A non-Kähler structure on an S^{2}-bundle over a ruled surface ^{∗}

Masahide Kato

In this note, we shall give an explicit construction of a non-Kähler structure Z on a manifold diffeomorphic to an S^{2}-bundle over S^{2} × C_{g}, where S^{2} is the 2-sphere and C_{g} is a compact Riemann surface with genus g ≥ 2. The complex manifold Z is the flat twistor space over a conformally flat real 4-manifold defined by Kuiper [Ku, page 489(d)]. We shall show that the 2nd Chern class of the SU(2)-bundle associated with the twistor space is equal to 2g − 2, where g is the genus of C_{g}, (see Theorem 1) and that Z admits no non-constant meromorphic functions (Theorem 2)^{1} . The manifold Z gives a negative answer to a question of F.Catanese (see Remark 3). In section 1, we shall construct the manifold and give the precise statements of our main result (Theorems 1 and 2). In sections 2 and 3, we shall give proofs of Theorem 1 and Theorem 2, respectively.

1 Construction of the manifold

Let P^{3} be a complex projective space of dimension 3. Let z = [z_{0} : z_{1} : z_{2} : z_{3}] be a standard system coordinates. Let L be the line in P^{3} defined by z_{2} = z_{3} = 0. We identify the set of quaternions with the set of 2 × 2 matrices

H = \{X = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) : x_{1}, x_{2} ∈ C\}.

Define a diffeomorphism

\begin{aligned}ϕ : & P^{3} \ L & → & S^{2} × H \\ & [z_{0} : z_{1} : z_{2} : z_{3}] & ↦ & (z′′, X) \end{aligned}

∗corrected version of ”A non-Kähler structure on the product of a compact Riemann surface and S^{2} × S^{2}”, preprint in 1990

1 This theorem follows also from a more general result of M. Pontecorvo [Po]

by

X = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) ,

x_{1} = (z_{0}z̄_{2} + z̄_{1}z_{3})(|z_{2}|^{2} + |z_{3}|^{2})^{−1},

x_{2} = (z_{1}z̄_{2} − z̄_{0}z_{3})(|z_{2}|^{2} + |z_{3}|^{2})^{−1},

z′′ = ( \begin{matrix}z_{2} \\ z_{3} \end{matrix} ) .

Note that

X = ( \begin{matrix}z_{0} & −z̄_{1} \\ z_{1} & z̄_{0} \end{matrix} ) ( \begin{matrix}z_{2} & −z̄_{3} \\ z_{3} & z̄_{2} \end{matrix} )^{−1}.

We introduce the orientation on H from the natural orientation on C^{2} by identifying

X = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) with ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) ∈ C^{2}. On the 2-sphere S^{2} = \{(t_{1}, t_{2}, t_{3}) ∈ R^{3} :

t_{1}^{2} + t_{2}^{2} + t_{3}^{2} = 1}, we consider the complex structure of a complex projective line by introducing a system of homogeneous coordinates [ζ_{0} : ζ_{1}], where

ζ_{0} = \{ \begin{matrix}t_{1} + \sqrt{−1}t_{2} & if & t_{3} ≠ −1 \\ 1 − t_{3} & if & t_{3} ≠ 1 \end{matrix}

and

ζ_{1} = \{ \begin{matrix}1 + t_{3} & if & t_{3} ≠ −1 \\ t_{1} − \sqrt{−1}t_{2} & if & t_{3} ≠ 1. \end{matrix}

The 2-vector z′′ is regarded as a point on the projective line with the homogeneous coordinates [z_{2} : z_{3}]. On the product space S^{2} × H, we introduce the orientation defined by the natural orientation of the projective line and the orientation of H defined above. It is easy to see that the map ϕ is orientation preserving. Let I denote the identity matrix of size 2 × 2. It is easy to check the following

Lemma 1 . For σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, R), the following diagram of diffeomor-

phisms is commutative:

\begin{aligned}& ϕ \\ P^{3} \ L & → & S^{2} × H \\ σ ↓ & & ↓ σ′ \\ P^{3} \ L & → & S^{2} × H \\ & ϕ \end{aligned} ,

where σ′(z′′, X) = ((cX + dI)z′′, (aX + bI)(cX + dI)^{−1}) for (z′′, X) ∈ S^{2} × H.

Put

K = {z ∈ P^{3} : z_{1}z̄_{2} − z̄_{0}z_{3} = Im (z_{0}z̄_{2} + z̄_{1}z_{3}) = 0}.

