}) ⟶^{δ} H^{1}(⊕_{i=1}^{r}N_{Y_{i}/S}). (27)

where δ denotes the connected homomorphism.

Note that since N_{Y_{i}/S} = O_{Y_{i}}(−2), we have

H^{0}(⊕_{i=1}^{r}N_{Y_{i}/S}) = {0}, H^{1}(⊕_{i=1}^{r}N_{Y_{i}/S}) ≃ C^{r}.

Moreover, one can easily see that

Θ_{D} ≃ ν_{∗}(⊕Θ_{Y_{i}}(−t_{i})) ≃ ν_{∗}(⊕O_{Y_{i}}(2 − t_{i}))

where t_{i} is the number of intersections of Y_{i} with the other Y_{j}s. On the other hand, since D · Y_{i} = t_{i} − 2, we see that

H^{0}(Θ_{D} ⊗ N_{D}) ≃ H^{0}(⊕_{i=1}^{r}O_{Y_{i}}),

hence

H^{0}(Θ_{D} ⊗ N_{D}) ≃ C^{r}.  (28)

From this, the connecting homomorphism δ

δ : H^{0}(Θ_{D} ⊗ N_{D}) ⟶ ⊕_{i=1}^{r}H^{1}(N_{Y_{i}/S})  (29)

can be identified with a linear map δ : C^{r} ⟶ C^{r} and we have an isomorphism

H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) ≃ ker δ.  (30)

The following proposition is the main theorem of [T]. Proposition 3.2. Let (S, Y) be as in Theorem 3.1. A matrix representation of the linear map δ : H^{0}(Θ_{D} ⊗ N_{D}) ⟶ ⊕_{i=1}^{r}H^{1}(N_{Y_{i}/S}) in (29) is equal to the ± of the intersection matrix of D = ∑_{i=1}^{r} Y_{i}, that is,

δ = ((Y_{i} · Y_{j}))_{1≤i,j≤r}.

Since the intersection matrix ((Y_{i} · Y_{j}))_{1≤i,j≤r} has exactly one-dimensional kernel corresponding to the space of Y = ∑_{1=1}^{r} m_{i}Y_{i}, we have

H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) ≃ ker δ = C.

Remark 3.1. We have the following exact sequence of sheaves

0 ⟶ Θ_{S}(− log D) ⟶ Θ_{S}(− log D) ⊗ O_{S}(D) ⟶ Θ_{S}(− log D) ⊗ N_{D} ⟶ 0.  (31)

Combining this with the local cohomology exact sequence (17) and the natural inclusion (24), we have the following commutative diagram

C ≃ H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) ⟶^{δ} H^{1}(Θ_{S}(− log D)) ⟶^{φ} H^{1}(Θ_{S}(− log D) ⊗ O_{S}(D))
↓  ∥  ↓
0 →  H_{D}^{1}(Θ_{S}(− log D))  ⟶ H^{1}(Θ_{S}(− log D)) ⟶^{res}  H^{1}(S − D, Θ_{S−D})

(32)

4. Reviews on Kodaira–Spencer theory

In this section, we review on Kodaira–Spencer theory of complex analytic deformation. A main reference is [KS].

Let X be a compact complex manifold of dimension n. We can take a locally finite open covering {U_{i}}_{i∈I} of X such that each open subset U_{i} admits local coordinates z_{i} = (z_{i}^{1}, ⋯, z_{i}^{n}):

X = ∪_{i∈I}U_{i}.

For a point p ∈ U_{i} ∩ U_{j}, we have two local coordinates z_{i}(p) and z_{j}(p) whose coordinate transformation are given by

z_{i} = (z_{i}^{1}, z_{i}^{2}, ⋯, z_{i}^{n}) = f_{ij}(z_{j}),

or more precisely for α = 1, ⋯, n,

z_{i}^{α} = f_{ij}^{α}(z_{j}^{1}, · · · .z_{j}^{n}).  (33)

Here f_{ij}^{α}(z_{j}) are holomorphic functions defined on U_{i} ∩U_{j}. Note that one can give the compatibility conditions for U_{i} ∩ U_{j} ∩ U_{k} ≠ ∅

f_{ik}^{α}(z_{k}) = f_{ij}^{α}(f_{jk}^{1}(z_{k}), ⋯, f_{jk}^{n}(z_{k}))  (34)

Complex structure of X can be deformed by changing these coordinate transformations keeping the compatibility conditions.

Let B be a complex manifold with a (global) coordinate system (t_{1}, ⋯, t_{m}) and a specific marked point 0 = (0, ⋯, 0) ∈ B. We may think that B is an affine variety or a complex analytic small open ball around the origin. Definition 4.1. A deformation of X with a parameter space B ∋ (t_{1}, ⋯, t_{m}) is a proper smooth holomorphic map π : X ⟶ B such that the following diagram is commutative:

X ↩^{ι} X_{0} ≃ X
π ↓  ↓
B ∋ 0 .

Definition 4.2. A deformation π : X ⟶ B of X is said to have a finite covering relative to B if X is covered by {Ũ_{i} = U_{i} × B} such that the following diagram is commutative:

X = ∪_{i∈I}(U_{i} × B)
π ↓  ↓
B =  B
.

Let us assume that π : X ⟶ B has a finite covering relative to B and take the local coordinate of Ũ_{i} = U_{i} × B by (z_{i}^{1}, ⋯, z_{i}^{n}, t_{1}, ⋯, t_{m}). The coordinate transformation for Ũ_{i} ∩ Ũ_{j} is given by

z_{i}^{α} = f_{ij}^{α}(z_{j}^{1}, ⋯, z_{j}^{n}, t_{1}, ⋯, t_{m}).

We may assume that for t = 0, we have f_{ij}^{α}(z_{j}^{1}, ⋯, z_{j}^{n}, 0, ⋯, 0) = f_{ij}^{α}(z_{j}^{1}, ⋯, z_{j}^{n}).

Now we can introduce the Kodaira–Spencer class of the deformation π : X ⟶ B for each t ∈ B. For simplicity we assume that B is one dimensional, hence t = t_{1} is the global parameter of B.

Let h be a holomorphic function on an open subset V of X. Then on Ũ_{i} ∩ V, h is a function in a local coordinate h(z_{i}^{1}, ⋯, z_{i}^{n}, t). Assume that Ũ_{i} ∩ Ũ_{j} ∩ V ≠ ∅. Regarding as

h(z_{i}, t) = h(f_{ij}^{1}(z_{j}, t), ⋯, f_{ij}^{n}(z_{j}, t), t),

from the chain rule, we obtain

(\frac{∂h}{∂t})_{j} = (\frac{∂h}{∂t})_{i} + ∑_{α=1}^{n} \frac{∂f_{ij}^{α}(z_{j}, t)}{∂t} \frac{∂h}{∂z_{i}^{α}} .  (35)

This implies that, as a vector field on Ũ_{i} ∩ Ũ_{j}, we have the following identity:

(\frac{∂}{∂t})_{j} = (\frac{∂}{∂t})_{i} + ∑_{α=1}^{n} \frac{∂f_{ij}^{α}(z_{j}, t)}{∂t} \frac{∂}{∂z_{i}^{α}} .  (36)

Let us set {θ_{ij}(t)} by

θ_{ij}(t) = ∑_{α=1}^{n} \frac{∂f_{ij}^{α}(z_{j}, t)}{∂t} \frac{∂}{∂z_{i}^{α}} .  (37)

From the compatibility conditions for Ũ_{i} ∩ Ũ_{j} ∩ Ũ_{k} ≠ ∅

f_{ik}^{α}(z_{k}, t) = f_{ij}^{α}(f_{jk}^{1}(z_{k}, t), ⋯, f_{jk}^{n}(z_{k}, t), t),  (38)

we obtain the identity

θ_{ik}(t) = θ_{ij}(t) + θ_{jk}(t).

This implies that {θ_{ij}(t)} defines a Čech 1-cocycle with values in Θ_{X_{t}}, hence defines a cohomology class

θ(t) ∈ H^{1}(X_{t}, Θ_{X_{t}}),

which is called the Kodaira–Spencer class .

If the dimension of B is greater than one, we can define the cohomology class for each \frac{∂}{∂t_{μ}}. More precisely one can define the Kodaira-Spencer map

ρ : T_{t}(B) ⟶ H^{1}(X_{t}, Θ_{X_{t}})  (39)
v ↦ ρ(v) = θ_{v}(t)  (40)

by

θ_{v,ij} = \{θ_{v,ij}(t) = ∑_{α=1}^{n} v(f_{ij}^{α}(z_{j}, t))\frac{∂}{∂z_{i}^{α}} \}.

Here for

v = ∑_{μ=1}^{m} A_{μ}(t)\frac{∂}{∂t_{μ}} ,

we set

v(f_{ij}^{α}(z_{j}, t)) = ∑_{μ=1}^{m} A_{μ} \frac{∂f_{ij}^{α}(z_{j}, t)}{∂t_{μ}}.

Definition 4.3. A deformation π : X ⟶ B is called locally trivial, if for each point t ∈ B there exists an open neighborhood I of t such that X_{|I} ⟶ I is complex analytically isomorphic to the product X_{t} × I. Proposition 4.1. ([KS]) Let π : X ⟶ B be a deformation of a compact complex manifold with parameter space B ∋ t = (t_{1}, ⋯, t_{m}). If for every point t ∈ B dim H^{1}(X_{t}, Θ_{X_{t}}) is constant and Kodaira–Spencer map ρ is the zero map, then π : X ⟶ B is a locally trivial fibration.

For a proof in detail, we refer the reader to [Theorem 5.1, [KS]]. Since we will use the idea of the proof later, we explain an outline of the proof of theorem when dim B = 1. By replacing B with a neighborhood of t ∈ B, we may assume that a deformation π : X ⟶ B has a finite covering {Ũ_{i} = U_{i} × B} relative to B. Then the Kodaira–Spencer class

ρ(\frac{∂}{∂t}) = θ(t) ∈ H^{1}(X_{t}, Θ_{X_{t}}).

is represented by Čech cocycles {θ_{ij}(t)} given in (37). Since θ(t) is cohomologus to zero, for each t we can find

θ_{i}(t) ∈ Γ(Ũ_{i} ∩ X_{t}, Θ_{U_{i}}).

such that

θ_{ij}(t) = θ_{j}(t) − θ_{i}(t) on Ũ_{i} ∩ Ũ_{j} ∩ X_{t}

From (36), we obtain the following identities of vector fields on each Ũ_{i} ∩ Ũ_{j} ∩ X_{t}

(\frac{∂}{∂t})_{j} = (\frac{∂}{∂t})_{i} + (θ_{j}(t) − θ_{i}(t)),  (41)

and hence

(\frac{∂}{∂t})_{j} − θ_{j}(t) = (\frac{∂}{∂t})_{i} − θ_{i}(t).  (42)

At this moment, it is not obvious that the dependence of

θ_{i}(t) = ∑_{α=1}^{n} θ_{i}^{α}(z_{i}, t)\frac{∂}{∂z_{i}^{α}}  (43)

with respect to t is in C^{∞} class. However under the condition that dim H^{1}(X_{t}, Θ_{X_{t}}) is constant on B, one can prove that θ_{i}(t) can be chosen as a vector field on Ũ_{i} = U_{i} × B in C^{∞} class.

