DEFORMATION OF OKAMOTO–PAINLEVÉ PAIRS AND PAINLEVÉ EQUATIONS

MASA-HIKO SAITO

0. abstract

In this note, we will explain how Painlevé equations can be understood from algebro-geometric viewpoint of deformation of Okamoto– Painlevé pairs.

1. Painlevé equations

In this section, we review briefly how the Painlevé equations are discovered and why it is important. Main reference will be [3.1, Ch. 3, [IKSY]].

Let us consider the following algebraic ordinary equation

F (t, x, \frac{dx}{dt}, \frac{d^{2}x}{dt^{2}}, ⋯, \frac{d^{n}x}{dt}) = 0  (1)

where

F (t, x_{0}, x_{1}, x_{2}, ⋯, x_{n}) ∈ C(t)[x_{0}, x_{1}, ⋯, x_{n}]

is a polynomial in x = (x_{0}, x_{1}, x_{2}, ⋯, x_{n}) with coefficients rational in t.

Take (t_{0}, c_{0}) = (t_{0}, c_{0}, c_{1}, ⋯, c_{n}) ∈ {(t_{0}, c_{0}) ∈ C^{n+2}|F (t_{0}, c_{0}, c_{1}, c_{2}, ⋯, c_{n}) = 0} and consider the Cauchy problem for (1) to find a solution x(t) = φ(t) such that

\frac{d^{i}φ}{dt^{i}}(t_{0}) = c_{i}, (i = 0, …, n).  (2)

The function obtained by an analytic continuation of the local solution x = φ(t) is also denoted by φ(t).

If an ODE (1) is a linear ordinary differential equation, that is, if F (t, x_{0}, x_{1}, ⋯, x_{n}) is linear with respect to the variables x_{i} ( i = 0, ⋯, n), then the sigularities of the solution φ(t) can be detected from the differential equation and do not depend on the initial values (t_{0}, c_{0}). These kinds of singularities of the solutions are called \underline{non-movable singularities} of the solutions of (1). On the other hand, the solutions of non-linear algebraic ODE may have \underline{movable singulairities}. For example, let us consider the following famous ODE:

(x′)^{2} = 4x^{3} − g_{2}x − g_{3}, g_{2}, g_{3} ∈ C, g_{2}^{3} − 27g_{3}^{2} ≠ 0.  (3)

(Here we denote by ′ the derivation with respect to t.)

Let τ ∈ H = {z ∈ C| Im z > 0} be the normalized period of the elliptic curve y^{2} = 4x^{3} − g_{2}x − g_{3} so that 1, τ is the fundamental periods of the elliptic curve and let

℘(z) = \frac{1}{z^{2}} + ∑_{(m,n)∈Z^{2}−(0,0)} (\frac{1}{(z − m − nτ)^{2}} − \frac{1}{(m + nτ)^{2}} )

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

This note is based on a joint work [STT] with Taro Takebe and Hitomi Terajima.

be the Weierstrass ℘-function with the periods Z + Zτ.

If we consider the solution of (3) with the initial values (t, x, x′) = (0, a, b) ∈ C^{3} such b^{2} = 4a^{3} − g_{2}a − g_{3}, the solution can be written as

x = ℘(t + c)

with a = ℘(c) and b = ℘′(c). It is well-known that ℘(t) has poles of order 2 at t ≡ 0 mod Z + Zτ. Therefore the solution x(t) = ℘(t + c) has poles of order 2 at t ≡ −c mod Z + Zτ. The singularities of the solution x(t) do depend on the initial values and this is an example of movable singularities of a non-linear ODE.

There are many examples of non-linear algebraic ODE whose solutions have movable essential singularities or movable branch points and one can easily realized that it is rather rare that the movable singularities of ODE are only poles. Definition 1.1. An algebraic ODE (1) has \underline{Painlevé property} if the generic solution of (1) has only poles as its movable singularities. Theorem 1.1. (L. Fuchs, H. Poincaré). For n = 1, an algebraic ODE (1) has Painlevé property if and only if (1) can be transformed, by a holomorphic change of the variable t and by a linear fractinal change of the unknown x with coefficients in holomorphic functions of t, into one of the following equations:

(1) The equation of the Weierstrass ℘ function .

(x′)^{2} = 4x^{3} − g_{2}x − g_{3}  (4)

(2) Riccati equation

x′ = a(t)x^{2} + b(t)x + c(t).  (5)

Remark 1.1. By the change of unknown

x = −\frac{1}{a(t)} \frac{d}{dt} log(u) = −\frac{1}{a(t)} \frac{u′}{u} ,  (6)

the Riccati equation (5) is transformed into the linear equation

u′′ − [\frac{a′(t)}{a(t)} + b(t)]u′ + a(t)c(t)u = 0.  (7)

Hence the solutions u(t) of (7) has only nonmovable singularities and only movable singularities of x(t) is the zero of u(t). Since the zero of u(t) has a finite order, then the movable singularities x(t) are only poles.

