A note to Teleman’s paper on Class VII_{o} surfaces

Masahide Kato ^{∗}

1 Introduction

In his paper [T], Teleman gave a proof of the following

Theorem. No compact complex surfaces S have the following three properties simultaneously.

(A) the first Betti number b_{1}(S) is equal to 1, and the second Betti numer b_{2}(S) is equal to 0,

(B) S contains no compact analytic curves,

(C) dim H^{0}(S, Ω^{1}(F)) = 0 for any flat line bundle F.

This theorem implies that all Class VII_{o} surfaces with b_{2} = 0 are Hopf surfaces and Inoue surfaces S_{M}, S_{N}^{+}, S_{N}^{−}. In this note, we shall give an elementary explanation of Teleman’s proof, starting from Buchdahl’s existence theorem of Hermitian-Einstein metric on S.

2 Existence of Hermitian-Einstein metric

Let X be a complex manifold of dimension n. We consider a system of coordinate neighborhoods {(U_{α}, z_{α})}, z_{α} = (z_{α}^{1}, …, z_{α}^{n}), on X. Let E = {E_{αβ}} be a holomorphic vector bundle on X of rank E = r and h = {h_{α}} a hermitian metric on E. Then {h_{α}} satisfies the equality

tE_{αβ}h_{α}Ē_{αβ} = h_{β}, (1)

where the (i, j)-component h_{αij̄} of the matrix

h_{α} = ( \begin{matrix}h_{α1¯1}, & ⋯ & h_{α1r̄} \\ ⋮ & ⋯ & ⋮ \\ h_{αr¯1}, & ⋯ & h_{αrr̄} \end{matrix} )

∗The author is partly supported by the Grants-in-Aid for Co-operative Reserch as well as Scientific Research, the Ministry of Education, Science and Culture, 1997, Japan.

1991 Mathematics Subject Classification. Primary 32J15; Secondary 53C10,53C56.

is given by h_{αij̄} = h (e_{αi}, e_{αj}), where {e_{αi}, i = 1, … r}, is a local frame of E defined on U_{α}. The hermitian connection ω = {ω_{α}} is defined by ω_{α} = ^{t}(∂h_{α} · h_{α}^{−1}) (2) on U_{α}. Then {ω_{α}} satisfies the equality E_{αβ}^{−1}dE_{αβ} = ω_{β} − E_{αβ}^{−1}ω_{α}E_{αβ} (3) on U_{α} ∩ U_{β}. The curvature form F (h) of the hermitian connection is given by F (h)_{α} = dω_{α} + ω_{α} ∧ ω_{α} (4) on U_{α}. The Chern forms c_{j}(E, h), j = 1, 2, … r, associated with the hermitian metric is defined by

∑_{j} c_{j}(E, h)t^{j} = det (I_{r} − \frac{t}{2π\sqrt{−1}}F (h)) .

In particular, we have

c_{1}(E, h) = −\frac{1}{2π\sqrt{−1}}\overline{∂}∂ log det h_{α}.

Suppose that there is a g-Hermitian-Einstein metric h = {h_{α}} on the tangent bundle T = {T_{αβ}} of X. Then (1) is written by tT_{αβ}h_{α}T̄_{αβ} = h_{β}. (5) Put Ω(h) = ^{t}F (h). By (2) and (4) Ω(h) = \overline{∂}∂h_{α} · h_{α}^{−1} + ∂h_{α} · h_{α}^{−1} ∧ \overline{∂}h_{α} · h_{α}^{−1} (6) on U_{α}. We shall define the torsion of the connection h. Put dz_{α} = ^{t}(dz_{α}^{1}, …, dz_{α}^{n}). Then, by (3), we have T_{αβ}^{−1}dT_{αβ} ∧ dz_{β} = ω_{β} ∧ dz_{β} − T_{αβ}^{−1}ω_{α}T_{αβ} ∧ dz_{β} = ω_{β} ∧ dz_{β} − T_{αβ}^{−1}ω_{α} ∧ dz_{α} (7) Since dT_{αβ} ∧ dz_{β} = d(dz_{α}) = 0, we have T_{αβ}ω_{β} ∧ dz_{β} = ω_{α} ∧ dz_{α}. (8) We write the connection ω_{α} as

ω_{α} = ( \begin{matrix}ω_{α1}^{1}, & ⋯ & ω_{αn}^{1} \\ ⋮ & ⋯ & ⋮ \\ ω_{α1}^{n}, & ⋯ & ω_{αn}^{n} \end{matrix} ) ,

where

ω_{αν}^{λ} = ω_{αμν}^{λ}dz_{α}^{μ}, and ω_{αμν}^{λ} = \frac{∂h_{ανκ̄}}{∂z_{α}^{μ}} h_{α}^{κ̄λ}.