This implies that the vector field

\{(\frac{∂}{∂t})_{i} − θ_{i}(t)\}_{i∈I}  (44)

on Ũ_{i} can be glued together and defines a global C^{∞}–vector field, say, ṽ on the total space X. We see that ṽ is a lift of vector field \frac{∂}{∂t} by π. Then on each open set Ũ_{i}, we can consider the ordinary differential equation

\frac{dz_{i}^{α}}{dt} = −θ_{i}^{α}(z_{i}, t) α = 1, ⋯, n.  (45)

And these set of differential equations can be patched together on whole X. Starting from an initial conditions (a_{1}, ⋯, a_{n}, t_{0}) ∈ X_{t_{0}}, the solution (z_{1}(a_{j}, t), ⋯, z_{n}(a_{j}, t)) of differential equation (45) defines a C^{∞}–curve which is transversal to each fiber X_{t}. Then the whole solutions of (45) define a foliation on X and define C^{∞}-defeomorphisms φ_{t} : X_{0} →^{≃} X_{t}. Moreover, one can show that this defeomorphism φ_{t} is a complex biholomorphic morphism for each t ∈ B.

This implies the following. If we have a family of compact complex manifolds π : X ⟶ B with a parameter t ∈ B such that the Kodaira–Spencer map ρ_{t} is zero, we will obtain a differential equation as in (45) defined on the total space X.

Summarizing these, we have the following implications (cf. Figure 3).

Deformation π : X ⟶ B of complex manifolds with zero Kodaira–Spencer map

⇓

There exists a vector field ṽ on X which is a lift of \frac{∂}{∂t}.

⇓

Differential Equation on π : X ⟶ B given by ṽ

⇓

Local trivializations of the deformation X ⟶ B .

Figure 3

5. Global Deformations of Okamoto–Painlevé pairs

Affine coverings and Symplectic Structures on S − D

Let (S, Y) be a generalized rational Okamoto–Painlevé pair. Then by definition, S has a rational 2-form ω_{Y} whose pole divisor is Y. Setting D = Y_{red}, the rational 2-form ω_{Y} induces a nondegenerate holomorphic 2-forms on the open surface S−D, hence induces a holomorphic symplectic structure on S − D.

In [O1], Okamoto introduced the space of initial conditions of Painlevé equation of each type, which can be written as S −D for an Okamoto-Painlevé pair (S, Y). The main reason why Painlevé equations can be written as Hamiltonian systems is this holomorphic symplectic structure. For Painlevé equations P_{J}, (J = II, III, IV, V, V I), Takano et al. [ST], [MMT] constructed a good family of Okamoto–Painlevé pairs (S_{α,t}, Y_{α,t}) depending on the time variable and a system of auxiliary parameters α = (α_{1}, ⋯, α_{s}) appeared in each Painlevé equation.

Summarizing results in [ST], [MMT], let us explain the situation of spaces of initial conditions of classical Painlevé equations in the way of our setting. Let R = R(Y) be a type of the root systems corresponding to a Painlevé equation. Then there exist an affine open subset M_{R} of C^{s} = Spec C[α] = Spec C[α_{1}, ⋯, α_{s}], an affine open subset B_{R} of C = Spec C[t] and the following deformation of non-singular pair

\begin{matrix}S & ↩ & D \\ π ↓ & ↙ & φ \\ M_{R} × B_{R} \end{matrix}  (46)

where S ⟶ M_{R} × B_{R} is a smooth family of rational surface and D ↪ S is a normal crossing divisor. In order to relate this diagram to Okamoto–Painlevé pair, let Ω_{S/M_{R}×B_{R}}^{2}(∗D) denote the sheaf of germs of relative rational two forms on S which have poles only along D. There exists a section

ω_{S} ∈ Γ(S, Ω_{S/M_{R}×B_{R}}^{2}(∗D))

which induces a rational 2-form ω_{S_{α,t}} for each fiber S_{α,t}. The pole divisor ω_{S} is denoted by Y, and with suitable choice of ω_{S} we may assume that each fiber (S_{α,t}, Y_{α,t}) is an Okamoto–Painlevé pair of type R = R(Y) and Y_{red} = D. (Note that on S − D the relative rational 2-form ω_{S} is holomorphic and non-degenerate on each fiber S_{α,t} − D_{α,t}). Assuming that the family (46) is effectively parameterized and semiuniversal at each point of M_{R} × B_{R}, or equivalently the Kodaira–Spencer map

ρ : T_{α,t}(M_{R} × B_{R}) ⟶ H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}))  (47)

is an isomorphism at each point, we have the equality

dim M_{R} = dim H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t})) − 1.  (48)

The dimensions of M_{R} for generalized rational Okamoto–Painlevé pairs.

R = R(Y)  Ẽ_{8} D̃_{8}  Ẽ_{7}  D̃_{7}  D̃_{6}  Ẽ_{6} D̃_{5} D̃_{4}

Painlevé equation  P_{I} P D̃_{8} ; III P_{II} P D̃_{7} ; III P D̃_{6} ; III P_{IV} P_{V} P_{V I}

s = s(R) = dim M_{R}  0  0  1  1  2  2  3  4

(= ♯ of auxiliary parameters.)

Table 6

(Note that the Okamoto–Painlevé pairs of type D̃_{8}, D̃_{7} did not appear in the classical literatures (cf. [O1], [O2], [MMT]).)

More notably, Takano et al. [ST], [MMT] constructed an affine open covering {Ũ_{i}}_{i=1}^{l+m} of S for the classical Okamoto–Painlevé pair of Painlevé equation P_{J} (J = II, ⋯, V I), which is relative to π and so that

Ũ_{i} = M_{R} × B_{R} × U_{i}

where U_{i} = Spec C[x_{i}, y_{i}] ≃ C^{2}. Moreover, we may assume that {Ũ_{i}}_{i=1}^{l} covers S − D and for 1 ≤ i ≤ l, we have

ω_{S|Ũ_{i}} = dx_{i} ∧ dy_{i}  (49)

In this sense, the restricted morphism

π : S − D ⟶ M_{R} × B_{R}

is a deformation of open symplectic surfaces.

By using the results in Appendix B of [Sakai], we can generalize the result of Takano, et al. as follows. Proposition 5.1. Let R = R(Y) be one of types of the root systems appeared in Proposition 1.2 which is additive type, so that

dim H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) = 1

for corresponding generalized rational Okamoto–Painlevé pair (S, Y) (cf. Theorem 3.1). Moreover denote by r the number of irreducible components of D = Y_{red}.

Let M_{R} be an affine open subscheme in C^{s} = Spec C[α_{1}, ⋯, α_{s}] of dimension s = s(R) = 9−r and B_{R} be an affine open subscheme of C = Spec C[t]. Then there exists the following commutative diagram satisfying the conditions below.

\begin{matrix}S & ↩ & D \\ π ↓ & ↙ & φ \\ M_{R} × B_{R} & & . \end{matrix}  (50)

(1) The above diagram is a deformation of non-singular pair of projective surfaces and normal crossing divisors in the sense of Definition 2.1

(2) There exists a rational relative 2-form

ω_{S} ∈ Γ(S, Ω_{S/M_{R}×B_{R}}^{2}(Y))

whose pole divisor is Y with Y_{red} = D.

(3) If we denote by Y the pole divisor of ω_{S}, then for each point (α, t) ∈ M_{R} ×B_{R}, (S_{α,t}, Y_{α,t}) is a generalized Okamoto–Painlevé pair of non fibered type with type R = R(Y). Moreover Y_{red} = D.

(4) The family is semiuniversal at a general point (α, t) ∈ M_{R} × B_{R}, that is, the Kodaira– Spencer map

ρ : T_{α,t}(M_{R} × B_{R}) ⟶ H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}))  (51)

is an isomorphism for a general point (α, t).

The Kodaira–Spencer class ρ(\frac{∂}{∂t}) of t-direction (= B_{R}) lies in the image of the natural map (cf. (32)):

δ : C ≃ H^{0}(D_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}) ⊗ N_{D_{α,t}}) ↪ H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t})).  (52)

(5) Let M_{R} and B_{R} denote the affine coordinate rings of M_{R} and B_{R} respectively so that M_{R} = Spec M_{R} and B_{R} = Spec B_{R}. (Note that M_{R} and B_{R} is obtained by some localization’s of C[α_{1}, ⋯, α_{s}] and C[t] respectively. ) There exists a finite affine covering {Ũ_{i}}_{i=1}^{l+k} of S relative to M_{R} × B_{R} such that there exists an isomorphism for each i

Ũ_{i} ≃ Spec(M_{R} ⊗ B_{R})[x_{i}, y_{i}, \frac{1}{f_{i}(x_{i}, y_{i}, α, t)}] ⊂ Spec C[α, t, x_{i}, y_{i}] ≃ C^{s+3} ≃ C^{12−r} (53)

Here f_{i}(x_{i}, y_{i}, α, t) is a polynomial in (M_{R} ⊗ B_{R})[x_{i}, y_{i}]. Moreover we may assume that S − D can be covered by {Ũ_{i}}_{i=1}^{l}. Moreover for each i the restriction of the rational two form ω_{S} can be written as

ω_{S|Ũ_{i}} = \frac{dx_{i} ∧ dy_{i}}{f_{i}(x_{i}, y_{i}, α, t)^{m_{i}}}  (54)

(6) For each pair i, j such that Ũ_{i} ∩ Ũ_{j} ≠ ∅, the coordinate transformation functions

x_{i} = f_{ij}(x_{j}, y_{j}, α, t), y_{i} = g_{ij}(x_{j}, y_{j}, α, t)  (55)

are rational functions in variables x_{j}, y_{j}, α, t.

Here we will give a sketch of the proof of Proposition 5.1. (See [Sa-Te] for explicit constructions). For a generalized rational Okamoto–Painlevé pair (S, Y), we see that S can be obtained as a blowing up of P^{2} at (possibly infinitely near ) 9-points which lie on anti-canonical divisors. Then one can parameterize these 9-points in a suitable way, and this leads to a special time parameter t and other parameter α_{1}, ⋯, α_{s}, hence we obtain affine schemes M_{R} ⊂ C^{s} and B_{R} ⊂ C deleting the points which do not correspond to Okamoto–Painlevé pairs of non-fibered type with type R. ( We can show that non-fiberedness and type of (S, Y) just depends on the parameters α and not depends on t ∈ B_{R}. ) Moreover we can construct a semiuniversal family π : S ⟶ M_{R} × B_{R} of rational surfaces by blowings–up of P^{2} × M_{R} × B_{R}. Moreover by these explicit constructions, we can obtain the affine coverings of the total space S as above.