E. Picard took a pessimistic view of finding ODEs with with Painlevé property in case of n ≥ 2. However it is Paul Painlevé (1863–1933) who attacked te classification problem for n = 2. Definition 1.2. Painlevé equation is a second order algebraic ODE of rational type, that is,

x′′ = R(x, x′, t), R(x, y, t) ∈ C(x, y, t)  (8)

satisfying Painlevé property.

Painlevé and his student B.O. Gambier showed that Painlevé equation reduces, by an approptiate transformation of the variables, to an equation which can be integrated by quadrature, or to a linear equation, or to P_{J}, J = I, II, III, IV, V, V I. (See Table 1). Here α, β, γ and δ are complex constants.

\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 1

It is known that Painlevé equations appear in many fields of mathematics and physics. For example, Painlevé equations can be derived from isomonodromic deformations of linear ordinary equations over P^{1}. ([IKSY]). Besides the important relation to correlation functions of Ising models, the later is deeply related to the notion of Frobenius structure or Frobenius manifolds due to Dubrovin [Du]. Frobenius structure is a kind of generalization of WDVV equations, which is equivalent to the fact that related quantum cohomology rings are associative. (Note that the Frobenius structure is essentially same as the flat structure, originally introduced by Kyoji Saito in case of versal deformation of isolated singularities. )

2. Okamoto’s spaces of initial conditions for Painlevé equations

In this section, we recall a series of works of Okamoto which shows the importance of study of the space of initial conditions of Painlevé equation. Later we will introduce the notion of Okamoto–Painlevé pairs, which is a generalization of Okamoto’s space of intial conditions.

First, let us recall that each P_{J} is equivalent to a Hamiltonian system H_{J} (cf. [Mal], [O1], [IKSY], [MMT]):

(H_{J}) : \{ \begin{matrix}\frac{dx}{dt} & = & \frac{∂H_{J}}{∂y} , \\ \frac{dy}{dt} & = & −\frac{∂H_{J}}{∂x} , \end{matrix}  (9)

where the Hamiltonians H_{J} = H_{J}(x, y, t) are given in Table 2.

\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_{∞}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 2

Let us consider the Hamiltonian system H_{J} from now on. Let Σ_{J} = {a_{1}, …, a_{l}} ⊂ C be the set of non-movable singularities for (H_{J}) and set

B_{J} = C − Σ_{J} = Spec C[t, \frac{1}{t − a_{1}} , …, \frac{1}{t − a_{l}}].

Note that the non-movable singularities can be easily detected from the differential equation itself.

Consider the product space C^{2} × B_{J} ∋ (x, y, t) and the projection map π : C^{2} × B_{J} ⟶ B_{J}. On the product space C^{2} × B_{J} ∋ (x, y, t), we can consider the Hamiltonaian system (H_{J}). Let us take a relative compactification of π:

\begin{matrix}C^{2} × B_{J} & ↪ & P^{2} × B_{J} \\ ↓ & & ↓ \\ B_{J} & = & B_{J} \end{matrix}  (10)

If (H_{J}) satisfies Painlevé property, all movable singularities of the solutions of (H_{J}) are just poles. The solution curve (x(t), y(t), t) of (H_{J}) starting from an initial vaule (x_{0}, y_{0}, t_{0}) ∈ C^{2} × B_{J} may have singularities at some point at t_{1} ∈ B_{J} depending on the initial vaule (x_{0}, y_{0}, t_{0}). However Painlevé property ensures that the solutions x(t), y(t) have only poles as it singularities, hence they have limits in P^{2} × B_{J} thanks to the properness of P^{2}. Setting L = P^{2}−C^{2}, the solution curve (x(t), y(t), t) starting from (x_{0}, y_{0}, t_{0}) ∈ C^{2} ×B_{J} passes through a point at the boundary L×{t_{1}}. Such a singulairity of solution curve is called an accesible singularity. For a classical Painlevé equation P_{J}, or equivalently Hamiltonian system (H_{J}), for each fixed t = t_{1},

t t_{0}  t_{1}  t_{2} (x_{0}, y_{0}, t_{0})  P^{2} × B_{IV} L  P^{2} − L = C^{2}

t t_{0}  t_{1}  t_{2} (x_{0}, y_{0}, t_{0})  S B_{IV} π

Figure 1

there are only finitely many accesible singular points at the boundary L × t_{1}. At those points, infinitely many solution curves come together.