For any element

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, R),

we associate the element

( \begin{matrix}aI & bI \\ cI & dI \end{matrix} ) ∈ P GL(4).

Naturally SL(2, R) acts on P^{3}. Lemma 2 . The set K is SL(2, R)-invariant.

Proof. Let τ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) be any element of SL(2, R). Then a point z = [z_{0} : z_{1} : z_{2} : z_{3}] ∈ P^{3} is mapped by τ to the point z′ = [z′_{0} : z′_{1} : z′_{2} : z′_{3}], where

\{ \begin{matrix}z′_{0} = az_{0} + bz_{2}, \\ z′_{1} = az_{1} + bz_{3}, \\ z′_{2} = cz_{0} + dz_{2}, \\ z′_{3} = cz_{1} + dz_{3}. \end{matrix}

Hence we have the equations

z′_{0}z̄′_{3} − z̄′_{1}z′_{2} = (az_{0} + bz_{2})(cz̄_{1} + dz̄_{3}) − (az̄_{1} + bz̄_{3})(cz_{0} + dz_{2})
= (ad − bc)(z_{0}z̄_{3} − z̄_{1}z_{2})
= z_{0}z̄_{3} − z̄_{1}z_{2},

and

Im (z′_{0}z̄′_{2} + z̄′_{1}z′_{3}) = Im ((az_{0} + bz_{2})(cz̄_{0} + dz̄_{2}) + (az̄_{1} + bz̄_{3})(cz_{1} + dz_{3}))
= (ad − bc)Im (z_{0}z̄_{2} + z̄_{1}z_{3})
= Im (z_{0}z̄_{2} + z̄_{1}z_{3}).

Thus we conclude that K is SL(2, R)-invariant. Put

l = \{X = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) ∈ H : Im x_{1} = x_{2} = 0 \}

and

H^{∗} = H \ l.

Then we have easily Lemma 3 . K = ϕ^{−1}(S^{2} × l) and L ⊂ K hold.

Put

Z̃ = P^{3} \ K

and

ψ = ϕ|Z̃.

Then ψ is a diffeomorphism onto S^{2} × H^{∗}. Let D denote the upper half-plane of a complex plane. We define a projection

π : H → D

by π(X) = y = y_{1} + \sqrt{−1}y_{2} (1)

where

y_{1} = Re x_{1},
y_{2} = |X′|,
|X′| = ((Im x_{1})^{2} + |x_{2}|^{2})^{\frac{1}{2}}

for X = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) ∈ H. Then the map

Ψ :  H^{∗}  →  S^{2} × D
X = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) ↦ ((Im \frac{x_{1}}{|X′|}, Re \frac{x_{2}}{|X′|}, Im \frac{x_{2}}{|X′|}), π(X))

is an orientation preserving diffeomorphism. Recall that σ ∈ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, R)

acts on D by

σ(y) = (ay + b)(cy + d)^{−1}, y ∈ D.

Lemma 4 . For any σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, R), the following diagram of diffeo-

morphisms is commutative:

\begin{aligned}& & Ψ \\ & H^{∗} & → & S^{2} × D \\ σ & ↓ & & ↓ & id_{S^{2}} × σ \\ & H^{∗} & → & S^{2} × D \\ & & Ψ \end{aligned} ,

where σ ∈ SL(2, R) acts on H^{∗} as X ↦ (aX + bI)(cX + dI)^{−1}.

Proof. For X = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) ∈ H^{∗}, (x′_{1}, x′_{2}) = σ(x_{1}, x_{2}) ∈ H^{∗} is given by

Re x′_{1} = (ac(|x_{1}|^{2} + |x_{2}|^{2}) + (ad + bc)Re x_{1} + bd) /D,
(2)  Im x′_{1} = Im x_{1}/D,
(3)  x′_{2} = x_{2}/D,

where D = |cx_{1} + d|^{2} + |cx_{2}|^{2}. Thus the first component of ΨσΨ^{−1} acts on S^{2} identically. For the second component of ΨσΨ^{−1}, we can verify also the commutativity by calculations. Combining Lemmas 1, 3, and 4, we have easily

Lemma 5 . The following diagram of orientation preserving diffeomorphisms is commutative:

ψ  id_{S^{2}} × Ψ
Z̃ → S^{2} × H^{∗}  →  S^{2} × S^{2} × D
σ ↓  ↓ σ′  ↓ σ′′
Z̃ → S^{2} × H^{∗}  →  S^{2} × S^{2} × D,
ψ  id_{S^{2}} × Ψ

where

\begin{aligned}σ & = & ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, R), \\ σ′(z′′, X) & = & ((cX + dI)z′′, (aX + bI)(cX + dI)^{−1}), \\ σ′′(z′′, p, y) & = & ((cΨ^{−1}(p, y) + dI)z′′, p, σ(y)). \end{aligned}

and

(z′′, X) ∈ S^{2} × H^{∗}, (z′′, p, y) ∈ S^{2} × S^{2} × D.