Remark 5.1. We can construct a similar family of generalized Okamoto–Painlevé pairs of type Ã_{r−1}, 2 ≤ r ≤ 9 (multiplicative type). However as proved in [T], we see that

H_{Y}^{1} (S, Θ_{S}(− log Y)) = {0}.

This result implies that we can not obtain a differential equation from the generalized Okamoto– Painlevé pair of type Ã_{r} as in the way above.

6. From Global Deformations to Hamiltonian systems

In this section, we will explain how one can derive differential equations from the one dimensional global deformation of generalized rational Okamoto-Painlevé pairs of additive type which arises from the image of linear map

δ : C ≃ H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) ⟶ H^{1}(S, Θ_{S}(− log D))

or more generally the kernel of the restriction map res in (17). (Recall that Im(δ) ⊂ ker(res)). The idea of the proof is essentially same as in Kodaira–Spencer theory (Section 4). We will be able to obtain a global rational vector field ṽ on S ⟶ M_{R} × B_{R} whose poles are only supported on D, which is a lift of \frac{∂}{∂t}. Hence the restriction of ṽ to S − D gives a global regular algebraic vector field. Moreover we will show that such a lift ṽ of \frac{∂}{∂t} is unique. The restriction of ṽ to the open affine subset Ũ_{i} of S − D gives an explicit system of algebraic ODEs of 1st order.

The second aim of this section is to explain the reason why the systems obtained as above are Hamiltonian systems. This fact is known classically ([O1], [O2], [MMT]) and from the uniqueness of the ODE we can check the fact from explicit equations. In this section, we will show that global vector field ṽ preserves the relative 2 forms ω_{S}. This fact can be expressed (cf. Theorem 6.2) as

d_{S/M}(ṽ · (ω_{S} ∧ dt)) = 0.  (56)

In classical cases (that is, R ≠ D̃_{7}, D̃_{8} ), as in ([MMT]) we have at least one open embedding C^{2} ⊂ S − D and its nice coordinates (x, y) such that ω_{S|C^{2}} = dx ∧ dy. By using such nice coordinates, the equation (56) shows that the differential equation obtained as above becomes a Hamiltonian system with a polynomial Hamiltonian. However since we can not always cover S −D by copies of C^{2} in case of D̃_{7}, D̃_{8} and Ẽ_{8} and the restriction of ω_{S} to some open affine set looks like as in (54), we can not always written the systems in Hamiltonian systems in the affine coordinates.

We start to explain how to obtain the differential equations.

Let R = R(Y) be one of types of additive affine root systems appeared in Proposition 1.2 and let

\begin{matrix}S & ↩ & D \\ π ↓ & ↙ & φ \\ M_{R} × B_{R} \end{matrix}  (57)

be a global deformation of generalized Okamoto–Painlevé pairs of type R as in Proposition 5.1. The total space S has a finite affine covering {Ũ_{i}}_{i=1}^{l+k} such that

Ũ_{i} ≃ Spec(M_{R} ⊗ B_{R})[x_{i}, y_{i}, \frac{1}{f_{i}(x_{i}, y_{i}, α, t)}] ⊂ Spec C[α, t, x_{i}, y_{i}]  (58)

as in (53). Moreover, we may assume that S − D can be covered by {Ũ_{i}}_{i=1}^{l}, that is,

S − D = ∪_{i=1}^{l}Ũ_{i}.

Let us recall that the coordinate transformations in (55) for Ũ_{i} ∩Ũ_{j} ≠ ∅ are given by the rational functions

x_{i} = f_{ij}(x_{j}, y_{j}, α, t), y_{i} = g_{ij}(x_{j}, y_{j}, α, t)  (59)

The Kodaira–Spencer class ρ(\frac{∂}{∂t}) can be represented by the Čech 1-cocycles

ρ(\frac{∂}{∂t}) = \{ θ_{ij} = \frac{∂f_{ij}}{∂t} \frac{∂}{∂x_{i}} + \frac{∂g_{ij}}{∂t} \frac{∂}{∂y_{i}} ∈ Γ(Ũ_{i} ∩ Ũ_{j}, Θ_{S/M_{R}×B_{R}}(− log D)) \}  (60)

Recall that (cf. Proposition 5.1, 3) on M_{R} × B_{R} the corresponding Okamoto–Painlevé pairs (S_{α,t}, Y_{α,t}) are of non-fibered type.

From (52) of Proposition 5.1, we may assume that ρ(\frac{∂}{∂t}) is non-zero element of the image of δ:

C ≃ H^{0}(D, Θ_{S_{α,t}}(− log D_{α,t}) ⊗ N_{D}) ⊂ H_{D_{α,t}}^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t})).
δ ↘  ↓
H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}))

(61)

Since the local cohomology group is the kernel of the natural restriction map (cf. Proposition 3.1)

res : H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t})) ⟶ H^{1}(S_{α,t} − D_{α,t}, Θ_{S_{α,t}}(− log D_{α,t})),  (62)

the Kodaira–Spencer class ρ(\frac{∂}{∂t}) is cohomologus to zero in H^{1}(S_{α,t} − D_{α,t}, Θ_{S_{α,t}}(− log D_{α,t})). More precisely, from (32), we see that the 1-cocycle ρ(\frac{∂}{∂t}) is cohomologus to zero in

H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}) ⊗ O_{S}(D_{α,t})).  (63)

Since dimensions of these cohomology groups are constant as functions of (α, t), by an argument using the base change theorem, we can see that for 1 ≤ i ≤ l + k there exist rational vector fields

θ_{i}(x_{i}, y_{i}, α, t) = η_{i}(x_{i}, y_{i}, α, t)\frac{∂}{∂x_{i}} + ζ_{i}(x_{i}, y_{i}, α, t)\frac{∂}{∂y_{i}} ∈ Γ(Ũ_{i}, ΘŨ_{i}(− log D) ⊗ O(D)) (64)

such that

θ_{ij}(x_{i}, y_{i}, α, t) = θ_{j}(x_{j}, y_{j}, α, t) − θ_{i}(x_{i}, y_{i}, α, t).  (65)

Note that for 1 ≤ i ≤ l one has Ũ_{i} ∩D = ∅. Hence for 1 ≤ i ≤ l, θ_{i}(x_{i}, y_{i}, α, t) is a regular algebraic vector field, that is, η_{i}(x_{i}, y_{i}, α, t) and ζ(x_{i}, y_{i}, α, t) are regular algebraic functions on Ũ_{i}.

As in (36) of §4, for any pair i, j such that Ũ_{i} ∩ Ũ_{j} ≠ ∅, we have the identity on Ũ_{i} ∩ Ũ_{j}

(\frac{∂}{∂t})_{j} = (\frac{∂}{∂t})_{i} + θ_{ij}(α, t),  (66)

and hence we have

(\frac{∂}{∂t})_{j} = (\frac{∂}{∂t})_{i} + (θ_{j}(x_{j}, y_{j}, α, t) − θ_{i}(x_{i}, y_{i}, α, t)),  (67)

or

(\frac{∂}{∂t})_{j} − θ_{j}(x_{j}, y_{j}, α, t) = (\frac{∂}{∂t})_{i} − θ_{i}(x_{i}, y_{i}, α, t).  (68)

This means that the vector fields

\{(\frac{∂}{∂t})_{i} − θ_{i}(x_{i}, y_{i}, α, t)\}_{1≤i≤l+k}  (69)

can be patched together and define a global rational vector field

ṽ ∈ Γ(S, Θ_{S}(− log D) ⊗ O(D)).

Note that this global rational vector field ṽ is a lift of \frac{∂}{∂t} via π : S ⟶ M_{R} × B_{R} and

ṽ_{|S−D}  (70)

is a regular algebraic vector field. Moreover since for each (α, t) ∈ M_{R} × B_{R} we have H^{0}(S_{α,t} − D_{α,t}, Θ_{S_{α,t}−D_{α,t}}^{alg}) = 0 (cf. Proposition 2.1), we can show the uniqueness of ṽ of lifting of \frac{∂}{∂t} to the rational vector field to S. Now we obtain the following Theorem 6.1. Let R = R(Y), S, D, M_{R} × B_{R} be as above. Then there exists a unique global rational vector field

ṽ ∈ Γ(S, Θ(− log D) ⊗ O_{S}(D))

on S which is a lift of \frac{∂}{∂t}. The restriction ṽ_{S−D} to S−D is a regular algebraic vector field. Moreover the restriction ṽ to each open covering Ũ_{i} for 1 ≤ i ≤ l (corresponding to the open coverings of S − D) can be written as

ṽ_{|Ũ_{i}} = \frac{∂}{∂t} − θ_{i} = \frac{∂}{∂t} − η_{i} \frac{∂}{∂x_{i}} − ζ_{i} \frac{∂}{∂y_{i}}  (71)

and defines a system of differential equations

\{ \begin{matrix}\frac{dx_{i}}{dt} & = & −η_{i}(x_{i}, y_{i}, α, t) \\ \frac{dy_{i}}{dt} & = & −ζ_{i}(x_{i}, y_{i}.α, t) \end{matrix} .  (72)

Here the functions η_{i}, ζ_{i} are regular algebraic functions on Ũ_{i}. Remark 6.1. (1) The argument above shows that there exists a differential equation as above

at least locally for any direction corresponding to the kernel of the restriction map

res : H^{1}(S, Θ_{S}(− log D)) ⟶ H^{1}(S − D, Θ_{S}(− log D)).

(2) Let us recall the so-called Painlevé property which is states as follows. If (x(t), y(t)) is a local solution of (72) determined by an arbitrary initial conditions (x_{0} = x(t_{0}), y_{0} = y(t_{0})) ∈ Ũ_{i} with fixed t_{0} ∈ B_{R} then both solutions x(t) and y(t) can be meromorphic-ally continued along any curve in B_{R} with a starting point t_{0}. For non-classical Okamoto– Painlevé pair, it is not clear that the differential equation in (72) has the Painlevé property. In general, the proof of Painlevé property for classical Painlevé equation is not so easy. We hope that there is an easy geometric proof of the Painlevé property for differential equation in (72).

It is well-known that each classical Painlevé differential equation P_{J}, J = I, II, ⋯, V I is equivalent to a Hamiltonian system (H_{J}) whose Hamiltonian H_{J}(x, y, α, t) is a polynomial in (x, y) ∈ C^{2} (cf. [O1], [MMT], §7).

(H_{J}) : \{ \begin{matrix}\frac{dx}{dt} & = & \frac{∂H_{J}}{∂y} \\ \frac{dy}{dt} & = & −\frac{∂H_{J}}{∂x} \end{matrix} .  (73)

This fact can be checked by explicit calculations for each differential equations (cf. [Sa-Te]). However, the reason why the equations obtained in (72) become Hamiltonian systems is not clear.

