Three-Dimensional Complex Homogeneous Complex Contact Manifolds

Brendan Foreman

Abstract In this paper, we classify all three-dimensional complex contact manifolds which have global complex contact forms and which are complex homogeneous with transitive groups of holomorphic contactomorphisms. We compare these results to previous results by W. Boothby and J. Wolf.

Mathematics Subject Classification: 53T58 Key words: complex contact manifolds, complex homogeneous manifolds.

1 Introduction

A complex contact manifold is a complex manifold M of complex dimension 2n + 1 with an open atlas U = {O} such that

1. On each O ∈ U, there is a holomorphic 1-form η with η ∧ (dη)^{n} ≠ 0 everywhere on O.

2. On O ∩ O′, there exists a holomorphic function f : O ∩ O′ → C^{∗} such that η = fη′.

In particular, H^{1,0} = ∪_{O∈U}(ker η) is a well-defined holomorphic subbundle of T^{1,0}M with maximal rank. We say that M is complex homogeneous, if there is a complex Lie group G acting transitively as a space of biholomorphic contactomorphisms (i.e. preserving H) on M. For a general reference on these types of manifolds, see [3] and [13].

In [4] and [5], W. Boothby classified all compact, simply-connected complex homogeneous complex contact manifolds. By later work of J. Wolf [20], S. Salamon [17], and B. Bérard-Bergery [1], it was found that this list consisted of the twistor spaces of homogeneous quaternionic-Kähler manifolds (See [2]). However, this list excluded a sizable portion of complex contact manifolds. In particular, those with global complex contact forms are left out, since their universal covers are not compact.

In this paper we will be classifying these manifolds. We will make heavy use of the fact that all homogeneous complex 2-manifolds and 3-manifolds have been classified in [16] and [19], respectively. In the first section, we discuss an important class of complex contact manifolds, the so-called complex Boothby-Wang fibrations. In the third section, we present the main results. Finally, we end the paper with some remarks.

Balkan Journal of Geometry and Its Applications, Vol.4, No.1, 1999, pp. 53-67 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2 Complex Boothby-Wang fibrations

In this section, we describe a newly discovered class of complex contact manifolds, called the complex Boothby-Wang fibrations. Recall that in [6], W. Boothby and H.C. Wang proved the following theorems.

Theorem 2.1 (Boothby-Wang Fibration, part i) Let B be a symplectic manifold with fundamental 2-form Ω such that [Ω] ∈ H^{2}(M, Z). Then the principal S^{1}- bundle M corresponding to [Ω] has a connection form η such that dη is the pull-back of Ω and η is a global real contact form on M.

Theorem 2.2 (Boothby-Wang Fibration, part ii) Let M be a compact, regular contact manifold with contact form η′. Then M is the total space of an principal S^{1}- bundle π : M → B, where B is a symplectic manifold with symplectic form Ω. Also, there is a nowhere-zero function τ on M such that η = τη′ is a global contact form on M and a connection form for the fibration such that dη = (π)^{∗}(Ω). Lastly, the free S^{1}-action on M is defined by the characteristic vector field ξ of η, given by the equations η(ξ) = 1 and ι(ξ)dη = 0.

The resulting real contact manifolds are called real Boothby-Wang fibrations. In [9], the author proved complex analogues of these results.

Theorem 2.3 (Complex Boothby-Wang Fibration, part i) Let B be a complexsymplectic manifold with a complex symplectic form Ω = Ω_{1} + iΩ_{2} such that both Ω_{1} and Ω_{2} are integral classes. Then M the S^{1} × S^{1} -bundle defined by ([Ω_{1}], [Ω_{2}]) ∈ H^{2}(B, Z) ⊕ H^{2}(B, Z) (or [Ω] ∈ H^{2}(B, Z + iZ)) has an integrable complex structure and also a complex contact structure given by a holomorphic connection form whose curvature form is given by ω.

Theorem 2.4 (Complex Boothby-Wang Fibration, part ii) Let M be a (2n+1)- dimensional compact complex contact manifold with a global holomorphic contact form π such that the resulting vertical vertical vector fields U and JU are regular in M. Then π generates a free S^{1} × S^{1}-action on M, and M is an principal S^{1} × S^{1}-bundle over a complex symplectic manifold B such that π is a connection form of this fibration and the symplectic form Ω on B is given by π^{∗}Ω = dπ.

In our context, we are curious as to which of the above manifolds (called complex Boothby-Wang Fibrations) are complex homogeneous. In particular, we need to know which complex symplectic surfaces are complex homogeneous.

In [10], H. Geiges classified all complex symplectic surfaces.