Put

τ_{α} = \frac{∂}{∂z_{α}^{λ}} ⊗ ω_{ανμ}^{λ}dz_{α}^{ν} ∧ dz_{α}^{μ}. (9)

Define

τ_{αμν}^{λ} = \frac{1}{2} (ω_{αμν}^{λ} − ω_{ανμ}^{λ}) . (10)

Then we have

τ_{α} = τ_{αμν}^{λ} \frac{∂}{∂z_{α}^{λ}} ⊗ dz_{α}^{μ} ∧ dz_{α}^{ν}.

From (8), the equality τ_{α} = τ_{β} (11)

follows. Hence τ = τ_{α} defines a C^{∞} global tensor field of type (1,2), which is called the torsion of the metric h. Taking the determinants of (5), we have

(12) det h_{α}| det T_{αβ}|^{2} = det h_{β}.

Assume that

D. the 1-cocycle {det T_{αβ}} consists of constant functions.

Then, applying ∂ log to (12), we have

(13) ∂ log det h_{α} = ∂ log det h_{β}.

Put φ_{α} = (det h_{α})^{− \frac{1}{n}} . (14)

The equation (13) shows that ∂ log φ = {∂ log φ_{α}} is a C^{∞} (1, 0)-form defined globally on X. Put

σ_{α} = \frac{∂}{∂z_{α}} ⊗ I_{n} · ∂ log φ_{α} ∧ dz_{α}.

Then σ = {σ_{α}} is a C^{∞} tensor field of type (1, 2). We write σ_{α} as

σ_{α} = σ_{αμν}^{λ} \frac{∂}{∂z_{α}^{λ}} ⊗ dz_{α}^{μ} ∧ dz_{α}^{ν},

where

σ_{αμν}^{λ} = \frac{1}{2} (δ_{ν}^{λ} \frac{∂ log φ_{α}}{∂z_{α}^{μ}} − δ_{μ}^{λ} \frac{∂ log φ_{α}}{∂z_{α}^{ν}} ) . (15)

Define another C^{∞} tensor field τ̃ = {τ̃_{α}} of type (1, 2) by

τ̃_{α} = τ_{α} + σ_{α} (16)

Put

τ̃_{α} = τ̃_{αμν}^{λ} \frac{∂}{∂z_{α}^{λ}} ⊗ dz_{α}^{μ} ∧ dz_{α}^{ν}

with

τ̃_{αμν}^{λ} = −τ̃_{ανμ}^{λ}.

Then we have

τ̃_{αμν}^{λ} = τ_{αμν}^{λ} + σ_{αμν}^{λ}.

Put

θ_{α} = τ_{αμλ}^{λ}dz_{α}^{μ}

and θ̃_{α} = τ̃_{αμλ}^{λ}dz_{α}^{μ}. (17)

Then θ = {θ_{α}} and θ̃ = {θ̃_{α}} are C^{∞} (1, 0)-forms defined globally on X, and we have

θ̃_{α} = θ_{α} + \frac{n − 1}{2}∂ log φ_{α}. (18)

The following fact is a part of a theorem of Lübke [L] (see also [Ko] page 114).

Theorem 2.1 (Lübke) If a vector bundle E with rankE = r admits a g-Hermitian-Einstein metric h such that ∫_{X}((r − 1)c_{1}^{2}(E, h) − 2rc_{2}(E, h)) ∧ Φ(g)^{n−2} = 0, (19)

then, there is a scalar-valued (1, 1)-form η(h) defined globally on X such that

F (h) = \sqrt{−1}η(h) ⊗ I_{r}. (20)

Here Φ(g) denotes the Kähler form of the Gauduchon metric g.