In what follows, we will give a geometric explanation for this fact. For the purpose, we will introduce sheaves of infinitesimal symplectic automorphisms. Lemma 6.1. Let (S, Y) be a rational Okamoto–Painlevé pair of non-fibered type with the root system R of additive type and set D = Y_{red} and E = Y − D. Fix a non-zero rational 2-form ω with the pole divisor Y. Then there exists a commutative diagram of exact sequences of sheaves

0  0  0
↓  ↓  ↓
0 ⟶  Γ_{ω}  ⟶  Θ_{S}(− log D)  ⟶^{d_{ω}}  Ω_{S}^{2}(Y) ≃ O_{S}  ⟶ 0
↓  ↓  ↓
0 ⟶ Γ(D)_{ω} ⟶ Θ_{S}(− log D) ⊗ O_{S}(D) ⟶^{d_{ω}} Ω_{S}^{2}(Y + D) ≃ O_{S}(D) ⟶ 0
↓ ↓ ↓
0 ⟶ Γ(D)_{ω}/Γ_{ω} ⟶ Θ_{S}(− log D) ⊗ N_{D}  ⟶^{d_{ω}} Ω_{S}^{2}(Y + D)/Ω_{S}^{2}(D) ≃ N_{D} ⟶ 0
↓  ↓  ↓
0  0  0

(74)

Here the map d_{ω} is given by θ ↦ d_{ω}(θ) = d(θ · ω) and Γ_{ω} and Γ(D)_{ω} are the kernels of d_{ω}’s in the above diagram. Proof. Since Y is the pole of the rational two form ω and K_{S} = −Y, we see that Ω_{S}^{2}(Y) ≃ O_{S} and the contractions with ω give the isomorphisms

Θ_{S}(− log D) ⟶^{≃} Ω_{S}^{1}(log D) ⊗ O_{S}(E), Θ_{S}(− log D) ⊗ O_{S}(D) ⟶^{≃} Ω_{S}^{1}(log D) ⊗ O_{S}(Y) (75)

Then we only have to show that the following two maps of sheaves are well-defined and surjective.

Ω_{S}^{1}(log D) ⊗ O_{S}(E) ⟶^{d} Ω_{S}^{2}(log D) ⊗ O(E) ≃ Ω_{S}^{2}(D + E) ≃ Ω_{S}^{2}(Y)
Ω_{S}^{1}(log D) ⊗ O_{S}(Y) ⟶^{d} Ω_{S}^{2}(log D) ⊗ O(Y) ≃ Ω_{S}^{2}(Y + D)

(76)

We can consider the problem analytic locally. At a point p ∈ S − D, the above two maps are just usual exterior derivatives and so they are well-defined and surjective. Let us consider the point p ∈ D. We have the local coordinates (x, y) near p such that the local equation of Y is given by x^{m}y^{n} = 0. First we consider the case of m = 0 and n > 0. Then the equations of D and E near p is given by y = 0 and y^{n−1}. Local generators of Ω_{S}^{1}(log D) ⊗ O(E) and Ω_{S}^{1}(log D) ⊗ O(Y) are given by

\frac{1}{y^{n−1}} dx, \frac{1}{y^{n}} dy, \frac{1}{y^{n}} dx, \frac{1}{y^{n+1}} dy

respectively. Moreover in case that both of n, m are positive, local generators of these sheaves are given by

\frac{1}{x^{m}y^{n−1}} dx, \frac{1}{x^{m−1}y^{n}} dy, \frac{1}{x^{m+1}y^{n}} dx, \frac{1}{x^{m}y^{n+1}} dy.

By using these local generators, we see that the exterior derivatives in (76) are well-defined and surjective.

Definition 6.1. The kernel Γ_{ω} of d_{ω} : Θ_{S}(− log D) ⟶ Ω_{S}^{2}(Y) is called the sheaf of germs of local symplectic logarithmic vector fields with respect to ω. Note that Γ_{ω} depends on the choice of ω.

Now let us consider the relative situation. Let π : S ⟶ M × B be the global semiuniversal family of Okamoto–Painlevé pair of type R (cf. Proposition 5.1). Note that we have a relative rational 2-form ω_{S} with the pole divisor Y, that is, ω_{S} ∈ Γ(S, Ω_{S/M×B}^{2}(Y)) and D = Y_{red}. Let dt denote the restriction of the canonical generator of Ω_{Spec C[t]}^{1} to B ⊂ Spec C[t]. It is easy to see that

Ψ = ω_{S} ∧ dt  (77)

defines a relative rational 3-form in Γ(S, Ω_{S/M}^{3}(Y)) with the pole divisor Y. (Note that Ψ defines an isomorphism O_{S} ≃ Ω_{S/M}^{3}(Y).) We shall prove the following fundamental theorem. Theorem 6.2. Let ṽ ∈ Γ(S, Θ_{S}(− log D)(D)) be the global rational vector field obtained in Theorem 6.1. Then we have

d_{S/M}(ṽ · Ψ) = 0.  (78)

Without loss of generality, we may assume that M = Spec C in the following proof of Theorem 6.2.

In order to prove Theorem 6.2, we need some preparation. Let us consider the following commutative diagram of exact sequences of tangent sheaves. (Note that since π : S ⟶ B is smooth and D is a divisor with normal crossings Θ_{S/B}(− log D) and Θ_{S}(− log D) are locally free O_{S}-sheaves.) n

0  0  0 ↓  ↓  ↓ 0 ⟶  Θ_{S/B}(− log D)  ⟶  Θ_{S}(− log D)  ⟶  π^{∗}(Θ_{B})  ⟶ 0 ↓  ↓  ↓ 0 ⟶ Θ_{S/B}(− log D)(D) ⟶ Θ_{S}(− log D)(D) ⟶ π^{∗}(Θ_{B})(D) ⟶ 0 ↓  ↓  ↓ 0 ⟶ Θ_{S/B}(− log D) ⊗ N_{D} ⟶ Θ_{S}(− log D) ⊗ N_{D} ⟶ π^{∗}(Θ_{B}) ⊗ N_{D} ⟶ 0 ↓  ↓  ↓ 0  0  0 . (79)

Lemma 6.2. Let π : S ⟶ B be the family of rational Okamoto–Painlevé pairs of type R of non-fibered type as above. Then we have the following:

(1) π_{∗}(O_{S}) ≃ π_{∗}(O_{S}(D)) ≃ O_{B}, R^{1}π_{∗}O_{S} = 0.

(2) π_{∗}(N_{D}) = 0.

(3) π_{∗}(Θ_{S/B}(− log D)) ≃ π_{∗}(Θ_{S/B}(− log D)(D)) ≃ 0.

(4)

O_{B} ≃ π_{∗}(Θ_{S/B}(− log D)) ⊗ N_{D}) ≃ π_{∗}(Θ_{S}(− log D)) ⊗ N_{D}).

(5) The connecting homomorphism

ρ : Θ_{B} ⟶ R^{1}π_{∗}(Θ_{S/B}(− log D))

is the Kodaira–Spencer map and injective.

(6) π_{∗}(Θ_{S}(− log D)) = 0.

Proof. Since all fibers of π : S ⟶ B are rational surfaces of non-fibered type, (1) of Proposition 2.1 shows the first assertion, from which the second assertion follows. The third assertion follows from (3) of Proposition 2.1. The fourth and fifth assertion follows from (4) of Proposition 5.1. Taking the direct images of the first row of (79), the assertion (6) follows from the assertion (5). □

Taking the direct images of the diagram (79), from Lemma 6.2, we obtain the commutative diagram:

0 ↓ 0  0  π_{∗}(Θ_{S/B}(− log D)) ⊗ N_{D}) ≃ O_{B} ↓  ↓  ↓ μ 0  ⟶ Θ_{B} ⟶^{ρ}  R^{1}π_{∗}(Θ_{S/B}(− log D)) ↓  ||  ↓ λ 0 ⟶ π_{∗}(Θ_{S}(− log D)(D)) ⟶^{τ} Θ_{B} ⟶^{ρ_{1}}  R^{1}π_{∗}(Θ_{S/B}(− log D)(D)) ↓ ν  ↓  ↓ O_{B} ≃ π_{∗}(Θ_{S}(− log D)) ⊗ N_{D}) ⟶ 0 ⟶ ↓ . (80)

Lemma 6.3. (1) The image of ρ and μ in (80) coincides, i.e.,

μ(π_{∗}(Θ_{S/B}(− log D) ⊗ N_{D}) = ρ(Θ_{B}) ⊂ R^{1}π_{∗}(Θ_{S/B}(− log D)).  (81)

(2) The sheaf map ρ_{1} in (80) is the zero map and τ induces an isomorphism

τ : π_{∗}(Θ_{S}(− log D)(D)) ⟶^{≃} Θ_{B}.  (82)

Hence the vector field ṽ in Theorem 6.1 is the lift of \frac{∂}{∂t} via the natural isomorphism

τ̃ : H^{0}(S, Θ_{S}(− log D)(D)) ⟶^{≃} H^{0}(B, Θ_{B}).  (83)

(3) The map

ν : π_{∗}(Θ_{S}(− log D)(D)) ⟶ π_{∗}(Θ_{S}(− log D)) ⊗ N_{D}) ≃ O_{B}

is an isomorphism.

Proof. The first assertion follows from (52) of Proposition 5.1. Then we have the second assertion because

ρ_{1}(Θ_{B}) = λ(ρ(Θ_{B})) = λ(μ(π_{∗}(Θ_{S/B}(− log D) ⊗ N_{D}))) = 0.

The rest of assertions follow from this easily. □

By the same argument as in Lemma 6.1, we obtain the following commutative diagram of sheaves:

0  0  0 ↓  ↓  ↓ 0 ⟶  Γ̃_{Ψ}  ⟶  Θ_{S}(− log D)  ⟶^{d_{Ψ}}  Ω_{S}^{3}(Y) ≃ O_{S}  ⟶ 0 ↓  ↓  ↓ 0 ⟶ Γ̃(D)_{Ψ}  ⟶^{φ} Θ_{S}(− log D)(D) ⟶^{d_{Ψ}} Ω_{S}^{3}(Y + D) ≃ O_{S}(D) ⟶ 0 ↓ ν_{1}  ↓ ν  ↓ 0 ⟶ Γ̃(D)_{Ψ}/Γ̃_{Ψ} ⟶^{φ_{1}} Θ_{S}(− log D) ⊗ N_{D} ⟶^{d_{Ψ}} Ω_{S}^{3}(Y + D)/Ω_{S}^{3}(D) ≃ N_{D} ⟶ 0 ↓  ↓  ↓ 0  0  0 .(84)

Here d_{Ψ} can be defined for a local section θ of Θ_{S}(− log D) as

d_{Ψ}(θ) = d_{S}(θ · Ψ),

and Γ̃_{Ψ} and Γ̃(D)_{Ψ} are defined to be the kernels of d_{Ψ}’s in (84).