Put

Z = Z̃/Γ

and

M = H^{∗}/Γ.

We denote by K_{X} the canonical line bundle of a complex manifold X.

Theorem 1 . Let Γ ⊂ SL(2, R) be a discrete subgroup such that its action on D is free and the quotient space D/Γ is compact. Then the quotient space Z is a compact complex 3-fold and is diffeomorphic to the projective bundle over P^{1} ×(D/Γ) associated with the rank 2 vector bundle L ⊕ L^{−1}, where L is a holomorphic line bundle on P^{1} × (D/Γ) such that L^{⊗2} = K_{P^{1}×D/Γ}.

We shall give a proof of this theorem in section 2. It is well-known (and in fact, in this case, easily seen from the above construction) that on twistor spaces there are no holomorphic q-forms with q > 0. Therefore there is no non-trivial map of Z to a curve with positive genus. More strongly, we have the following.

Theorem 2 . The manifold Z admits no non-constant meromorphic functions.

We shall give a proof of this theorem in section 3. Remark 1. By our construction, Z̃/Γ is the flat twistor space over the conformally flat manifold M ≅ S^{2} × (D/Γ). The metric on M is given by ((Im x_{1})^{2} + |x_{2}|^{2})^{−1}(dx_{1}dx̄_{1} + dx_{2}dx̄_{2}) on H^{∗}. Remark 2. After the completion of this note, C. LeBrun kindly taught me that Theorem 2 follows from a theorem of M. Pontecorvo [Po, Theorem 4.6].

2 Proof of Theorem 1

We have easily the following

Proposition 1 The vector bundle E is isomorphic to an SU(2)-bundle as a differentiable vector bundle. In particular, the first Chern class c_{1}(E) of E vanishes.

Proof. The transition functions of E have values in H^{∗}. Hence those of the determinant line bundle of E are real and positive valued. Therefore we can choose a 1-cocycle which represents E with values in SU(2). Now we shall calculate the second Chern class c_{2}(E) of E. Let f = f(z) = u(z) + \sqrt{−1}v(z) be a complex-valued C^{∞} function on a domain Ω in D, where u(z) and v(z) are real functions. Using an idea of Fueter [Fu], we associate f with an H-valued C^{∞} function Hf defined on π^{−1}(Ω) ⊂ H^{∗} whose value Hf(X) at X is given by

( \begin{matrix}u(π(X)) + \sqrt{−1}v(π(X))\frac{Im x_{1}}{|X′|} & −v(π(X))\frac{x̄_{2}}{|X′|} \\ v(π(X))\frac{x_{2}}{|X′|} & u(π(X)) − \sqrt{−1}v(π(X))\frac{Im x_{1}}{|X′|} \end{matrix} ) ,

where

X = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) ,

and

|X′| = ((Im x_{1})^{2} + |x_{2}|^{2})^{\frac{1}{2}} .

By the construction of Z (see Lemma 4), we see easily that there is a locally finite open covering U = {U_{α}} of D/Γ with the following properties;

1. On each π^{−1}(U_{α}) there is a system of coordinates

X_{α} : π^{−1}(U_{α}) → H^{∗}

such that

X_{α} = (a_{αβ}X_{β} + b_{αβ}I)(c_{αβ}X_{β} + d_{αβ}I)^{−1} on π^{−1}(U_{α}) ∩ π^{−1}(U_{β}), (4)

where

σ_{αβ} = ( \begin{matrix}a_{αβ} & b_{αβ} \\ c_{αβ} & d_{αβ} \end{matrix} ) ∈ SL(R, 2).