Okamoto analyses these kinds of singularities in details [O1]. He blowed up at the accesible singular points including infinitely near points of boundaries, so that all of solution curves of (H_{J}) can be separated (see Figure 1). More precisely, he constructed a smooth morphism π : S ⟶ B_{J} by blowing up P^{2} × B_{J}.

\begin{matrix}P^{2} × B_{J} & ←^{τ} & S \\ ↓ & ↙ π \\ B_{J} \end{matrix}  (11)

From the construction of Okamoto, one can obtain the following theorem (cf. [O1], see also [Sakai], [STa], [STT]).

Proposition 2.1. For a general t ∈ B_{J}, a fiber S = S_{t} of π : S ⟶ B_{J} over t is obtained by blowing ups of P^{2} at 9 points of boundary L (including infinitely near points). The anti-canonical divisor −K_{S} of S = S_{t} is an effective divisor Y = ∑_{i=1}^{r} m_{i}Y_{i}. Moreover the configuration of an effective divisior Y is same as in the list of Kodaira–Néron’s minimal model of singular elliptic curves of additive types (Figure 2).

We remark that the cases D̃_{7}, D̃_{8} did not appear in Okamoto’s paper [O1]. These cases really appear when parameters in Painlevé III equations take special values. Moreover we can show that for the case D̃_{8} the S − Y_{red} does not contains C^{2} but S can be obtained by blowing up of P^{2}.

Note that in Figure 2, the real line shows that a smooth rational curve C ≃ P^{1} with C^{2} = −2 and the number near the each rational curve denotes the multiplicity in Y = −K_{S}.

Let us set

−K_{S} = Y = ∑_{i=1}^{r} m_{i}Y_{i}.

Under the assumption that S is a rational surface, one can show that Y has the same configuration as Kodaira–Néron’s minimal model of singular elliptic curves if and only if the following conditions are satisfied (cf. [Kod], [STa]).

deg −(K_{S})_{|Y_{i}} = −K_{S} · Y_{i} = Y · Y_{i} = 0 for all i, 1 ≤ i ≤ m .  (12)

Here we should remark that rational surfaces S appeared above are not elliptic surface. Actually, one can show that the dimension of the linear system |−K_{S}| = |Y | is zero, while for an rational elliptic surface S′ one should have | − K_{S′}| = |Y | ≃ P^{1}.

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; 2; 1; 1; D̃_{8}]

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

[figure: 1; 1; 2; 2; 1; 1; D̃_{5}; 2; 1; 1; 1; 1; D̃_{4}]

Figure 2

3. Okamoto–Painlevé pairs

From the construction of Okamoto, one can understand that the following definition is natural. Definition 3.1. 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.  (13)

An Okamoto–Painlevé pair (S, Y) is called rational, if S is a rational surface. We will consider only rational Okamoto–Painlevé pairs from now on.

The first assertion of the following propositon is proved in [Sakai] and the rest is proved by Riemann–Roch theorem and standard arguments. Proposition 3.1. Let (S, Y) be a rational Okamoto–Painlevé pair. Then S can be obtained by 9 points blowings-up of P^{2}. Moreover

(1) dim | − nK_{S}| = dim |nY | ≤ 1 for all n ≥ 1.

(2) If dim | − nK_{S}| = dim |nY | = 1 for some n ≥ 1, there exists an elliptic

fibraiton f : S ⟶ P^{1} with f^{∗}(∞) = nY. Definition 3.2. A rational Okamoto–Painlevé pair (S, Y) is called of fibered–type if there exists an elliptic fibration f : S ⟶ P^{1} such that f^{∗}(∞) = nY for some positive integer n ≥ 1. If (S, Y) is not of fibered type, we call (S, Y) “of non-fibered type ”.

Note that a rational Okamoto–Painlevé pair (S, Y) is of fibered–type if and only if dim |nY | ≥ 1 for some positive integer n ≥ 1. Moreover, if (S, Y) is of fibered type and φ : S ⟶ P^{1} is an elliptic fibration 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.

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. (The root system R(Y) is of affine type.)

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 types of generalized Jacobians Pic^{0}(Y) of Y.

For the classification of ratinal Okaomoto–Painlevé pairs (S, Y) with normal crossing divisor Y_{red}, we can easily show the following theorem. Proposition 3.2. Let (S, Y) be a rational Okamoto–Painlevé pair such that Y_{red} is a divisor with only normal crossings. Then the type of Y is same as one in the list of Table 3.

The following proposition is technically important (cf. [STT]). Proposition 3.3. Let (S, Y) be a 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.

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

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

r  9 9  8  8  7  7 6 5  r  1

Table 3

4. Deformation of 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}.

We will consider the deformation of smooth pairs (S, D) = (S, Y_{red}) [Kaw], [SSU] and [STT]. By a generalization of Kodaira–Spencer theory due to Kawamata [Kaw], the set of infinitesimal deformations of (S, D) is isomorphic to the cohomology group

H^{1}(S, Θ_{S}(− log D))  (14)

and the set of obstructions to lift the infinitesimal deformations to higher order ones are contained in

H^{2}(S, Θ_{S}(− log D)).  (15)

The following results are proved in [STT]. (See also [AL], [SU]). Proposition 4.1. For a 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.