The (1, 1)-form η(h) in (20) can be expressed in terms of h. Indeed, on U_{α}, we have

r\sqrt{−1}η(h)_{α} = trace F (h)_{α}
= trace (dω_{α} + ω_{α} ∧ ω_{α})
= trace (dω_{α})
= dtrace (∂h_{α} · h_{α}^{−1})
= d∂ log det h_{α}
= \overline{∂}∂ log det h_{α}.

For E = T, we have

(21) η(h)_{α} = \frac{1}{n\sqrt{−1}} \overline{∂}∂ log det h_{α} = \sqrt{−1} \overline{∂}∂ log φ_{α}

on U_{α}, where n = dimX.

Proposition 2.1 . If the assumption D is satisfied and if (20) holds, then \overline{∂}θ̃ = 0. Proof. Take any system of local coordinates (U_{α}, z_{α}). For a little while, we omit the label α for simplicity. Applying \overline{∂} to (10), we have

2\frac{∂τ_{μλ}^{λ}}{∂z̄^{l}} = \frac{∂}{∂z̄^{l}} (ω_{μλ}^{λ} − ω_{λμ}^{λ})
= \frac{∂}{∂z̄^{l}} (\frac{∂h_{λκ̄}}{∂z^{μ}} h^{κ̄λ} − \frac{∂h_{μκ̄}}{∂z^{λ}} h^{κ̄λ})
= \frac{∂^{2}h_{λκ̄}}{∂z^{μ}∂z̄^{l}} h^{κ̄λ} + \frac{∂h_{λκ̄}}{∂z^{μ}} \frac{∂h^{κ̄λ}}{∂z̄^{l}} − \frac{∂^{2}h_{μκ̄}}{∂z^{λ}∂z̄^{l}} h^{κ̄λ} − \frac{∂h_{μκ̄}}{∂z^{λ}} \frac{∂h^{κ̄λ}}{∂z̄^{l}} .

Let o be any point in U_{α}. Taking a suitable affine transformation with respect to z_{α}, we can assume that h_{ij̄}(o) = h^{j̄i}(o) = δ_{ij}. Then we have

\frac{∂h_{jk̄}}{∂z̄^{l}} + \frac{∂h^{j̄k}}{∂z̄^{l}} = 0 (22)

at o. Therefore, we have

(23)  2\frac{∂τ_{μλ}^{λ}}{∂z̄^{l}} = \frac{∂^{2}h_{λλ̄}}{∂z^{μ}∂z̄^{l}} − \frac{∂h_{λκ̄}}{∂z^{μ}} \frac{∂h_{κλ̄}}{∂z̄^{l}} − \frac{∂^{2}h_{μλ̄}}{∂z^{λ}∂z̄^{l}} + \frac{∂h_{μκ̄}}{∂z^{λ}} \frac{∂h_{κλ̄}}{∂z̄^{l}} ,

at o, where λ and κ are summed. Comparing the (i, j)-components of both sides of (6), we have

Ω(h)_{j}^{i} = (−\frac{∂^{2}h_{ij̄}}{∂z^{μ}∂z̄^{l}} + \frac{∂h_{iκ̄}}{∂z^{μ}} \frac{∂h_{κj̄}}{∂z̄^{l}} ) dz^{μ} ∧ dz̄^{l} (24)

at o, where μ, κ and l are summed. On the other hand, by (20), Ω(h)_{j}^{i} = \sqrt{−1}η(h)δ_{j}^{i}. (25) Comparing (24) and (25), we have

\frac{∂^{2}h_{ij̄}}{∂z^{μ}∂z̄^{l}} = −\sqrt{−1}η(h)_{μl̄}δ_{ij̄} + \frac{∂h_{iκ̄}}{∂z^{μ}} \frac{∂h_{κj̄}}{∂z̄^{l}} (26)

at o, where κ is summed. Here, we put

η(h) = η(h)_{μl̄}dz^{μ} ∧ dz̄^{l}.