Theorem 6.2 follows from the following Lemma 6.4. The natural map φ in (84) induces an isomorphism

φ̃ : H^{0}(S, Γ̃(D)_{Ψ}) ⟶^{≃} H^{0}(S, Θ_{S}(− log D)(D)).  (85)

In fact, since ṽ ∈ H^{0}(S, Θ_{S}(− log D)(D)), Lemma 6.4 implies that ṽ ∈ H^{0}(S, Γ̃(D)_{Ψ}) and by definition

d_{Ψ}(ṽ) = d_{S}(ṽ · Ψ) = 0.

Proof of Lemma 6.4. We only have to show that the natural morphism

φ : π_{∗}(Γ̃(D)_{Ψ}) ⟶ π_{∗}(Θ_{S}(− log D)(D))  (86)

is an isomorphism.

Taking the direct images of the sheaves in (84), together with Lemma 6.2, we obtain the following commutative diagram:

0 ↓ 0 ⟶  0  ⟶  0  ⟶^{d_{Ψ}} O_{B} ↓  ↓  ↓ 0 ⟶  π_{∗}(Γ̃(D)_{Ψ}) ⟶^{φ} π_{∗}(Θ_{S}(− log D)(D)) ≃ O_{B}  ⟶^{d_{Ψ}} O_{B} ↓ ν_{1}  ↓ ν  ↓ 0 ⟶ π_{∗}(Γ̃(D)_{Ψ}/Γ̃_{Ψ}) ⟶^{φ_{1}} π_{∗}(Θ_{S}(− log D) ⊗ N_{D}) ≃ O_{B} ⟶^{d_{Ψ}} 0 ↓ μ_{1}  ↓  ↓ 0 ⟶ O_{B} ⟶  R^{1}π_{∗}(Γ̃_{Ψ})  ⟶  R^{1}π_{∗}Θ_{S}(− log D)  ⟶ 0 . (87)

Since we have proved that the map ν and φ_{1} are isomorphism of sheaves, in order to show that φ is an isomorphism, we only have to show that the map

μ_{1} : π_{∗}(Γ̃(D)_{Ψ}/Γ̃_{Ψ}) ⟶ R^{1}π_{∗}(Γ̃_{Ψ})  (88)

is the zero map.

Combining the commutative diagram (84) with (79), we obtain the the following commutative diagram:

0  0  0 ↓  ↓  ↓ 0 ⟶  Γ_{Ψ}  ⟶  Θ_{S/B}(− log D)  ⟶^{d_{Ψ}}  Ω_{S}^{3}(Y) ≃ O_{S}  ⟶ 0 ↓  ↓  ↓ 0 ⟶ Γ(D)_{Ψ} ⟶ Θ_{S/B}(− log D) ⊗ O_{S}(D) ⟶^{d_{Ψ}} Ω_{S}^{3}(Y + D) ≃ O_{S}(D) ⟶ 0 ↓  ↓  ↓ 0 ⟶ Γ(D)_{Ψ}/Γ_{Ψ} ⟶ Θ_{S/B}(− log D) ⊗ N_{D}  ⟶^{d_{Ψ}} Ω_{S}^{3}(Y + D)/Ω_{S}^{3}(D) ≃ N_{D} ⟶ 0 ↓  ↓  ↓ 0  0  0 .

(89)

Again, here Γ_{Ψ} and Γ(D)_{Ψ} denote the kernels of d_{Ψ}’s in the above diagram. Then it is easy to see the following diagrams are commutative:

0  0 ↓  ↓ 0 ⟶  Γ_{Ψ}  ⟶  Γ̃_{Ψ}  ⟶ π^{∗}(Θ_{B}) ⟶ 0 ↓  ↓  || 0 ⟶ Θ_{S/B}(− log D) ⟶ Θ_{S}(− log D) ⟶ π^{∗}(Θ_{B}) ⟶ 0 ↓  ↓ O_{S}  =  O_{S} ↓  ↓ 0  0 .  (90)

0  0  0 ↓  ↓  ↓ 0 ⟶  Γ_{Ψ}  ⟶  Γ̃_{Ψ}  ⟶  π^{∗}(Θ_{B})  ⟶ 0 ↓  ↓  ↓ 0 ⟶ Γ(D)_{Ψ} ⟶ Γ̃(D)_{Ψ} ⟶ π^{∗}(Θ_{B})(D) ⟶ 0 ↓  ↓  ↓ 0 ⟶ Γ(D)_{Ψ}/Γ_{Ψ} ⟶ Γ̃(D)_{Ψ}/Γ̃_{Ψ} ⟶ π^{∗}(Θ_{B}) ⊗ N_{D} ⟶ 0 ↓  ↓  ↓ 0  0  0 .  (91)

By Lemma 6.2, we easily see the following.

π_{∗}(Γ_{Ψ}) ≃ π_{∗}(Γ(D)_{Ψ}) = 0
π_{∗}(Γ̃_{Ψ}) = 0
π_{∗}(Θ_{S/D}(− log D) ⊗ N_{D}) ≃ π_{∗}(Γ(D)_{Ψ}/Γ_{Ψ}) ≃ π_{∗}(Γ̃(D)_{Ψ}/Γ̃_{Ψ}) ≃ O_{B}

(92)

By using these facts, from (89) and (90), we obtain the commutative diagram:

0  0
↓  ↓
π_{∗}(Γ(D)_{Ψ}/Γ_{Ψ}) ≃ π_{∗}(Θ_{S/B}(− log D) ⊗ N_{D}) ≃ O_{B}
↓ μ_{2}  ↓ μ
0 ⟶ O_{B} ⟶ R^{1}π_{∗}Γ ⟶ R^{1}π_{∗}Θ_{S/B}(− log D)
↑ ρ  ↑ ρ
Θ_{B} = Θ_{B}
↑  ↑
0  0

(93)

Since Im μ = Im ρ (cf. (81)), the diagram (93) shows that

μ_{2}(π_{∗}(Γ(D)_{Ψ}/Γ_{Ψ})) = ρ(Θ_{B}).  (94)

On the other hand, from (91) and (92), we obtain

0  ⟶ π_{∗}(Γ̃(D)_{Ψ}) ↓  ↓ ν_{1} 0 ⟶ π_{∗}(Γ(D)_{Ψ}/Γ_{Ψ}) ≃^{ϕ} π_{∗}(Γ̃(D)_{Ψ}/Γ̃_{Ψ}) ↓ μ_{2}  ↓ μ_{1} 0 ⟶ Θ_{B} ⟶^{ρ}  R^{1}π_{∗}(Γ_{Ψ})  ⟶^{ϕ_{1}}  R^{1}π_{∗}Γ̃_{Ψ} .  (95)

Now from (95) and (94), we obtain

μ_{1}(π_{∗}(Γ̃(D)_{Ψ}/Γ̃_{Ψ})) = μ_{1}(ϕ(π_{∗}(Γ(D)_{Ψ}/Γ_{Ψ}))) = ϕ_{1}μ_{2}((π_{∗}(Γ(D)_{Ψ}/Γ_{Ψ}))) = ϕ_{1}(ρ(Θ_{B})) = 0. (96)

Therefore μ_{1} is the zero map, which shows Lemma 6.4.

□.

In what follows, we shall see the consequence of Theorem 6.2. Recall that S − D is covered by {Ũ_{i}}_{i=1}^{l} and

Ũ_{i} = Spec(M_{R} ⊗ B_{R})[x_{i}, y_{i}, \frac{1}{f_{i}(x_{i}, y_{i}, α, t)}] ⊂ M_{R} × B_{R} × C^{2},

and the restriction of the relative two form ω_{S} to Ũ_{i} can be written as

ω_{S|Ũ_{i}} = \frac{dx_{i} ∧ dy_{i}}{f_{i}(x_{i}, y_{i}, α, t)^{m}} .  (97)

Let ṽ be the global rational vector fields obtained in Theorem (6.1) and let Ψ = ω_{S} ∧ dt be the relative 3-form in Ω_{S/M}^{3}(Y) defined in (77). Then the contraction gives a global rational 2-form

Ω := ṽ · Ψ ∈ Γ(S, Ω_{S/M}^{2}(log D)(Y)).  (98)

On Ũ_{i}, the restriction of 2-form Ω to Ũ_{i} is given by

Ω_{i} := Ω_{|Ũ_{i}} = (\frac{∂}{∂t} − θ_{i}) · (ω_{S|Ũ_{i}} ∧ dt) = ω_{S|Ũ_{i}} − θ_{i} · ω_{S|Ũ_{i}} ∧ dt.  (99)

Here, for 1 ≤ i ≤ l, the contraction of θ_{i} and ω_{S|Ũ_{i}} is given by

θ_{i} · ω_{S|Ũ_{i}} = \frac{1}{f_{i}^{m_{i}}} (η_{i}dy_{i} − ζ_{i}dx_{i}).

Theorem 6.2 can be restated as follows. Proposition 6.1.

d_{S/M}(Ω) = d_{S/M}(ṽ · Ψ) = 0

For Ũ_{i} ⊂ S − D (1 ≤ i ≤ l), we have

d_{S/M}(Ω_{|Ũ_{i}}) = d_{S/M}(\frac{dx_{i} ∧ dy_{i}}{f_{i}(x_{i}, y_{i}, α, t)^{m_{i}}} − (θ_{i} · ω_{S}) ∧ dt)
= (\frac{∂}{∂t} (\frac{1}{f_{i}(x_{i}, y_{i}, α, t)^{m_{i}}} ) dx_{i} ∧ dy_{i} − d_{π}(θ_{i} · ω_{S})) ∧ dt = 0

where we set d_{π} = d_{S/M×B}.

Therefore, Proposition 6.1 implies the following Corollary 6.1. For each i, 1 ≤ i ≤ l, we have the fundamental equation

\frac{∂}{∂t} (\frac{1}{f_{i}(x_{i}, y_{i}, α, t)^{m_{i}}} ) dx_{i} ∧ dy_{i} − d_{π}(θ_{i} · ω_{S}) = 0  (100)

Finally, we obtain the following result which explains the reason why ODE (72) becomes a Hamiltonian system in the classical cases. Proposition 6.2. For i, 1 ≤ i ≤ l such that

Ũ_{i} = M_{R} × B_{R} × Spec C[x_{i}, y_{i}] ≃ M_{R} × B_{R} × C^{2}, ω_{S|Ũ_{i}} = dx_{i} ∧ dy_{i},  (101)

we have

d_{π}(θ_{i} · dx_{i} ∧ dy_{i}) = d_{π}(η_{i}dy_{i} − ζ_{i}dx_{i}) = 0.

Since H_{DR}^{1}(C^{2}) = 0, we have a regular function H_{i}(x_{i}, y_{i}, α, t) ∈ (M_{R} ⊗ B_{R})[x_{i}, y_{i}] such that

d_{π}H_{i} = \frac{∂H_{i}}{∂x_{i}} dx_{i} + \frac{∂H_{i}}{∂y_{i}} dy_{i} = −(θ_{i} · dx_{i} ∧ dy_{i}) = −η_{i}dy_{i} + ζ_{i}dx_{i}.

