Compact complex 3-manifolds of Class L with finite cyclic fundamental groups

Masahide Kato

November 19, 1988; revised December 26, 2008

In this note, we shall construct compact complex 3-manifolds with finite cyclic fundamental groups. Since any two Class L manifolds can be connected complex analytically like connected sums in differential topology, we see, in particular, that PSL(2, Z) ≅ Z_{2}∗Z_{3} can be realized the fundamental group of a compact 3-manifold of Class L.

1

Let P^{3} be a projective space of dimension three and [z_{0} : z_{1} : z_{2} : z_{3}] a system of homogeneous coordinates. Let ℓ′ (resp. ℓ′′) be the lines defined by z_{0} = z_{1} = 0 (resp. z_{2} = z_{3} = 0). Put Ω = P^{3} \ (ℓ′ ∪ ℓ′′). Let g be an element of PGL(4, C) defined by

( \begin{matrix}αa & 0 & 0 & 0 \\ 0 & αā & 0 & 0 \\ 0 & 0 & a & 0 \\ 0 & 0 & 0 & ā \end{matrix} ) ,

where 0 < α < 1 and a = exp(2πi/p), p ∈ N. Now consider the line

ℓ : \{ \begin{matrix}z_{0} & = & λz_{1} \\ z_{2} & = & μz_{3} \end{matrix} , λμ ≠ 0.

If p is odd then put q = p else put q = p/2. Then

g^{q} = ± ( \begin{matrix}α^{q} & 0 & 0 & 0 \\ 0 & α^{q} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) ∈ P GL(4, C)

acts on ℓ. Put M_{1} = Ω/⟨g⟩ and M_{q} = Ω/⟨g^{q}⟩, which are compact complex 3-manifolds of Class L. It is easy to check that g^{k}(ℓ) ∩ ℓ = ∅ for any k ≠ 0 mod q. Since ⟨g⟩ acts on the set ⋃_{ν=0}^{q−1} g^{ν}(ℓ) ∩ Ω, we see that the quotient

E_{1} := ⋃_{ν=0}^{q−1}(g^{ν}(ℓ) ∩ Ω)/⟨g⟩

is a connected elliptic curve in M_{1}. Similarly, since ⟨g^{q}⟩ acts on ℓ, the quotient E_{q} := (ℓ ∩ Ω)/⟨g^{q}⟩ is a (connected) elliptic curve in M_{q}.

It is easy to see that E_{1} has a tubular neighborhood T_{ε} in M_{1} which is biholomorphic to a neighborhood T_{qε} of E_{q}. The canonical projection π̄ : M_{q} → M_{1} sends T_{qε} onto T_{ε} biholomorphically. Note that E_{q} coincides with the fibre π^{−1}([λ : 1] × [μ : 1]) of the fibre bundle structure

\begin{aligned}π : & M_{q} & → & P^{1} × P^{1} \\ & [[z_{0} : z_{1} : z_{2} : z_{3}]] & → & [z_{0} : z_{1}] × [z_{2} : z_{3}]. \end{aligned}

Let γ be the automorphism of P^{1} × P^{1} defined by

γ([z_{0} : z_{1}], [z_{2} : z_{3}]) = ([az_{0} : āz_{1}], [az_{2} : āz_{3}]).

Then M_{1} is an elliptic fibre space over a singular surface,

ϖ : M_{1} → (P^{1} × P^{1}) /⟨γ⟩.

We can assume that the tubular neighborhood T_{qε} of E_{q} is given by

T_{qε} = { [[z_{0} : z_{1} : z_{2} : z_{3}]] ∈ M_{q} : |z_{0}/z_{1} − λ|^{2} + |z_{2}/z_{3} − μ|^{2} < ε^{2}, z_{1}z_{3} ≠ 0}.

Let B_{ε} denote the open ball in C^{2} centered at the origin with radius ε > 0. Then

\begin{aligned}φ : & B_{ε} × E_{q} & → & T_{qε} \\ & (w_{1}, w_{2}, [ξ]) & ↦ & [[(w_{1} + λ)ξ : ξ : w_{2} + μ : 1]] \end{aligned}

gives a biholomorphic map of B_{ε} × E_{q} onto T_{qε}, where E_{q} is identified with C^{∗}/⟨α^{q}⟩.

2

Put W = C^{2} − {(0, 0)}. Let S = W/⟨β · I_{2}⟩ be a Hopf surface, where β > 1 and I_{2} is the identity matrix of size 2. Consider the line bundle L on S which is the quotient space (W × C)/⟨h⟩ , where h is an automorphism of W × C defined by

h : (u_{1}, u_{2}, ξ) ↦ (βu_{1}, βu_{2}, β^{−1}ξ).

The zero-section of L is identified with S. In L, we consider the subdomain

L_{ε} := { [u_{1}, u_{2}, ξ] ∈ L : |u_{1}ξ|^{2} + |u_{2}ξ|^{2} < ε^{2} }.

Note that L_{ε} is a tubular neighborhood of S. Note that the map

\begin{aligned}ψ : & L_{ε} \ S & → & B_{ε}^{∗} × (C^{∗}/⟨β⟩) \\ & [u_{1}, u_{2}, ξ] & ↦ & (u_{1}ξ, u_{2}ξ, [ξ]) \end{aligned}

is biholomorphic, where B_{ε}^{∗} = B_{ε} \ {(0, 0)}.