Substituting (26) into (23) and using (22), we have

2 ∑_{λ} \frac{∂τ_{μλ}^{λ}}{∂z̄^{l}} = ∑_{λ} (−\sqrt{−1}η(h)_{μl̄} + ∑_{k} \frac{∂h_{λκ̄}}{∂z^{μ}} \frac{∂h_{κλ̄}}{∂z̄^{l}} ) − ∑_{λ k} \frac{∂h_{λκ̄}}{∂z^{μ}} \frac{∂h_{κλ̄}}{∂z̄^{l}}

− ∑_{λ} (−\sqrt{−1}η(h)_{λl̄}δ_{μλ̄} + ∑_{k} \frac{∂h_{μκ̄}}{∂z^{λ}} \frac{∂h_{κλ̄}}{∂z̄^{l}} ) + ∑_{λ k} \frac{∂h_{μκ̄}}{∂z^{λ}} \frac{∂h_{κλ̄}}{∂z̄^{l}}

= −(n − 1)\sqrt{−1}η(h)_{μl̄}

Hence we have

\overline{∂}θ = \frac{∂τ_{μλ}^{λ}}{∂z̄^{l}} dz̄^{l} ∧ dz^{μ}
= \frac{\sqrt{−1}(n − 1)}{2}η(h).

or, by (21),

\overline{∂}θ = −\frac{n − 1}{2}\overline{∂}∂ log φ.

Therefore by (18), we have

\overline{∂}θ̃ = \overline{∂}θ + \frac{n − 1}{2}\overline{∂}∂ log φ = 0. (27)

3 An elementary proof of Teleman’s theorem

Teleman’s proof is based on the following important fact, which is a part of a theorem of Buchdahl. Theorem 3.1 (Buchdahl [B]). Let X be a complex surface with a Gauduchon metric g (i.e., ∂\overline{∂}Φ(g) = 0 ). Suppose that E be a indecomposable vector bundle on X which is stable with respect to g. Then there is a hermitian metric h on E such that the associated connection ω = {ω_{α}}, ω_{α} = ^{t}(∂h_{α} · h_{α}^{−1}) is of g-Hermitian-Einstein.

Assume that a compact complex surface S has the following three properties (A) the first Betti number b_{1}(S) is equal to 1, and the second Betti numer b_{2}(S) is equal to 0, (B) S contains no compact analytic curves, (C) dim H^{0}(S, Ω^{1}(F)) = 0 for any flat line bundle F. We shall disprove the existence of S. We recall some known facts on S. Fact 1. Every line bundle on S is holomorphically equivalent to a flat bundle. Fact 2 (Inoue [I]). The canonical line bundle is holomorphically equivalent to a flat bundle with

real constant transition functions. Fact 3 (Inoue-Kobayashi-Ochiai [IKO]). S admits a flat affine structure. Fact 4. All Chern numbers of S vanish.

Fact 4 follows immediately from Fact 3. By Fact 3, we can choose an open covering U = {U_{α}} of S and a system of local coordinates {(U_{α}, z_{α})}, z_{α} = (z_{α}^{1}, z_{α}^{2}), on each U_{α} such that every transition function of local coordinates are affine transformations;

(28)  z_{α}^{1} = p_{0αβ}^{1} + p_{1αβ}^{1}z_{β}^{1} + p_{2αβ}^{1}z_{β}^{2}
(29)  z_{α}^{2} = p_{0αβ}^{2} + p_{1αβ}^{2}z_{β}^{1} + p_{2αβ}^{2}z_{β}^{2},

where the p_{jαβ}^{i} are constants. Put

φ_{αβ} = z_{α} ∘ z_{β}^{−1}

and

T_{αβ} = ( \begin{matrix}p_{1αβ}^{1} & p_{2αβ}^{1} \\ p_{1αβ}^{2} & p_{2αβ}^{2} \end{matrix} ) .

In what follows, we assume that the local coordinate systems {(U_{α}, z_{α})} satisfy (28) and (29). By the assumptions (C), it is easy to see that T has no coherent subsheaf of rank 1. Hence, trivially, T is stable with respect to any Gauduchon metric g on S. Therefore we can apply Theorems 3.1 and 2.1 to the surface X = S and its tangent bundle E = T using Fact 4.

Proposition 3.1 . There is a g-Hermitian-Einstein metric h on S such that the curvature form F (h) satisfies

Ω(h) = ^{t}F (h) = \sqrt{−1}η(h) ⊗ I_{2}.

By Fact 3, the assumption D is satisfied. Therefore it is possible to define σ, and consequently, τ̃ and θ̃ ( see (15) and (17) ).

Lemma 3.1 θ̃ = 0.