Theorem 2.5 (Geiges) Let M be a closed 4-manifold. Then M admits a complex symplectic structure if and only if M is diffeomorphic to one of the following manifolds:

1. A complex torus,

2. A primary Kodaira surface,

3. A K3 Surface. Furthermore, in [16], K. Oeljeklaus and W. Richthofer give a list of all complex homogeneous surfaces: Theorem 2.6 Let M be a complex homogeneous surface.

1. (Tits [18]) If M is compact, then it is diffeomorhphic to one of the following: (a) CP^{2}

(b) CP^{1} × CP^{1}

(c) A complex torus

(d) A homogeneous Hopf surface

(e) The product of an elliptic curve with CP^{1}

2. (Huckleberry, Livorni [11]) If M is non-compact, then it is one of the following:

(a) A product of complex homogeneous Riemann surfaces

(b) A topologically trivial C^{∗}-bundle over an elliptic curve

(c) An elliptic curve bundle over C

(d) A certain nontrivial C^{∗}-bundle over C^{∗}

(e) A C^{∗}-bundle over CP^{1}

(f) A positive line bundle over CP^{1}

(g) The affine quadric

(h) The complement of the quadric curve in CP^{2} with the affine quadric as its

universal cover

The intersection of these two lists is simply the set of all complex torii. In [9], the author shows that only a certain class of complex torii have complex Boothby-Wang fibrations and that the universal cover of all such fibrations is the complex Heisenberg group

HC = \{( \begin{matrix}1 & z_{2} & z_{1} \\ 0 & 1 & z_{3} \\ 0 & 0 & 1 \end{matrix} ) : z_{1}, z_{2}, z_{3} ∈ C\} .

The complex contact structure of this manifold is given by the left-invariant form ω = dz_{1} − z_{2}dz_{3}. For further details on this group, see [7].

3 Main theorem

Given a complex contact manifold M with a global complex contact form ω, we define U ∈ T^{1,0}M to be the unique holomorphic vector field given by the equations:

ω(U) = 1, ι(U)dω = 0.

We then let V = spanC(U). Then V is a holomorphic subbundle of T^{1,0}M, which is transerve to H, i.e. T^{1,0}M = V ⊕ H. In general, the author shows in [8] that such transerve subbundles can be constructed on any complex contact manifold, although the resulting “vertical bundle” V in these cases will not necessarily split T^{1,0} = V ⊕H holomorphically.

Theorem 3.1 If M is a three-dimensional complex-homogeneous complex contact manifold with global complex contact form, then M is of the form M = G/Γ where G is a simply-connected three-dimensional complex Lie group and Γ ⊂ G is a discrete subgroup.

1. Suppose G is unimodular. Then

(a) G = Sl(2, C), if rk(ad(V)) = 2,

(b) G = ẼC(2), the universal cover of EC(2) the rigid motions of the complex euclidean plane, if rk(ad(V)) = 1,

(c) G = HC, if rk(ad(V)) = 0.

2. Suppose G is not unimodular. Then G is necessarily solvable; rk(ad(V)) = 1; and G is one of the following complex Lie groups:

(a) The semidirect product G_{α} = C ×_{τ_{α}} C^{2}, for any α ∈ C^{∗} \ 1, where τ_{α} is a certain representation of C in Gl(2, C).

(b) G = \{( \begin{matrix}e^{t} & te^{t} & u \\ 0 & e^{t} & v \\ 0 & 0 & 1 \end{matrix} ) : t, u, v ∈ C\}

Proof. We prove this theorem in two lemmas. The first lemma will classify all leftinvariant complex contact structures on complex Lie groups; the last will show that there are no other possible three-dimensional complex-homogeneous complex contact manifolds with a global complex contact form.

Lemma 3.2 Suppose G is a three-dimensional, simply-connected complex Lie group with a left-invariant complex contact structure H. Then G is one of the complex Lie groups as described above.

Proof. Suppose G is a 3-dimensional complex Lie group with a left-invariant complex contact structure. Let g be the Lie algebra of G. Then there is a 2-dimensional complex subbundle H = ⟨e_{2}, e_{3}⟩ ⊂ g such that [e_{2}, e_{3}] = 2f_{1} ∉ H. Define the left-invariant 1-form η ∈ g^{∗} by η = \frac{1}{2}f_{1}^{∗} with respect to the dual basis \{f_{1}, e_{2}, e_{3}\} of g^{∗}. Then there is a unique left-invariant vector field U ∈ g such that η(U) = \frac{1}{2} and ι(U)dη = 0. And, thus, we have η = \frac{1}{2}U^{∗} with respect to the dual basis \{U^{∗}, e_{2}^{∗}, e_{3}^{∗}\}.