(5) If furthermore S is of non-fibered type, H^{0}(S, Θ_{S}(− log D)(H)) = 0 for any effective divisor H supported on D.

From the above results, one can obtain the following proposition [STT]. Proposition 4.2. 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,  (16)
b_{2}(S) = rank H^{2}(S, Z) = 10,  (17)
dim H^{1}(S, Θ_{S}) = 10,  (18)

and

dim H^{1}(S, Θ_{S}(− log D)) = 10 − r  (19)

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.

5. 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)) → (20)
H^{1}(S, Θ_{S}(− log D)) →^{res} H^{1}(S − D, Θ_{S}(− log D)) .  (21)

Since (S, Y) is of non-fibered type, from (2), Proposition 4.1, we see that

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

Hence, we have the following Proposition 5.1. 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))

(22)

The following theorem is proved in [T]. Theorem 5.1. 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.  (23)

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.  (24)

This theorem plays an important role to understand the Painlevé equation related to (S, Y). Note that for a generic rational Okamoto–Painlevé pair (S, Y) in Theorem 5.1, one can show that

H_{D}^{1}(Θ_{S}(− log D)) ≃ H^{0}(D, Θ_{S}(− log D) ⊗ N_{D}) ≃ C.  (25)

From the exact sequence (22), 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} \} .

We can construct semi-universal families of Okamoto–Painlevé pairs with nice coordinate systems ([Sakai], [STT], [ShT]).

Proposition 5.2. Let R = R(Y) be one of types of the root systems appeared in Proposition 3.2 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 5.1). 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}  (26)

(1) The above diagram is a deformation of non-singular pair of projective surfaces and normal crossing divisors.

(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}))  (27)

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 :

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})). (28)

(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}

(29)

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}}}  (30)

(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)  (31)

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

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 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)).

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, which is equivalent to Painlevé equation.

Let R = R(Y) be one of types of additive affine root systems appeared in Proposition 3.2 and let

\begin{matrix}S & ↩ & D \\ π ↓ & ↙ & φ \\ M_{R} × B_{R} \end{matrix}  (32)

be a global deformation of generalized Okamoto–Painlevé pairs of type R as in Proposition 5.2. 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}]  (33)

as in (29). 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 (31) 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)  (34)

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)) \} (35)

Recall that (cf. Proposition 5.2, 3) on M_{R} × B_{R} the corresponding Okamoto– Painlevé pairs (S_{α,t}, Y_{α,t}) are of non-fibered type.

From (28) of Proposition 5.2, 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})) (36)

Since the local cohomology group is the kernel of the natural restriction map (cf. Proposition 5.1)

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})), (37)

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, 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})).  (38)

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)) (39)

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).  (40)

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}.

For any pair i, j such that Ũ_{i} ∩ Ũ_{j} ≠ ∅, we can obtain the identity on Ũ_{i} ∩ Ũ_{j}

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

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)),  (42)

or

(\frac{∂}{∂t})_{j} − θ_{j}(x_{j}, y_{j}, α, t) = (\frac{∂}{∂t})_{i} − θ_{i}(x_{i}, y_{i}, α, t).  (43)

This means that the vector fields

\{(\frac{∂}{∂t})_{i} − θ_{i}(x_{i}, y_{i}, α, t)\}_{1≤i≤l+k}  (44)

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}  (45)

is a regular algebraic vector field. 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}}  (46)

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} .  (47)

Here the functions η_{i}, ζ_{i} are regular algebraic functions on Ũ_{i}.

Moreover we will be able to explain the reason why the systems obtained as above are Hamiltonian systems as in (9). Our proof is geometrically clear. In fact we will be able to show that global vector field ṽ preserves the relative 2 forms ω_{S} in (30) . This fact can be expressed as (cf. [STT])

d_{S/M}(ṽ · (ω_{S} ∧ dt)) = 0,  (48)

where d_{S/M} is the relative differential with respect to the morphism S ⟶ M and ṽ · (ω_{S} ∧ dt) denotes the contraction of vector fields and 3-forms.

On an affine open subset U ×M_{R} ×B_{R} with U ≃ C^{2} and coordinate systems (x, y) such that ω_{S|C^{2}} = dx ∧ dy, the equation (48) implies that the differential equation obtained as above becomes a Hamiltonian system with a polynomial Hamiltonian.

We note that in the case of R = D̃_{7}, D̃_{8}, Ẽ_{7} S−D can not be covered by affine open sets all of which are isomorphic to C^{2}. On such affine open subsets, the differential equation (47) can not be written in Hamiltonian systems globaly.

Furthermore, one can obtain explicit description of differential equations 47 by using the explicit affine coordinates (cf. [STT], [STe]).

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

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