Proof. By Propositions 3.1 and 2.1, we see that θ̃ is a holomorphic 1-form defined globally on S. Since S has no non-zero holomorphic 1-form, we have the lemma.

Proposition 3.2 τ̃ = 0.

Proof. By Lemma 3.1 and (17), we have

τ̃_{11}^{1} + τ̃_{12}^{2} = 0
τ̃_{21}^{1} + τ̃_{22}^{2} = 0.

Since τ̃_{μν}^{λ} = −τ̃_{νμ}^{λ}, the proposition is an easy consequence of these equalities.

On each U_{α}, we fix a point o. Now we consider the following differential equation on U_{α}.

F_{α}^{−1}∂F_{α} = ω̃_{α}, F_{α}(o) = I_{2}, (30) where

ω̃_{α} = ^{t}(∂h̃_{α} · h̃_{α}^{−1}),
h̃_{α} = φ_{α}h_{α},
φ_{α} = (det h_{α})^{− \frac{1}{2}} ,
F_{α} = ( \begin{matrix}\frac{∂f_{α}^{1}}{∂z_{α}^{1}} & \frac{∂f_{α}^{1}}{∂z_{α}^{2}} \\ \frac{∂f_{α}^{2}}{∂z_{α}^{1}} & \frac{∂f_{α}^{2}}{∂z_{α}^{2}} \end{matrix} ) ,

with unknown functions f_{α}^{1}, f_{α}^{2}.

Proposition 3.3 dω̃_{α} + ω̃_{α} ∧ ω̃_{α} = 0.

Proof. Note that

(31)  ω̃_{α} = ^{t}(∂h̃_{α} · h̃_{α}^{−1})
= I_{2} · ∂ log φ_{α} +^{t} (∂h_{α} · h_{α}^{−1})
= I_{2} · ∂ log φ_{α} + ω_{α}.

Then we have dω̃_{α} + ω̃_{α} ∧ ω̃_{α} = I_{2} · \overline{∂}∂ log φ_{α} + F (h), (32)

which is equal to zero by Proposition 3.1 and (21).

By the proposition above we see in particular that \overline{∂}ω̃_{α} = 0. Hence the coefficients of ω̃_{α} are holomorphic.

Lemma 3.2 The differential equation (30) has a holomorphic solution (f_{α}^{1}, f_{α}^{2}).

Proof. It is enough to check the integrability condition of Gunning [G] Lemma 2, page 41. The vanishing of torsion τ̃ = 0 of Proposition 3.2 implies λ_{j_{1},j_{2}}^{k} = λ_{j_{2},j_{1}}^{k} in [G]. The flatness of Proposition 3.3 implies the symmetry of λ_{j_{1},j_{2},j_{3}}^{i} in [G]. By an easy calculation, we have

Lemma 3.3 The equality F_{α}^{−1}∂F_{α} = ω̃_{α} is equivalent to ∂(^{t}F_{α}^{−1}h̃_{α}) = 0 .

In view of the lemma above, there is a 2 × 2-matrix-valued holomorphic function E_{α} such that

tF_{α}^{−1}h̃_{α} = Ē_{α}. (33)

Since h̃_{α} is a hermitian matrix, we have

tF_{α}Ē_{α} = h̃_{α} = ^{t}E_{α}F̄_{α}.

Hence A_{α} := ^{t}E_{α}^{−1}^{t}F_{α} = F̄_{α}Ē_{α}^{−1} (34)

is a constant matrix, since ^{t}E_{α}^{−1}^{t}F_{α} is holomorphic and F̄_{α}Ē_{α}^{−1} is anti-holomorphic. Moreover the constant matrix A_{α} is positive definite. Take a positive definite constant matrix B_{α} such that ^{t}B_{α}B̄_{α} = A_{α}^{−1}. Put

g_{α}(z_{α}) = B_{α} · ( \begin{matrix}f_{α}^{1}(z_{α}) \\ f_{α}^{2}(z_{α}) \end{matrix} ) (35)

Let G_{α} denote the Jacobian matrix of g_{α}. Then G_{α} = B_{α}F_{α}. By (33), we have

h̃_{α} = ^{t}G_{α}Ḡ_{α}. (36)

Since B_{α} is constant, we have G_{α}^{−1}∂G_{α} = ω̃_{α}. (37)

The equation (5) is written by

(38)  ^{t}T_{αβ}h̃_{α}T̄_{αβ} = \frac{φ_{α}}{φ_{β}} h̃_{β}
= | det T_{αβ}| · h̃_{β}

Therefore, letting S_{αβ} = G_{α}T_{αβ}G_{β}^{−1}, (39)

we have tS_{αβ}S̄_{αβ} = | det T_{αβ}| · I_{2}. (40)

Lemma 3.4 dS_{αβ} = 0.