Set e_{1} = U. Then, for any Z ∈ g,

0 = (ι(e_{1})dη)(Z) = −\frac{1}{2}η([e_{1}, Z]).

Hence,

[e_{2}, e_{3}] = 2e_{1} − α_{1}e_{2} − β_{1}e_{3}
[e_{3}, e_{1}] =  α_{2}e_{2} + β_{2}e_{3}
[e_{1}, e_{2}] =  α_{3}e_{2} + β_{3}e_{3}.

An application of Jacobi’s identity gives that

α_{3} = β_{2}.

and also that

\{ \begin{matrix}0 & = & −α_{1}α_{3} & + & β_{1}α_{2} \\ 0 & = & −α_{1}β_{3} & + & β_{1}β_{2}. \end{matrix}

These last two equations can be put in the form

( \begin{matrix}−β_{2} & α_{2} \\ −β_{3} & β_{2} \end{matrix} ) ( \begin{matrix}α_{1} \\ β_{1} \end{matrix} ) = ( \begin{matrix}0 \\ 0 \end{matrix} ) .

Then we have

ad(e_{1}) = ( \begin{matrix}0 & 0 & 0 \\ 0 & −α_{2} & −β_{2} \\ 0 & β_{2} & β_{3} \end{matrix} ) .

We now split into the various possible cases of ad(e_{1}) according to its rank. Case 1: rank(ad(e_{1})) = 2.

In this case,

det ( \begin{matrix}−α_{2} & −β_{2} \\ β_{2} & β_{3} \end{matrix} ) ≠ 0,

so that the equation

( \begin{matrix}−α_{2} & −β_{2} \\ β_{2} & β_{3} \end{matrix} ) ( \begin{matrix}β_{1} \\ −α_{1} \end{matrix} ) = ( \begin{matrix}0 \\ 0 \end{matrix} )

has a unique solution, namely α_{1} = β_{1} = 0. So, we have

[e_{2}, e_{3}] = 2e_{1}
[e_{3}, e_{1}] =  α_{2}e_{2} + β_{2}e_{3}
[e_{1}, e_{2}] =  β_{2}e_{2} + β_{3}e_{3}.

Since det(ad(e_{1})) ≠ 0, we know that ad(e_{1}) has at least one eigenvector X with corresponding nonzero eigenvalue λ. Furthermore, ad(e_{1})g ⊂ ⟨e_{2}, e_{3}⟩ , so that X ∈ H. Also,

ad(e_{1}) = \frac{1}{2}ad([e_{2}, e_{3}])
= \frac{1}{2}ad(e_{2}) ∘ ad(e_{3}) − \frac{1}{2}ad(e_{3}) ∘ ad(e_{2}).

In particular, tr(ad(e_{1})) = \frac{1}{2}tr(ad(e_{2}) ∘ ad(e_{3}) − ad(e_{3}) ∘ ad(e_{2})) = 0. So, ad(e_{1}) has another eigenvector Y in H with eigenvalue −λ. By rescaling and renaming if necessary, we then have:

\begin{aligned}[e_{2}, e_{3}] & = & 2e_{1} \\ [e_{3}, e_{1}] & = & & & λe_{3} \\ [e_{1}, e_{2}] & = & & −λe_{2} \end{aligned} .

Thus, g is isomorphic to

sl(2, C) = ⟨( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , ( \begin{matrix}0 & 0 \\ \frac{λ}{2} & 0 \end{matrix} ) , ( \begin{matrix}0 & \frac{λ}{2} \\ 0 & 0 \end{matrix} )⟩ .

Thus, G = Sl(2, C) and hence is unimodular. Case 2: rank(ad(e_{2})) = 1

In this case,

det ( \begin{matrix}−α_{2} & −β_{2} \\ β_{2} & β_{3} \end{matrix} ) = 0,

yet one of the rows is non-zero. Let us assume that ( \begin{matrix}β_{2} \\ β_{3} \end{matrix} ) ≠ ( \begin{matrix}0 \\ 0 \end{matrix} ) . Then

( \begin{matrix}−α_{2} \\ −β_{2} \end{matrix} ) = k ( \begin{matrix}β_{2} \\ β_{3} \end{matrix} ) ,

so that −α_{2} = kβ_{2} = k^{2}β_{3}. Also,

( \begin{matrix}0 \\ 0 \end{matrix} ) = ( \begin{matrix}−kβ_{3} & −k^{2}β_{3} \\ β_{3} & kβ_{3} \end{matrix} ) ( \begin{matrix}α_{1} \\ β_{1} \end{matrix} )
= β_{3} ( \begin{matrix}−k(α_{1} + kβ_{1}) \\ α_{1} + kβ_{1} \end{matrix} ) .