From this, we have

−η_{i} = \frac{∂H_{i}}{∂y_{i}} , −ζ_{i} = −\frac{∂H_{i}}{∂x_{i}} .

Therefore, the differential equation (72) can be written in the Hamiltonian system

\{ \begin{matrix}\frac{dx_{i}}{dt} & = & \frac{∂H_{i}}{∂y_{i}} \\ \frac{dy_{i}}{dt} & = & −\frac{∂H_{i}}{∂x_{i}} . \end{matrix}  (102)

Remark 6.2. (1) If f_{i}(x_{i}, y_{i}, α, t) in (97) is independent of t, from the fundamental equation

(100),we obtain

d_{π}(θ_{i} · \frac{dx_{i} ∧ dy_{i}}{f_{i}(x_{i}, y_{i}, α)^{m_{i}}}) = 0.

Therefore, we may have a chance to obtain a regular function H_{i}(x_{i}, y_{i}, α, t) on Ũ_{i} such that

d_{π}H_{i} = −θ_{i} · \frac{dx_{i} ∧ dy_{i}}{f_{i}(x_{i}, y_{i}, α)^{m_{i}}} .

In this case, the differential equation in (72) can be written in

\{ \begin{matrix}\frac{dx_{i}}{dt} & = & (f_{i})^{m_{i}} · \frac{∂H_{i}}{∂y_{i}} \\ \frac{dy_{i}}{dt} & = & −(f_{i})^{m_{i}} · \frac{∂H_{i}}{∂x_{i}} , \end{matrix}  (103)

or equivalently,

\{ \begin{matrix}\frac{1}{(f_{i})^{m_{i}}} \frac{dx_{i}}{dt} & = & \frac{∂H_{i}}{∂y_{i}} \\ \frac{1}{(f_{i})^{m_{i}}} \frac{dy_{i}}{dt} & = & −\frac{∂H_{i}}{∂x_{i}} . \end{matrix}  (104)

(2) In general, we can not transform the differential equation in (72) into a Hamiltonian system in the global affine coordinates.

(3) Takano, et al. show that for any Okamoto–Painlevé pair (S, Y) of type D̃_{4}(= P_{V I}), D̃_{5}(= P_{V}), D̃_{6}(= P_{III}), Ẽ_{6}(= P_{IV}), Ẽ_{7}(= P_{II}) , the open surface S − Y_{red} is covered by a finite number of affine spaces U_{i} = C^{2} and regular 2-form ω_{S|Ũ_{i}} can be written as in dx_{i} ∧ dy_{i}. Hence from Proposition 6.2 we obtain the Hamiltonian systems for those Okamoto–Painlevé pairs on any affine chart U_{i} of S −D as proved in [O1], [MMT]. Note that for an Okamoto– Painlevé pair (S, Y) of type D̃_{8}, S − Y_{red} does not contain C^{2} (cf. Theorem 1.1 and Proposition 1.2). For explicit descriptions of Ẽ_{7} and D̃_{8}, see §8.

We summarize our results in this section as follows (cf. Figure 4).

Deformation D ↪ S ⟶ M_{R} × B_{R} ∋ (α, t) of Okamoto–Painlevé pairs such that for any (α, t) ∈ B_{R} the Kodaira–Spencer class ρ(\frac{∂}{∂t}) lies in the Kernel of the restriction map res : H^{1}(S_{α,t}, Θ_{S_{α,t}}(− log D_{α,t})) ⟶ H^{1}(S_{α,t} − D_{α,t}, Θ_{S_{α,t}}(− log D_{α,t}))

⇓

There exists a global holomorphic vector field ṽ on S − D which is a lift of \frac{∂}{∂t}

⇓

Differential Equations on π : S − D ⟶ M_{R} × B_{R} defined by ṽ

⇓ Painlevé property

Local trivializations of the deformation S − D ⟶ B_{R} .

7. Painlevé Equations

Let us recall the classical Painlevé differential equations and Hamiltonian systems which are equivalent to the Painlevé equations ([IKSY], [T], [O1]).

Painlevé equations P_{J} (J = I, II, ⋯, V I) are given in Table 7:

\begin{aligned}P_{I} : & \frac{d^{2}x}{dt^{2}} & = & 6x^{2} + t, \\ P_{II} : & \frac{d^{2}x}{dt^{2}} & = & 2x^{3} + tx + α, \\ P_{III} : & \frac{d^{2}x}{dt^{2}} & = & \frac{1}{x} (\frac{dx}{dt})^{2} − \frac{1}{t} \frac{dx}{dt} + \frac{1}{t}(αx^{2} + β) + γx^{3} + \frac{δ}{x}, \\ P_{IV} : & \frac{d^{2}x}{dt^{2}} & = & \frac{1}{2x} (\frac{dx}{dt})^{2} + \frac{3}{2}x^{3} + 4tx^{2} + 2(t^{2} − α)x + \frac{β}{x} , \\ P_{V} : & \frac{d^{2}x}{dt^{2}} & = & (\frac{1}{2x} + \frac{1}{x − 1}) (\frac{dx}{dt})^{2} − \frac{1}{t} \frac{dx}{dt} + \frac{(x − 1)^{2}}{t^{2}} (αx + \frac{β}{x} ) + γ \frac{x}{t} + δ \frac{x(x + 1)}{x − 1}, \\ P_{V I} : & \frac{d^{2}x}{dt^{2}} & = & \frac{1}{2} (\frac{1}{x} + \frac{1}{x − 1} + \frac{1}{x − t}) (\frac{dx}{dt})^{2} − (\frac{1}{t} + \frac{1}{t − 1} + \frac{1}{x − t}) (\frac{dx}{dt}), \\ & & & +\frac{x(x − 1)(x − t)}{t^{2}(t − 1)^{2}} [α − β \frac{t}{x^{2}} + γ \frac{t − 1}{(x − 1)^{2}} + (\frac{1}{2} − δ) \frac{t(t − 1)}{(x − t)^{2}} ]. \end{aligned} Table 7

Here x and t are complex variables, α, β, γ and δ are complex constants. It is known that each P_{J} is equivalent to a Hamiltonian system (cf. [O1], [IKSY], [MMT]):

(H_{J}) : \{ \begin{matrix}\frac{dx}{dt} & = & \frac{∂H_{J}}{∂y} , \\ \frac{dy}{dt} & = & −\frac{∂H_{J}}{∂x} , \end{matrix}  (105)

where the Hamiltonians H_{J} are given in Table 8.

Moreover the relations between the constants in the equations P_{J} and the Hamiltonians H_{J} are given in Table 9.

For the meaning of the constants in Table 9, see [IKSY], [O2]. Note that these constants are not effective parameters. In some cases, we can normalize these constants further by coordinate transformations. Moreover, the equivalence of P_{J} and (H_{J}) means that if we eliminate the variable y in (H_{J}) then we obtain (P_{J}).

\begin{aligned}H_{I}(x, y, t) & = & \frac{1}{2}y^{2} − 2x^{3} − tx, \\ H_{II}(x, y, t) & = & \frac{1}{2}y^{2} − (x^{2} + \frac{t}{2}) y − (α + \frac{1}{2}) x, \\ H_{III}(x, y, t) & = & \frac{1}{t} [2x^{2}y^{2} − \{2η_{∞}tx^{2} + (2κ_{0} + 1)x − 2η_{0}t\} y + η_{∞} (κ_{0} + κ_{∞}) tx], \\ H_{IV} (x, y, t) & = & 2xy^{2} − \{x^{2} + 2tx + 2κ_{0}\} y + κ_{0}x, \\ H_{V} (x, y, t) & = & \frac{1}{t} [x(x − 1)^{2}y^{2} − \{κ_{0}(x − 1)^{2} + κ_{t}x(x − 1) − ηtx\} y + κ(x − 1)], \\ & & (κ := \frac{1}{4} \{(κ_{0} + κ_{t})^{2} − κ_{∞}^{2}\}), \\ H_{V I}(x, y, t) & = & \frac{1}{t(t − 1)} [x(x − 1)(x − t)y^{2} − \{κ_{0}(x − 1)(x − t) \\ & & +κ_{1}x(x − t) + (κ_{t} − 1)x(x − 1)\} y + κ(x − t)] \\ & & (κ := \frac{1}{4} \{(κ_{0} + κ_{1} + κ_{t} − 1)^{2} − κ_{∞}^{2}\}). \end{aligned} Table 8

α  β  γ  δ  number of aux. parameters
P_{I}  none  none  none  none  0
P_{II}  α  none  none  none  1
P_{III}  −4η_{∞}κ_{∞}  4η_{∞}(κ_{0} + 1)  4η_{∞}^{2}  −4η_{0}^{2}  2
P_{IV} −κ_{0} + 2κ_{∞} + 1  −2κ_{0}^{2}  none  none  2
P_{V}  \frac{1}{2}κ_{∞}^{2}  −\frac{1}{2}κ_{0}^{2}  −η(1 + κ_{t}) −\frac{1}{2}η^{2}/2  3
P_{V I}  \frac{1}{2}κ_{∞}^{2}  \frac{1}{2}κ_{0}^{2}  \frac{1}{2}κ_{1}^{2}  \frac{1}{2}κ_{t}^{2}  4

Table 9

Remark 7.1. For the Painlevé equation of type P_{III}, we have the following remark by Sakai in [Sakai]. The Painlevé equation of type P_{III} as in Table 7 can be transformed into

\frac{d^{2}x}{dt^{2}} = \frac{1}{x} (\frac{dx}{dt})^{2} − \frac{1}{t} \frac{dx}{dt} + \frac{x^{2}}{4t^{2}} (γx + α) + \frac{β}{4t} + \frac{δ}{4x}.  (106)

If γδ ≠ 0, then we can normalize γ = −δ = 4 without loss of generality. In this case we obtain the Painlevé equation of type P D̃_{6} ; III:

P D̃_{6} ; III : \frac{d^{2}x}{dt^{2}} = \frac{1}{x} (\frac{dx}{dt})^{2} − \frac{1}{t} \frac{dx}{dt} + \frac{x^{2}}{4t^{2}} (4x + α) + \frac{β}{4t} − \frac{1}{x}.  (107)

If one of γ and δ equals to zero (not both), then we have P D̃_{7} ; III.

P D̃_{7} ; III : \frac{d^{2}x}{dt^{2}} = \frac{1}{x} (\frac{dx}{dt})^{2} − \frac{1}{t} \frac{dx}{dt} − \frac{4x^{2}}{t^{2}} − \frac{1 + 2a}{t}.  (108)

Moreover when γ = δ = 0, we have P D̃_{8} ; III:

P D̃_{8} ; III : \frac{d^{2}x}{dt^{2}} = \frac{1}{x} (\frac{dx}{dt})^{2} − \frac{1}{t} \frac{dx}{dt} − \frac{x^{2}}{4t^{2}} − \frac{4}{t}.  (109)

These differential equations correspond to generalized rational Okamoto–Pailevé pairs of type D̃_{6}, D̃_{7}, D̃_{8} respectively.