2. On each U_{α}, there is a complex coordinate z_{α} which satisfies

z_{α} = π(X_{α})

and z_{α} = (a_{αβ}z_{β} + b_{αβ})(c_{αβ}z_{β} + d_{αβ})^{−1} on U_{α} ∩ U_{β}. (5)

Let F be the holomorphic line bundle defined by the 1-cocycle {c_{αβ}z_{β} + d_{αβ}} on D/Γ. It is easy to see that

(6)  F^{⊗2} = K_{D/Γ} and
(7)  c_{1}(F) = g − 1,

where g is the genus of D/Γ. There is a non-trivial meromorphic section a = {a_{α}} of F, where a_{α} = a_{α}(z_{α}) is a meromorphic function on U_{α} satisfying

a_{α}(z_{α}) = (c_{αβ}z_{β} + d_{αβ})a_{β}(z_{β}) on U_{α} ∩ U_{β}. (8)

Lemma 6 A_{α}(X_{α}) = (c_{αβ}X_{β} + d_{αβ}I)A_{β}(X_{β}), (9)

where

A_{α}(X_{α}) = Ha_{α}(X_{α}).

Proof. By (2) and (3), we have the relations

(10)  \frac{Im x_{α1}}{|X′_{α}|} = \frac{Im x_{β1}}{|X′_{β}|}
(11)  \frac{x_{α2}}{|X′_{α}|} = \frac{x_{β2}}{|X′_{β}|}.

Put u_{α} = Re a_{α}(z_{α}) and v_{α} = Im a_{α}(z_{α}). By (1),

(12)  z_{α} = Re x_{α1} + \sqrt{−1}|X′_{α}|,
(13)  z_{β} = Re x_{β1} + \sqrt{−1}|X′_{β}|.

Hence by (8), we have

(14)  u_{α}(z_{α}) = (c_{αβ}Re x_{β1} + d_{αβ})u_{β}(z_{β}) − c_{αβ}|X′_{β}|v_{β}(z_{β}),
(15)  v_{α}(z_{α}) = (c_{αβ}Re x_{β1} + d_{αβ})v_{β}(z_{β}) + c_{αβ}|X′_{β}|u_{β}(z_{β}).

Using (10),(11),(14) and (15), we obtain (9) by a simple calculation. Let E denote the rank 2 vector bundle on M defined by the 1-cocycle {ξ_{αβ}}, where

ξ_{αβ} = c_{αβ}X_{β} + d_{αβ}I.

It follows from (9) that

ξ_{αβ}^{−1}dξ_{αβ} = −dA_{β} · A_{β}^{−1} + ξ_{αβ}^{−1}dA_{α} · A_{α}^{−1}ξ_{αβ}

on π^{−1}(U_{α}) ∩ π^{−1}(U_{β}). Thus {−dA_{α} · A_{α}^{−1}} gives a connection of E outside the fibres over the points of Σ, where Σ is the set of zeros and poles of a. Taking a refinement of U if necessary, we can assume that each U_{α} contains at most one point of Σ and that Σ ∩ [U_{α}] ∩ [U_{β}] = ∅ for all distinct α and β, where [U] indicates the closure of a set U. We can choose the section a = {a_{α}} so that the zeros and poles are of order one. Take U_{α} such that the set Σ ∩ U_{α} consists of a point q. Write a_{α} as

a_{α}(z_{α}) = (z_{α} − q)^{m}ã_{α}(z_{α})

where m = ±1 and ã_{α}(z_{α}) is a nowhere vanishing holomorphic function on U_{α}. Put

X_{α} = ( \begin{matrix}x_{1} & −x̄_{2} \\ x_{2} & x̄_{1} \end{matrix} ) ,
|X′_{α}| = ((Im x_{1})^{2} + |x_{2}|^{2})^{\frac{1}{2}} ,
x = Re (x_{1} − q),
y = |X′_{α}| − Im q,
ũ_{α} = Re ã_{α}(z_{α}),
ṽ_{α} = Im ã_{α}(z_{α}),
û = ũ_{α}(π(X_{α})),
v̂ = ṽ_{α}(π(X_{α})).

First we consider the case where the point q is a zero of a_{α}, i.e., m = 1. Choose a small positive number ε such that the neighborhood

V_{α}^{ε} = {z ∈ U_{α} : |a_{α}(z)| < ε}

of q is relatively compact in U_{α} and that [V_{α}^{ε}] ∩ [U_{β}] = ∅ for every β ≠ α. Take an ε′ > ε such that V_{α}^{ε′} has the same properties. On π^{−1}(U_{α}), the matrix A_{α}(X_{α}) is given by

( \begin{matrix}xû − yv̂ + \sqrt{−1}(xv̂ + yû)\frac{Im x_{1}}{|X′_{α}|} & −(xv̂ + yû)\frac{x̄_{2}}{|X′_{α}|} \\ (xv̂ + yû)\frac{x_{2}}{|X′_{α}|} & xû − yv̂ + \sqrt{−1}(xv̂ + yû)\frac{Im x_{1}}{|X′_{α}|} \end{matrix} ) . (16)

For a matrix A, we denote ^{t}Ā by A^{∗}.