Proof. Since T_{αβ} is constant, applying ∂ to (39), we have

∂S_{αβ}G_{β} + S_{αβ}∂G_{β} = ∂G_{α}T_{αβ}.

Hence

∂S_{αβ}G_{β} = ∂G_{α}T_{αβ} − S_{αβ}∂G_{β}
= G_{α}ω̃_{α}T_{αβ} − G_{α}T_{αβ}G_{β}^{−1}∂G_{β}
= G_{α}(ω̃_{α}T_{αβ} − T_{αβ}ω̃_{β})
= G_{α}(ω_{α}T_{αβ} − T_{αβ}ω_{β} + ∂ log φ_{α} · T_{αβ} − T_{αβ} · ∂ log φ_{β})
= G_{α}T_{αβ}∂ log \frac{φ_{α}}{φ_{β}}
= G_{α}T_{αβ}∂ log | det T_{αβ}|
= 0.

Here (31) is used at the fourth equality. At the fifth equality, we have used the relation ω_{α}T_{αβ} = T_{αβ}ω_{β}, which holds since T_{αβ} is constant.

Now we put

w_{α} = g_{α}(z_{α})

on U_{α}, shlinking U_{α} slightly, if necessary. Then {(U_{α}, w_{α})} gives a local coordinate system on S.

Proposition 3.4 The local coordinate system {(U_{α}, w_{α})} defines a quaternionic structure on S, replacing S with its double cover if necessary. Namely, each transition function w_{α} ∘ w_{β}^{−1} is an affine transformation of C^{2} whose linear part is an element of R^{+} · SU(2), where R^{+} is the multiplicative group of positive numbers.

Proof. That w_{α} ∘w_{β}^{−1} is an affine transformation follows from Lemma 3.4. By Fact 2, replacing S with its double cover if necessary, we can assume that the transition functions of the canonical line bundle are positive numbers. Therefore det S_{αβ} > 0. Then (40) implies S_{αβ} ∈ R^{+} · SU(2).

On the other hand, we know the following.

Proposition 3.5 ([Ka], Proposition 7). Let S be a surface satisfying (A), (B) and (C). Then S does not admit quaternionic structures.

Proof of Theorem. By the two propositions 3.4 and 3.5, we have a contradiction. Hence there are no surfaces S satisfying the properties (A), (B), and (C). Thus the theorem of Teleman is proved.

References

[B] Buchdahl, N. P. : Hermitian-Einstein connections and stable vector bundles over compact complex surace, Math. Ann. 280 (1988), 625-648.

[G] Gunning, R. C. : On uniformization of complex manifolds : The role of connections, Math. Notes, Princeton Univ. Press, 1978.

[I] Inoue, M. : On surfaces of Class VII_{o} , Invent. Math. 24 (1974) 269-310.

[IKO] Inoue, M. ; Kobayashi, S, ; Ochiai, T. : Holomorphic affine connections on compact complex surfaces, J. Fac. Sci. Univ. Tokyo Sec.IA 27(1980)247-264.

[Ka] Kato, Ma. : Compact differentiable 4-folds with quaternionic structures, Math. Ann. 248 (1980) 79-96. Erratum : Math. Ann. 283(1989), 352.

[Ko] Kobayashi, S. : Differential geometry of complex vector bundles, Publ. of Math. Soc. Japan 15, Iwanami Shoten Publ. and Princeton Univ. Press. 1987.

[L] Lübke, M. : Chernklassen von Hermite-Einstein-Vektorbündeln, Math. Ann. 260 (1982), 133-141.

[T] Teleman, Andrei-Dumitru : Projectively flat surfaces and Bogomolov’s theorem on Class VII_{o} surfaces, International Journal of Mathematics (1994) 5-2 253-264

first version: December 18, 1997 misprints corrected: June 30, 1999