8. Examples

In this section, we will apply our methods for deriving the differential equation in (72) from the explicit deformations of Okamoto–Painlevé pairs. We shall give a full detail of the cases of Ẽ_{7}(= P_{II}) and D̃_{8}(= P D̃_{8} ; III). For other cases, see [Sa-Te].

Example 8.1. Ẽ_{7}–type: In this case, we will use the Takano’s description of the family of Okamoto–Painlevé pairs of type Ẽ_{7} (cf. [Theorem 4, [MMT]]). We will not consider all of the family S ⟶ M_{R} × B_{R}, but consider the family S − D ⟶ M_{R} × B_{R} which is constructed as follows (cf. [SU]). Let us set

M_{R} = Spec C[α] ≃ C, B_{R} = Spec C[t] ≃ C

and take three affine schemes i = 1, 2, 3

Ũ_{i} = Spec C[α, t, x_{i}, y_{i}] ≃ C^{4}.  (110)

The family S − D ⟶ M_{R} × B_{R} can be constructed by patching these affine schemes by the coordinate transformations given as follows (cf. [MMT]):

\begin{aligned}x_{0} & = \frac{1}{x_{1}} & = \frac{1}{x_{2}} \\ y_{0} & = x_{1}((−α − \frac{1}{2}) − x_{1}y_{1}) & = 2x_{2}^{−2} + t + (α − \frac{1}{2})x_{2} − y_{2}x_{2}^{2} \\ x_{1} & = x_{2} & = \frac{1}{x_{0}} \\ y_{1} & = −\frac{2}{x_{2}^{4}} − \frac{t}{x_{2}^{2}} − \frac{2α}{x_{2}} + y_{2} & = x_{0}((−α − \frac{1}{2}) − x_{0}y_{0}) \\ x_{2} & = \frac{1}{x_{0}} & = x_{1} \\ y_{2} & = 2x_{0}^{4} + tx_{0}^{2} + (α − \frac{1}{2})x_{0} − x_{0}^{2}y_{0} & = \frac{2}{x_{1}^{4}} + \frac{t}{x_{1}^{2}} + \frac{2α}{x_{1}} + y_{1} \end{aligned} .

The Kodaira–Spencer class ρ(\frac{∂}{∂t})_{S_{t}−D_{t}} is given by the Čech 1-cocycle

θ_{01} = 0, θ_{02} = \frac{∂}{∂y_{0}} , θ_{12} = −x_{1}^{−2} \frac{∂}{∂y_{1}} .  (111)

DEFORMATION OF OKAMOTO–PAINLEVÉ PAIRS AND PAINLEVÉ EQUATIONS 39

Setting

θ_{0} := [−y_{0} + x_{0}^{2} + \frac{t}{2}] \frac{∂}{∂x_{0}} − [2x_{0}y_{0} + α + \frac{1}{2}] \frac{∂}{∂y_{0}}  (112)
θ_{1} := \frac{1}{2} [−2 − tx_{1}^{2} − x_{1}^{3} − 2αx_{1}^{3} − 2x_{1}^{4}y_{1}] \frac{∂}{∂x_{1}}  (113)
+\frac{1}{4} [(1 + 2α + 4x_{1}y_{1})(t + x_{1}(1 + 2α + 2x_{1}y_{1}))] \frac{∂}{∂y_{1}}  (114)
θ_{2} := \frac{1}{2}(2 + tx_{2}^{2} + (2α − 1)x_{2}^{3} − 2x_{2}^{4}y_{2})\frac{∂}{∂x_{2}}
+\frac{1}{4}(−1 + 2α − 4x_{2}y_{2})(t + x_{2}(−1 + 2α − 2x_{2}y_{2}))\frac{∂}{∂y_{2}} ,  (115)

we have the relations

θ_{01} = θ_{1} − θ_{0}, θ_{02} = θ_{2} − θ_{0}, θ_{12} = θ_{2} − θ_{1},

that is,

θ_{0} = θ_{1}, θ_{2} = θ_{02} + θ_{0}.

Since on each Ũ_{i}, the relative 2-forms ω_{S−D} is given by

ω_{S−D|Ũ_{i}} = dx_{i} ∧ dy_{i},

by Proposition 6.2 the 1-forms θ_{i}dx_{i}∧dy_{i} is exact form, hence there exists a polynomial H_{i}(x_{i}, y_{i}, α, t) satisfying

−θ_{i}dx_{i} ∧ dy_{i} = d_{π}H_{i}.

The polynomials H_{i} are called the Hamiltonians and given by

H_{0}(x_{0}, y_{0}, α, t) = \frac{1}{2}y_{0}^{2} − (x_{0}^{2} + \frac{t}{2}) y_{0} − (α + \frac{1}{2}) x_{0},  (116)
H_{1}(x_{1}, y_{1}, α, t) = \frac{tx_{1}}{4} + \frac{αtx_{1}}{2} + \frac{x_{1}^{2}}{8} + \frac{αx_{1}^{2}}{2} + \frac{α^{2}x_{1}^{2}}{2} + y_{1} + \frac{1}{2}tx_{1}^{2}y_{1}  (117)

+\frac{x_{1}^{3}y_{1}}{2} + αx_{1}^{3}y_{1} + \frac{x_{1}^{4}y_{1}^{2}}{2} ,

H_{2}(x_{2}, y_{2}, α, t) = \frac{1}{8} [(1 − 2α)^{2}x_{2}^{2} − 8y_{2} − 4(−1 + 2α)x_{2}^{3}y_{2} + 4x_{2}^{4}y_{2}^{2}  (118)
−2tx_{2}(1 − 2α + 2x_{2}y_{2}))].  (119)

Hence the Hamiltonian system defined on Ũ_{0} is given by

\{ \begin{matrix}\frac{dx_{0}}{dt} & = & \frac{∂H_{0}}{∂y_{0}} = y_{0} − x_{0}^{2} − \frac{t}{2}, \\ \frac{dy_{0}}{dt} & = & −\frac{∂H_{0}}{∂x_{0}} = 2x_{0}y_{0} + α + \frac{1}{2}. \end{matrix}  (120)

Eliminating y_{0} in (120), we obtain

\frac{d^{2}x_{0}}{dt^{2}} = 2x_{0}^{3} + x_{0}t + α,  (121)

which is the Painlevé equation P_{II} in Table 7.

Example 8.2. D̃_{8}–type: The Okamoto–Painlevé pair (S, Y) of type D̃_{8} did not appear in the former literatures [IKSY], [O1] explicitly. Since S − Y_{red} does not contain C^{2} as a Zariski open set ( cf. Theorem 1.1), the situation is a little bit different from the classical cases.

We can construct a family of generalized rational Okamoto–Painlevé pair of type D̃_{8} π : S − D ⟶ B_{R} by blowings-ups as in Sakai [Sakai]. For detail, see [Sa-Te]. Here note that dim H^{1}(S_{t}, Θ_{S_{t}}(− log D_{t})) = 1 and dim M_{R} = 0 and

B_{R} = Spec C[t, t^{−1}] ≃ C^{×}.

The total space S − D is covered by the three affine open sets:

S − D = Ũ_{0} ∪ Ũ_{1} ∪ Ũ_{2}.

These affine open sets are given by:

Ũ_{0} = Spec C[x_{0}, y_{0}, \frac{1}{y_{0}} , t, t^{−1}] ≅ (C^{2} − \{y_{0} = 0\}) × C^{×},
Ũ_{1} = Spec C[x_{1}, y_{1}, \frac{1}{F_{1}(x_{1}, y_{1})}, t, t^{−1}] ≅ (C^{2} − \{F_{1}(x_{1}, y_{1}) = 0\}) × C^{×},
Ũ_{2} = Spec C[x_{2}, y_{2}, \frac{1}{F_{2}(x_{2}, y_{2}, t)}, t, t^{−1}] ≅ C^{3} − \{F_{2}(x_{2}, y_{2}, t) = 0, t = 0\},

where

F_{1}(x_{1}, y_{1}) = 1 + x_{1}y_{1}^{2},  (122)
F_{2}(x_{2}, y_{2}, t) = t − ty_{2} + x_{2}y_{2}^{2}.  (123)

The coordinate transformations are given as follows

\begin{aligned}x_{0} & = \frac{1}{y_{1}F_{1}(x_{1}, y_{1})} & = \frac{1}{y_{2}} , \\ y_{0} & = \frac{1}{y_{1}^{2}F_{1}(x_{1}, y_{1})} & = y_{2}^{2} (t + x_{2}y_{2}^{2} − ty_{2}), \\ x_{1} & = \frac{y_{0}^{2}(−x_{0}^{2} + y_{0})}{x_{0}^{4}} & = y_{2}^{6}(t + x_{2}y_{2}^{2} − ty_{2}), \\ y_{1} & = \frac{x_{0}}{y_{0}} & = \frac{1}{y_{2}^{3}(t(1 − y_{2}) + x_{2}y_{2}^{2})}, \\ x_{2} & = x_{0}(−tx_{0} + x_{0}^{3}y_{0} + t) & = \frac{1 + ty_{1}^{4}F_{1}^{3}(y_{1}^{2}F_{1} − 1)}{y_{1}^{6}F_{1}^{5}}, \\ y_{2} & = \frac{1}{x_{0}} & = y_{1}^{2}(1 + x_{1}y_{1}^{2}). \end{aligned}

The Kodaira–Spencer class ρ(\frac{∂}{∂t})_{S_{t}−D_{t}} ∈ H^{1}(S_{t} − D_{t}, Θ_{S_{t}}(− log D_{t})) is given by Čech cocycles

θ_{01} = 0, θ_{02} = \frac{−1 + x_{0}}{x_{0}^{3}} \frac{∂}{∂y_{0}} , θ_{21} = \frac{−1 + y_{2}}{y_{2}^{2}} \frac{∂}{∂x_{2}} .  (124)

Since ρ(\frac{∂}{∂t})_{S_{t}−D_{t}} = 0 ∈ H^{1}(S_{t} − D_{t}, Θ_{S_{t}}(− log D_{t})), we can obtain Čech coboundary \{θ_{i} ∈ Γ(Ũ_{i}, Θ_{Ũ_{i}/B_{R}}} such that

{θ_{ij}} = {θ_{j} − θ_{i}}.