Lemma 7 There is an H-valued C^{∞} function B_{α} = B_{α}(X_{α}) and a real-valued C^{∞} function φ_{α} = φ_{α}(X_{α}) on π^{−1}(U_{α}) such that

B_{α}(X_{α}) = φ_{α}(X_{α})A_{α}(X_{α})^{−1}, (17)

where

(18) φ_{α}(X_{α}) = \{ \begin{matrix}1 & for & X_{α} ∈ π^{−1}(U_{α} − V_{α}^{ε′}), \\ ε^{−2} det A_{α}(X_{α}) & for & X_{α} ∈ π^{−1}([V_{α}^{ε}]). \end{matrix}

Proof. Let ρ = ρ(t) be a C^{∞} function on R such that

ρ(t) = \{ \begin{matrix}1 & t ≥ ε′^{2} \\ 0 & t ≤ ε^{2} \end{matrix}

and 0 ≤ ρ(t) ≤ 1 for ε^{2} ≤ t ≤ ε′^{2}. Put

B_{α}(X_{α}) = ρ(det A_{α}(X_{α}))A_{α}(X_{α})^{−1} + (1 − ρ(det A_{α}(X_{α}))) ε^{−2}A_{α}(X_{α})^{∗}

and

φ_{α}(X_{α}) = ρ(det A_{α}(X_{α})) + ε^{−2} (1 − ρ(det A_{α}(X_{α}))) det A_{α}(X_{α}).

Since

det A_{α}(X_{α}) = |a_{α}(π(X_{α}))|^{2}

and

A_{α}(X_{α})^{∗} = det A_{α}(X_{α})A_{α}(X_{α})^{−1},

both B_{α}(X_{α}) and φ_{α}(X_{α}) have the desired properties.

Next we consider the case where the point q is a pole of a_{α}, i.e., m = −1. Choose a small positive number ε such that the neighborhood

V_{α}^{ε} = {z ∈ U_{α} : ε^{−1} < |a_{α}(z)| ≤ +∞}

of q is relatively compact in U_{α} and that [V_{α}^{ε}] ∩ [U_{β}] = ∅ for every β ≠ α. Take an ε > ε′ such that V_{α}^{ε′} has also the same properties. On π^{−1}(U_{α}), A_{α}(X_{α}) is given by

( \begin{matrix}\frac{xû + yv̂}{x^{2} + y^{2}} + \sqrt{−1}\frac{xv̂ − yû}{x^{2} + y^{2}} \frac{Im x_{1}}{|X′_{α}|} & −\frac{xv̂ − yû}{x^{2} + y^{2}} \frac{x̄_{2}}{|X′_{α}|} \\ \frac{xv̂ − yû}{x^{2} + y^{2}} \frac{x_{2}}{|X′_{α}|} & \frac{xû + yv̂}{x^{2} + y^{2}} − \sqrt{−1}\frac{xv̂ − yû}{x^{2} + y^{2}} \frac{Im x_{1}}{|X′_{α}|} \end{matrix} ) . (19)

Lemma 8 The H-valued functions A_{α}(X_{α})^{−1} and (A_{α}(X_{α})^{∗})^{−1} are of C^{∞} along π^{−1}(q).

Proof. It is sufficient to show that (A_{α}(X_{α})^{∗})^{−1} is of C^{∞}. By (19). the matrix (A_{α}(X_{α})^{∗})^{−1} = (det A_{α}(X_{α}))^{−1} A_{α}(X_{α}) is given by

(û^{2} + v̂^{2})^{−1} ( \begin{matrix}xû + yv̂ + \sqrt{−1}(xv̂ − yû)\frac{Im x_{1}}{|X′_{α}|} & −(xv̂ − yû)\frac{x̄_{2}}{|X′_{α}|} \\ (xv̂ − yû)\frac{x_{2}}{|X′_{α}|} & xû + yv̂ − \sqrt{−1}(xv̂ − yû)\frac{Im x_{1}}{|X′_{α}|} \end{matrix} ) .