Thus, 0 = α_{1} + kβ_{1}.

Hence, g has the following Lie algebra structure:

[e_{2}, e_{3}] = 2e_{1} − kβ_{1}e_{2} + β_{1}e_{3}
[e_{3}, e_{1}] =  k^{2}β_{3}e_{2} + −kβ_{3}e_{3}
[e_{1}, e_{2}] =  −kβ_{3}e_{2} + β_{3}e_{3}.

with β_{3} ≠ 0. Thus, [g, g] = ⟨e_{1}, −ke_{2} + e_{3}⟩ . Clearly, if we choose e_{2} and e_{3} well enough, we can assume k = 0. This will give us this Lie algebra structure:

[e_{2}, e_{3}] = 2e_{1}  − β_{1}e_{3}
[e_{3}, e_{1}] = 0
[e_{1}, e_{2}] =  β_{3}e_{3}.

Set f_{1} = 2e_{1} + β_{1}e_{3}, f_{3} = β_{3}e_{3}. So, [g, g] = ⟨f_{1}, f_{3}⟩ . It is then easily checked that

\begin{aligned}[e_{2}, f_{1}] & = & β_{1}f_{1} & − & 2f_{3} \\ [e_{2}, f_{3}] & = & β_{3}f_{1} \\ [f_{1}, f_{3}] & = & 0 \end{aligned} ,

so that, with respect to the basis {f_{1}, f_{3}}, ad(e_{2}) on [g, g] is given by the matrix

T̃ = ( \begin{matrix}β_{1} & β_{3} \\ −2 & 0 \end{matrix} ) .

The characteristic polynomial p(X) of T̃ is given by p(X) = X^{2} − β_{1}X + 2β_{3}. Let γ_{1} and γ_{2} be the roots of p. We then have two possibilities:

1. γ_{1} = γ_{2}.

2. γ_{1} ≠ γ_{2}.

Subcase 1: Suppose γ_{1} = γ_{2} = γ.

Then β_{1} = −2γ and γ^{2} = 2β_{3}. Thus,

T̃ = ( \begin{matrix}−2γ & \frac{1}{2}γ^{2} \\ −2 & 0 \end{matrix} ) .

Set f_{2} = \frac{1}{γ} e_{2}. Then with respect to \{f_{1}, f_{3}\}, ad(f_{2}) on [g, g] is given by the matrix

T = ( \begin{matrix}−2 & \frac{1}{2}γ \\ −\frac{2}{γ} & 0 \end{matrix} ) .

The characteristic polynomial of this matrix is q(X) = (X − 1)^{2}. Furthermore, the eigenspace of ad(f_{2}) corresponding to 1 is given by:

E_{1} = ⟨− (\frac{γ}{2} ) f_{1} + f_{3}⟩ .

Finally, it is easily checked that

ad(f_{2})(− (\frac{γ}{2} ) f_{1}) = − (\frac{γ}{2} ) f_{1} + (− (\frac{γ}{2} ) f_{1} + f_{3}) .

In particular, if we set

v_{1} = − (\frac{γ}{2} ) f_{1} + f_{3}
v_{2} = f_{2}
v_{3} = − (\frac{γ}{2} ) f_{1},

then

[v_{1}, v_{2}] = v_{1},
[v_{2}, v_{3}] = v_{1} + v_{3},
[v_{1}, v_{3}] = 0.

Furthermore,

v_{1} = −γe_{1} + \frac{3}{2}γ^{2}e_{3},
v_{2} = \frac{1}{γ} e_{2},
v_{3} = \frac{1}{2}γ^{2}e_{3}.

Thus, H = ⟨v_{2}, v_{3}⟩ .

Now, it easily checked that the Lie algebra given by

g = \{( \begin{matrix}x & x & y + z \\ 0 & x & z \\ 0 & 0 & 0 \end{matrix} ) : x, y, z ∈ C\}

satisfies the above criterion with

v_{1} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) ,

v_{2} = ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} ) ,

v_{3} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) .

Furthermore, having no center, the Lie group

G = \{( \begin{matrix}e^{t} & te^{t} & u \\ 0 & e^{t} & v \\ 0 & 0 & 1 \end{matrix} ) : t, u, v ∈ C\}

is the unique connected complex Lie algebra having gas its Lie algebra. Note that, using {v_{1}, v_{2}, v_{3}} as above, we have that tr(ad(v_{1})) = 1, meaning that G is not unimodular (by theorem in [15]) Case 2 Suppose γ_{1} ≠ γ_{2}.