In fact, we can choose the following holomorphic vector field θ_{i} on each open set Ũ_{i}

\{ \begin{matrix}θ_{0} = \frac{t − y_{0}^{2}}{t y_{0}} \frac{∂}{∂x_{0}} + \frac{−2 x_{0} y_{0}}{t} \frac{∂}{∂y_{0}} , \\ θ_{1} = \frac{f_{1}(x_{1}, y_{1}, t)}{t F_{1}(x_{1}, y_{1})} \frac{∂}{∂x_{1}} + \frac{g_{1}(x_{1}, y_{1}, t)}{t F_{1}(x_{1}, y_{1})} \frac{∂}{∂y_{1}} , \\ θ_{2} = \frac{f_{2}(x_{2}, y_{2}, t)}{t F_{2}(x_{2}, y_{2}, t)} \frac{∂}{∂x_{2}} + \frac{g_{2}(x_{2}, y_{2}, t)}{t F_{2}(x_{2}, y_{2}, t)} \frac{∂}{∂y_{2}} . \end{matrix}  (125)

where

\begin{aligned}f_{1}(x_{1}, y_{1}, t) = & −2y_{1}(t − 2x_{1}^{2} + 5tx_{1}y_{1}^{2} + 9tx_{1}^{2}y_{1}^{4} + 7tx_{1}^{3}y_{1}^{6} + 2tx_{1}^{4}y_{1}^{8}), \\ g_{1}(x_{1}, y_{1}, t) = & 1 − x_{1}y_{1}^{2} + ty_{1}^{4} + 3tx_{1}y_{1}^{6} + 3tx_{1}^{2}y_{1}^{8} + tx_{1}^{3}y_{1}^{10}, \\ f_{2}(x_{2}, y_{2}, t) = & −t^{2} + 3tx_{2} − 2t^{3}y_{2} + tx_{2}y_{2} − 2x_{2}^{2}y_{2} + 7t^{3}y_{2}^{2} − 8t^{3}y_{2}^{3} − 8t^{2}x_{2}y_{2}^{3}, \\ & +3t^{3}y_{2}^{4} + 18t^{2}x_{2}y_{2}^{4} − 10t^{2}x_{2}y_{2}^{5} − 10tx_{2}^{2}y_{2}^{5} + 11tx_{2}^{2}y_{2}^{6} − 4x_{2}^{3}y_{2}^{7}, \\ g_{2}(x_{2}, y_{2}, t) = & −t + t^{2}y_{2}^{4} − 2t^{2}y_{2}^{5} + t^{2}y_{2}^{6} + 2tx_{2}y_{2}^{6} − 2tx_{2}y_{2}^{7} + x_{2}^{2}y_{2}^{8}. \end{aligned} (126)

Then we actually have the following relation as required

θ_{0} = θ_{1}, θ_{2} = θ_{02} + θ_{0}.

Hence, we have the differential equation on S − D as in Theorem 6.1, and on each open set Ũ_{i}, i = 0, 1, 2 the differential equation can be written as follows (cf. (72)).

On Ũ_{0}  \{ \begin{matrix}\frac{dx_{0}}{dt} = −\frac{t − y_{0}^{2}}{ty_{0}} = y_{0} \frac{∂H_{0}}{∂y_{0}} \\ \frac{dy_{0}}{dt} = \frac{2 x_{0} y_{0}}{t} = −y_{0} \frac{∂H_{0}}{∂x_{0}} . \end{matrix}  (127)
On Ũ_{1}  \{ \begin{matrix}\frac{dx_{1}}{dt} = −\frac{f_{1}(x_{1}, y_{1}, t)}{t F_{1}(x_{1}, y_{1})} = F_{1}(x_{1}, y_{1})^{2} \frac{∂H_{1}}{∂y_{1}} \\ \frac{dy_{1}}{dt} = −\frac{g_{1}(x_{1}, y_{1}, t)}{t F_{1}(x_{1}, y_{1})} = −F_{1}(x_{1}, y_{1})^{2} \frac{∂H_{1}}{∂x_{1}} . \end{matrix}  (128)
On Ũ_{2}  \{ \begin{matrix}\frac{dx_{2}}{dt} = −\frac{f_{2}(x_{2}, y_{2}, t)}{t F_{2}(x_{2}, y_{2}, t)} \\ \frac{dy_{2}}{dt} = −\frac{g_{2}(x_{2}, y_{2}, t)}{t F_{2}(x_{2}, y_{2}, t)}. \end{matrix}  (129)

Here H_{0}, H_{1} are given by

H_{0} = (−\frac{x_{0}^{2}}{t} + \frac{y_{0}}{t} + \frac{1}{y_{0}} ) ,  (130)
H_{1} = (y_{1}^{2} + x_{1}y_{1}^{4} + \frac{x_{1}}{t(1 + x_{1}y_{1}^{2})^{2}} ) .  (131)

Moreover on each Ũ_{i}, i = 0, 1, 2, the relative 2-form ω_{S−D} are given by

ω_{S−D|Ũ_{0}} = \frac{1}{y_{0}} dx_{0} ∧ dy_{0}
ω_{S−D|Ũ_{1}} = \frac{1}{F_{1}(x_{1}, y_{1})^{2}} dx_{1} ∧ dy_{1}
ω_{S−D|Ũ_{2}} = \frac{1}{F_{2}(x_{2}, y_{2}, t)} dx_{2} ∧ dy_{2}.

For each i = 0, 1, 2, consider the 1-form on Ũ_{i}

θ_{i}(ω_{S−D|Ũ_{i}}).

Then since θ_{i}(ω_{S−D|Ũ_{i}}) does not depend on t for i = 0, 1, the fundamental equation (100) is reduced to

d_{π}(θ_{i}(ω_{S−D|Ũ_{i}})) = 0, for i = 0, 1 .  (132)

Though Ũ_{i} is not simply connected, we can integrate θ_{i}(ω_{S−D|Ũ_{i}}) and obtain H_{i} for i = 0, 1 defined in (130) and (131), that is,

d_{π}H_{i} = θ_{i}(ω_{S−D|Ũ_{i}}).  (133)

On the other hand, since θ_{2}(ω_{S−D|Ũ_{i}}) is really depend on t, the fundamental equation (100) becomes as follows.

\frac{∂}{∂t} (\frac{1}{F_{2}(x_{2}, y_{2}, t)}) dx_{2} ∧ dy_{2} − d_{π}(θ_{i}(ω_{S−D|Ũ_{i}})) = 0.  (134)

This last equation is equivalent to the following equation, which one can check by hand.

\frac{∂}{∂t} (\frac{1}{F_{2}} ) + \frac{∂}{∂x_{2}} (\frac{f_{2}}{tF_{2}^{2}} ) + \frac{∂}{∂y_{2}} (\frac{g_{2}}{tF_{2}^{2}} ) = 0.  (135)

Eliminating x_{0} from the differential equation (127), we obtain the differential equation

\frac{d^{2}y_{0}}{dt^{2}} = \frac{1}{y_{0}} (\frac{dy_{0}}{dt} )^{2} − \frac{1}{t} \frac{dy_{0}}{dt} + \frac{2y_{0}^{2}}{t^{2}} − \frac{2}{t}.  (136)

It is easy to see that this equation is equivalent to the equation P D̃_{8} ; III in (109).

Acknowledgments

We would like to thank Kazuo Okamoto, Masatoshi Noumi, Hidetaka Sakai, Kyoichi Takano, Akihiro Tuschiya, Hiroshi Umemura, Yasuhiko Yamada for their continuous interest in our work and helpful discussions. In particular, H. Umemura sent us the interesting preprint [AL], H. Sakai sent us his beautiful preprint [Sakai] which have been playing essential roles in our theory. Daily discussions in Kobe with M. Noumi, K. Takano and Y. Yamada have been really stimulating us to make progress in our research. We are really grateful to them. We would like thank Kenji Iohara for helping us to improve our manuscript.

Thanks are also due to Yuji Shimizu and Kenji Ueno. They organized the workshops for Painlevé equations and Mathematical Physics both in March 1999 and March 2000, which were very stimulating and productive.

References

[AL] D. Arinkin and S. Lysenko, Isomorphisms between moduli spaces of SL(2)-bundles with connections on P^{1}\{x_{1}, · · · .x_{4}}, Math. Res. Letters 4, (1997), 181–190.

[B-W] D. M. Burns, Jr. & J. M. Wahl, Local contributions to global deformations of surfaces, Invent. Math. 26

(1974), 67-88.

[D] P. Deligne, Theórie de Hodge, II, Publ. Math. IHES, 40, (1971), 5–57.

[Gr] A. Grothendieck, Local cohomology, (noted by R. Hartshorne), Lecture Notes in Math. 41, Springer-Verlag, Berlin, Heidelberg, New York (1967), 106 pp.

[IKSY] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991.

[Kaw] Y. Kawamata, On deformations of compactifiable manifolds, Math. Ann., 235, (1978), 247–265.

[Kod1] K. Kodaira, On deformations of some complex psuedo-group structures, Ann. of Math., 71, (1960), 224– 302.

[Kod2] K. Kodaira, On compact analytic surfaces, II, Ann. of Math., 77, (1963), pp. 563–626.

[KodT] K. Kodaira, Complex manifolds and deformations of complex structures, Springer–Verlag, 1985.

[KS] K. Kodaira and D.C. Spencer, On deformations of complex analytic structures, I, II, Ann. of Math., 67, (1958), pp. 328–466.

[NY] M. Noumi and Y. Yamada, Affine Weyl Groups, Discrete Dynamical Systems and Painlevè Equations, Comm Math Phys 199, (1998), 2, pp281-295

[MMT] T. Matano, A. Matumiya and K. Takano, On some Hamiltonian structures of Painlevé systems, II, J. Math. Soc. Japan, 51, No.4, 1999, 843–866.

[O1] K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Espaces des conditions initiales, Japan. J. Math., 5, 1979, 1–79.

[O2] K. Okamoto, Polynomial Hamiltonians associated with Painlevé equations, I, II, Proc. Japan Acad., 56, (1980), 264–268; ibid, 367–371.

[O3] K. Okamoto, Studies on the Painlevé equations I. Annali di Mathematica pura ed applicata CXLVI 1987, 337–381; II. Japan. J. Math., 13, (1987), 47–76; III. Math. Ann. 275 (1986), 221–255; IV. Funkcial. Ekvac. Ser. Int. 30 (1987), 305–332.

[SSU] M.-H. Saito, Y. Shimizu and S. Usui, Variation of Hodge Structure and the Torelli Problem, Advanced Studies in Pure Math. 10, 1987, 649–693.

[Sa-Tak] M.-H. Saito and T. Takebe, Classification of Okamoto–Painlevé pairs. preprint, Kobe 2000. math.AG

0006028

[Sa-Te] M.-H. Saito and H. Terajima, Semiuniversal families of generalized Okamoto–Painlevé pairs and explicit

descriptions of Painlevé equations. in preparation.

[SU] M.-H. Saito and H. Umemura, Painlevé equations and deformations of rational surfaces with rational double points. preprint, Nagoya, (2000).

[Sakai] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, preprint, Kyoto-Math 99-10.

[ST] T. Shioda and K. Takano, On some Hamiltonian structures of Painlevé systems I, Funkcial. Ekvac., 40, 1997, 271–291.

[T] Hitomi Terajima, Local cohomology of generalized Okamoto–Painlevé pairs and Painlevé equations. Preprints, Kobe, May, 2000, math.AG 0006027.

Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 657-8501, Japan

   E-mail address: mhsaito@math.kobe-u.ac.jp
   E-mail address: takebe@math.kobe-u.ac.jp
   E-mail address: terajima@math.kobe-u.ac.jp