Since û^{2} + v̂^{2} ≠ 0 along π^{−1}(q), the lemma is proved.

Lemma 9 There is an H-valued C^{∞} function B′_{α} = B′_{α}(X_{α}) and a real-valued C^{∞} function φ′_{α} = φ′_{α}(X_{α}) on π^{−1}(U_{α}) such that

B′_{α}(X_{α}) = φ′_{α}(X_{α})A_{α}(X_{α}), (20)

where

(21) φ′_{α}(X_{α}) = \{ \begin{matrix}1 & for & X_{α} ∈ π^{−1}(U_{α} − V_{α}^{ε′}), \\ ε^{−2} (det A_{α}(X_{α}))^{−1} & for & X_{α} ∈ π^{−1}([V_{α}^{ε}]). \end{matrix}

Proof. Note that

(det A_{α}(X_{α}))^{−1} A_{α}(X_{α}) = (A_{α}(X_{α})^{∗})^{−1}

and

(det A_{α}(X_{α}))^{−1}

are of C^{∞} on π^{−1}(U_{α}) by Lemma 8. Thus the lemma can be proved by the same method as Lemma 7.

Now we define a connection on E by modifying {−dA_{α} · A_{α}^{−1}} along the fibres over the points of Σ as follows. If Σ ∩ U_{α} = ∅, then we define

θ_{α} = −dA_{α} · A_{α}^{−1}.

If Σ ∩ U_{α} consists of a zero, then using Lemma 7, we define

θ_{α} = −dA_{α} · B_{α}. (22)

If Σ ∩ U_{α} consists of a pole, then using Lemma 9, we define

θ_{α} = −dB′_{α} · A_{α}^{−1}. (23)

Then the collection θ = {θ_{α}} satisfies

ξ_{αβ}^{−1}dξ_{αβ} = θ_{β} − ξ_{αβ}^{−1}θ_{α}ξ_{αβ}

on π^{−1}(U_{α})∩π^{−1}(U_{β}). Thus θ gives a connection of E. We remark that the curvature form

Θ = dθ + θ ∧ θ

of θ vanishes outside the set ⋃_{Σ∩U_{α}≠∅} π^{−1}(V_{α}^{ε′}).

Proposition 2 The second Chern class c_{2}(E) of E is equal to 2g − 2, where g is the gunus of the compact Riemann surface D/Γ.

Proof. Since c_{1}(E) = 0 by Proposition 1, we have the equality

c_{2}(E) = \frac{1}{8π^{2}} ∫_{M} Tr Θ ∧ Θ.

By the remark above and (7), it suffices to show that

(24) \frac{1}{8π^{2}} ∫_{π^{−1}(U_{α})} Tr Θ ∧ Θ = \{ \begin{matrix}2 & if Σ ∩ U_{α} consists of a zero \\ −2 & if Σ ∩ U_{α} consists of a pole. \end{matrix}

First consider the case where Σ ∩ U_{α} consists of a zero of a_{α}. For simplicity, we omit the subscript α for a little while. Put

A = ( \begin{matrix}λ & −μ̄ \\ μ & λ̄ \end{matrix} )

and

s = det A.

Note that φ of Lemma 7 is a C^{∞} function of s. By Lemma 7 and (22), we have

Tr Θ ∧ Θ = Tr (2(φ − φ^{2})dφ ∧ dA · A^{−1} ∧ dA · A^{−1} ∧ dA · A^{−1})
= 2(φ − φ^{2})dφ ∧ Tr (dA · A^{−1})^{3}
(25)  = 12s^{−2} (φ − φ^{2}) ∧ (μ̄dλ ∧ dλ̄ ∧ dμ + λ̄dλ ∧ dμ ∧ dμ̄) .

Put

s_{1} = xû − yv̂,
s_{2} = xv̂ + yû,
t_{1} = \frac{Im x_{1}}{|X′_{α}|} ,
t_{2} = \frac{Re x_{2}}{|X′_{α}|} ,
t_{3} = \frac{Im x_{2}}{|X′_{α}|} .