Put

β = exp(−4π^{2}/(q log α)) > 1.

Let η be the isomorphism

η : C^{∗}/⟨β⟩ → C^{∗}/⟨α^{q}⟩ = E_{q}

of elliptic curves defined by

η[ξ] = exp (\frac{q log α}{2πi} · log ξ) .

Then we have an isomorphism,

\begin{aligned}ψ_{η} : & L_{ε} \ S & → & B_{ε}^{∗} × E_{q} \\ & [u_{1}, u_{2}, ξ] & ↦ & (u_{1}ξ, u_{2}ξ, η[ξ]). \end{aligned}

Recall that the canonical projection π̄ : M_{q} → M_{1} sends T_{qε} onto T_{ε} biholomorphically. Now we form the union

Z = (M_{1} \ E_{1}) ∪ L_{ε},

where L_{ε} \ S is identified with T_{ε} \ E_{1} by π̄ ∘ φ ∘ ψ_{η}. Then Z is a compact complex 3-manifold. In the next section, we shall show that Z is the desired manifold.

3

The domain M_{1} \ E_{1} contains many lines. Therefore we have

Proposition 3.1 The manifold Z is of Class L.

Next we shall prove the following

Proposition 3.2 The fundamental group π_{1}(Z) is a finite cyclic group of order q.

Proof. By the construction and van Kampen’s theorem, π_{1}(Z) is generated by a generator σ_{1} ∈ π_{1}(M_{1} \ E_{1}) ≃ Z and a generator σ_{2} ∈ π_{1}(L_{ε}) ≃ Z. Fix sufficiently small δ > 0 such that |δ + λ| ≠ |λ| and |δ + μ| ≠ |μ|. By the assumptions |δ + λ| ≠ |λ| and |δ + μ| ≠ |μ|, the closed path

\begin{aligned}ω_{1} : & [0, 1] & → & M_{1} \\ & θ & ↦ & [[(δ + λ) (αa)^{θ} : (αā)^{θ} : (δ + μ) a^{θ} : ā^{θ}]] \end{aligned}

is contained in M_{1} \ E_{1} and represents σ_{1}. The closed path

\begin{aligned}ω_{2} : & [0, 1] & → & L_{ε} \\ & θ & ↦ & [δβ^{θ}, δβ^{θ}, β^{−θ}] \end{aligned}

represents σ_{2}.

Lemma 3.1 (ω_{1})^{q} is homotopic to zero in L_{ε}.

Proof. Note that the closed path (ω_{1})^{q} is homotopic in M_{1} \ E_{1} to the closed path

ω(θ) = [[(δ + λ) α^{qθ} : α^{qθ} : δ + μ : 1]], 0 ≤ θ ≤ 1

which is contained in the intersection of M_{1} \ E_{1} and L_{ε}. The path ψ_{η}^{−1} ∘ φ^{−1} ∘ ω is given by

[ δ exp(−2πiθ), δ exp(−2πiθ), exp(2πiθ) ], 0 ≤ θ ≤ 1,

which is homotopic to zero in L_{ε}.

Lemma 3.2 ω_{2} is homotopic to zero in M_{1} \ E_{1}.

Proof. The closed path π̄ ∘ φ ∘ ψ_{η} ∘ omega_{2} is geven by

[[ (δ + λ) exp(−2πiθ) : exp(−2πiθ) : δ + μ : 1 ]], 0 ≤ θ ≤ 1,

which is homotopic to zero in M_{1} \ E_{1}.

Proof of Proposition 3.2 By Lemmas 3.1 and 3.2, π_{1}(Z) is a quotient of Z_{q}. The manifold M_{q} is a q-fold unramified covering of M_{1}. Recall the unramified covering π̄ : M_{q} → M_{1}. By the construction, π̄^{−1}(E_{1}) is a disjoint union of q elliptic curves. Replace each of these elliptic curves by a copy of the Hopf surface S, then we obtain a manifold Z_{q} which is a q-fold unramified covering of Z. Hence π_{1}(Z) ≃ Z_{q}.

Summing up, we have

Theorem 3.1 For any p ∈ N, there is a compact complex 3-manifold Z of Class L such that π_{1}(Z) ≃ Z_{p}.

4

Let Sum(X, Y) denote the complex analytic connected sum of two 3-manifolds X, Y of Class L. Then Sum(X, Y) is also of Class L whose fundamental group π_{1}(Sum(X, Y)) is the free product of π_{1}(X) and π_{1}(Y). Suppose that we are given a finite sequence of positive integers p_{1}, p_{2}, …, p_{m}. Then, by the above theorem, we can find a compact complex 3-manifold Z of Class L satisfying π_{1}(Z) ≃ Z_{p_{1}} ∗ ⋯ ∗ Z_{p_{m}} considering successive complex analytic connected sums. In particular, we see that PSL(2, Z) ≃ Z_{2} ∗ Z_{3} can be realized as the fundamental group of a compact comples 3-manifold of Class L.

Using the same idea, we can construct compact complex 3-manifolds of Class L whose fundamental groups are binary regular polyhedra groups and/or their free products.

Department of Mathematics Sophia University Kioi-cho, Chiyodaku 102, Tokyo/ Japan

References

[1] Kato, Ma. : On compact complex 3-folds with lines, Japanse J. Math., 11(1985), 1-58.