Here we have the relations

(26)  s_{1}^{2} + s_{2}^{2} = s,
(27)  t_{1}^{2} + t_{2}^{2} + t_{3}^{2} = 1,

and

\{ \begin{matrix}λ & = & s_{1} + \sqrt{−1}s_{2}t_{1}, \\ μ & = & s_{2}t_{2} + \sqrt{−1}s_{2}t_{3}. \end{matrix}

Provided that ε and ε′ are sufficiently small, we can use the quadruple (s_{1}, s_{2}, t_{1}, t_{2}) as a system of local coordinates. By the relations (26) and (27) on π^{−1}(∂V^{s}), we have

μ̄dλ ∧ dλ̄ ∧ dμ + λ̄dλ ∧ dμ ∧ dμ̄ = −2ss_{2}ds_{1} ∧ \frac{dt_{1} ∧ dt_{2}}{t_{3}} . (28)

Thus (25) becomes

Tr Θ ∧ Θ = −24 (φ − φ^{2}) dφ ∧ s^{−1}s_{2}ds_{1} ∧ \frac{dt_{1} ∧ dt_{2}}{t_{3}} .

Taking parameters θ, θ_{1} and θ_{2} such that

s_{1} = s^{\frac{1}{2}} cos θ
s_{2} = s^{\frac{1}{2}} sin θ
t_{1} = cos θ_{1}
t_{2} = sin θ_{1} cos θ_{2}
t_{3} = sin θ_{1} sin θ_{2},

where 0 ≤ θ ≤ 2π, 0 ≤ θ_{1} ≤ π and 0 ≤ θ_{2} ≤ 2π, we have

∫_{π^{−1}(U_{α})} Tr Θ ∧ Θ = −24 ∫_{0}^{ε′} (φ − φ^{2}) \frac{dφ}{ds}ds ∫_{0}^{2π} (− sin^{2} θ) dθ ∫_{0}^{π} sin θ_{1}dθ_{1} ∫_{0}^{2π} dθ_{2}
= 16π^{2}.

Thus (24) is proved in the case of zeros. In the case of poles, the integration (24) can be done by the same method. In this case, the orientation of the integration along the circle s_{1}^{2} + s_{2}^{2} = s is opposite to the case of zeros.

Proposition 3 The differentiable vector bundle E is isomorphic to L ⊕ L^{−1}, where L is a holomorphic line bundle on P^{1} × (D/Γ) such that L^{⊗2} = K_{P^{1}×(D/Γ)}.

Proof. Let H be the holomorphic line bundle of deg = 1 on P^{1}. Put L = F ⊗ H^{−1}. Then we have

c_{1}(E) = 0 = c_{1} (L ⊕ L^{−1}) ,

and

c_{2}(E) = 2g − 2 = c_{2}(L ⊕ L^{−1}).

Hence E and L ⊕ L^{−1} are isomorphic as differentiable vector bundles by the characterization theorem of SU(2)-bundles on 4-manifolds. Proof of Theorem 1 is clear from the proposition above. Remark 3. In [Ca], F. Catanese proved among other things that the existence of a non constant holomorphic map of a compact Kähler manifold to a compact curve of genus ≥ 2 is dictated by topology (Theorem (1.10) of [Ca]). He asked whether this held true if one dropped the Kähler assumption. The above results show that our manifold Z provides a negative answer to his question. Indeed, the projective bundle over P^{1} × (D/Γ) associated with the holomorphic vector bundle L ⊕ L^{−1} is a projective algebraic manifold which is diffeomorphic to Z but not biholomorphic as Theorem 2 shows.

3 Proof of Theorem 2

Since the action of Γ on D is free, there is an element σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ such that

|a + d| > 2. Hence for some τ ∈ SL(2, R), we have τ^{−1}στ = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) , α, β ∈ R.

Therefore, replacing [z_{0} : z_{1} : z_{2} : z_{3}] by [w_{0} : w_{1} : w_{2} : w_{3}] defined by

( \begin{matrix}z_{0} \\ z_{1} \\ z_{2} \\ z_{3} \end{matrix} ) = ( \begin{matrix}τ_{11} & 0 & τ_{12} & 0 \\ 0 & τ_{11} & 0 & τ_{12} \\ τ_{21} & 0 & τ_{22} & 0 \\ 0 & τ_{21} & 0 & τ_{22} \end{matrix} ) ( \begin{matrix}w_{0} \\ w_{1} \\ w_{2} \\ w_{3} \end{matrix} ) ,

where τ = ( \begin{matrix}τ_{11} & τ_{12} \\ τ_{21} & τ_{22} \end{matrix} ), we can assume without loss of generality that Γ contains an

element of the form σ_{0} = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) with |α+β| > 2 and α, β ∈ R. Now we shall prove

the theorem by contradition. Suppose that Z admits a non-constant meromorphic function f. Pulling back f onto Z̃, we obtain a Γ-invariant meromorphic function f̃ on Z̃. Obviously, f̃ extends to a Γ-invatriant rational function F on P^{3}. Put

F = \frac{F_{0}}{F_{1}} , where F_{0} and F_{1} are coprime homogeneous polynomials of z_{0},z_{1},z_{2} and z_{3}

with degree m. Let φ_{t} = t_{0}F_{0} + t_{1}F_{1}, t = [t_{0} : t_{1}] ∈ P^{1}, be any member of the linear pencil spanned by F_{0} and F_{1}. Denote by D_{t} the divisor defined by φ_{t} = 0. Since D_{t} is Γ-invariant, we have the relation

σ^{∗}φ_{t} = c(σ)φ_{t}

for any σ ∈ Γ, where c(σ) ∈ C^{∗}. By the relation σ_{0}^{∗}φ_{t} = c(σ_{0})φ_{t}, we see that φ_{t} is of the form

φ_{t} = ∑_{μ=0}^{l} a_{μ}(z_{0}, z_{1})z_{2}^{l−μ}z_{3}^{μ},

where l is an integer satisfying 0 ≤ l ≤ m, and the a_{μ}(z_{0}, z_{1}), 0 ≤ μ ≤ l, are homogeneous polynomials of z_{0} and z_{1} with degree m − l. Further we have c(σ_{0}) = α^{m−l}β^{l}. If l satisfies 0 < l < m for a general member D_{t}, we see that two skew lines l_{0} : z_{0} = z_{1} = 0 and l_{1} : z_{2} = z_{3} = 0 are contained in the base locus B of the linear pencil. Since B is connected, there is a point [ξ_{0} : ξ_{1} : ξ_{2} : ξ_{3}] ∈ B \ (l_{0} ∪ l_{1}). Consider the line l defined by

\{ \begin{matrix}ξ_{1}z_{0} − ξ_{0}z_{1} & = & 0 \\ ξ_{3}z_{2} − ξ_{2}z_{3} & = & 0. \end{matrix}

It is easy to check that l is contained in D_{t}. Hence l ⊂ B. Thus there is a subgroup Γ_{1} of Γ with [Γ : Γ_{1}] < ∞ such that Γ_{1} leaves invariant two planes ξ_{1}z_{0} − ξ_{0}z_{1} = 0 and ξ_{3}z_{2} − ξ_{2}z_{3} = 0. If l = 0 (resp. l = m), then the plane of the form ξ_{1}z_{0} − ξ_{0}z_{1} = 0 (resp. ξ_{3}z_{2} − ξ_{2}z_{3} = 0) is left invariant by a subgroup Γ_{1} of Γ with [Γ : Γ_{1}] < ∞. First we consider the case where a plane of the form ξ_{1}z_{0} − ξ_{0}z_{1} = 0, [ξ_{0} : ξ_{1}] ∈ P^{1},

is left invariant by Γ_{1}. Then any element σ ∈ Γ_{1} is of the form ( \begin{matrix}a & 0 \\ c & d \end{matrix} ). Since Γ_{1}

is not abelian, we can choose σ_{1} ∈ Γ_{1} such that σ_{1} = ( \begin{matrix}a & 0 \\ c & d \end{matrix} ), c ≠ 0. Then, for

n ∈ Z, we have σ_{1}^{−1}σ_{0}^{−n}σ_{1}σ_{0}^{n} = ( \begin{matrix}1 & 0 \\ ac(1 − α^{2n}) & 1 \end{matrix} ) ∈ Γ_{1}. Since ac ≠ 0 and |α| ≠ 1,

this implies that Γ_{1} is not discrete in SL(2, R). This is a contradiction. Thus the theorem is proved in this case. The other case is settled by the same argument.

References

[Ca] Catanese, F. : Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations, Invent. Math. 104 (1991), 263-289.

[Fu] Fueter, Von Rud. : Die Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Varablen, Comm. Math. Helv. 7(1935), 307-330.

[Ku] Kuiper, N. H. : On compact conformally euclidean spaces of dimension > 2, Ann. Math., 52 (1950), 478-490.

[Po] Pontecorvo, M. : Algebraic dimension of twistor spaces and scalar curvature of anti-self-dual metric, Math. Ann. 291(1991), 113-122.

Department of Mathematics Sophia University 7 Kioicho, Chiyoda-ku 102 Tokyo Japan