The characteristic polynomial p(X) of ad(e_{2}) is then given by p(X) = (X−γ_{1})(X− γ_{2}), i.e. β_{1} = γ_{1} + γ_{2} and 2β = γ_{1}γ_{2}.

Set f_{2} = \frac{1}{γ_{1}} e_{2}. Then, with respect to the basis \{f_{1}, f_{3}\}, ad(f_{2}) on [g, g] is given

by the matrix

T = ( \begin{matrix}\frac{γ_{1} + γ_{2}}{γ_{1}} & \frac{γ_{1}}{2} \\ −\frac{2}{γ_{1}} & 0 \end{matrix} ) .

The characteristic polynomial of this matrix is given by p(X) = (X − 1)(X − α), where α = \frac{γ_{1}}{γ_{2}} . Furthermore, the eigenspaces of ad(f_{2}) are given by

E_{1} = ⟨− (\frac{γ_{1}}{2} ) f_{1} + f_{3}⟩ , E_{α} = ⟨− (\frac{γ_{2}}{2} ) f_{1} + f_{3}⟩ .

Set

\begin{aligned}v_{1} & = & f_{1} − (\frac{2}{γ_{1}} ) f_{3} ∈ E_{1} \\ v_{2} & = & −f_{1} + (\frac{2}{γ_{2}} ) f_{3} ∈ E_{α} \\ v_{3} & = & f_{2}. \end{aligned}

Then

[v_{3}, v_{1}] = v_{1}
[v_{3}, v_{2}] = αv_{2}
[v_{1}, v_{2}] = 0.

Finally, H = ⟨v_{1} + v_{2}, v_{3}⟩ .

For any α ∈ C, not equal to 1, define the action τ_{α} : C → Gl(2, C) by

t ↦ ( \begin{matrix}e^{−t} & 0 \\ 0 & e^{−αt} \end{matrix} ) .

Let G_{α} = C ×_{τ_{α}} C^{2} be the semi-direct product, whose multplication is given by

(t_{1}, w_{1}) · (t_{2}, w_{2}) = (t_{1} + t_{2}, w_{1} + τ_{α}(t_{1})w_{2}) for t_{1}, t_{2} ∈ C, w_{1}, w_{2} ∈ C^{2}.

Set

v_{1} = \frac{d}{dt} (t, ( \begin{matrix}0 \\ 0 \end{matrix} ))_{|t=0} ,

v_{2} = \frac{d}{dt} (0, ( \begin{matrix}t \\ 0 \end{matrix} ))_{|t=0} ,

v_{3} = \frac{d}{dt} (0, ( \begin{matrix}0 \\ t \end{matrix} ))_{|t=0} .

Then it is easily checked that the Lie algebra g_{α} of G_{α} is given by g_{α} = ⟨v_{1}, v_{2}, v_{3}⟩ that v_{1}, v_{2}, and v_{3} satisfy the Lie brackets given in the previous paragraph. Lastly note that

tr(ad(v_{1})) = 0
tr(ad(v_{2})) = 0
tr(ad(v_{3})) = α.

So, g_{α} is unimodular if and only if α = −1, in which case G_{α} is the universal cover of E(2, C), the space of rigid motions of the complex euclidean plane. Case 3: rank(ad(e_{1})) = 0

In this case, α_{2} = β_{2} = β_{3} = 0, so that g is the Lie algebra given by:

\begin{aligned}[e_{2}, e_{3}] & = & 2e_{1} \\ [e_{3}, e_{1}] & = & 0 \\ [e_{1}, e_{2}] & = & 0. \end{aligned}

That is, G is the complex Heisenberg group.

Lemma 3.3 Suppose M is a three-dimensional complex-homogeneous complex contact manifold with global complex contact form η. Then M = G/Γ for some complex Lie group G with discrete subgroup Γ ⊂ G such that G preserves η.

Proof. Let G be a complex Lie group of complex-contactomorphisms of M whose action on M is transitive. Let G = S ×_{f} R be the Levi decomposition of G where S is a semi-simple Lie group acting on a solvable group R via a representation f : S → Aut(R). Then M = G/L for some closed subgroup L ⊂ G. Case 1. Suppose G is semi-simple, i.e. R = (0).

By the proof of the theorem A on page 147 in [4], we know that being semi-simple, G preserves the global contact form η and hence also dη. Setting p : G → M = G/L to be the obvious submersion, we have the standard statements for this situation:

1. If ω is a G-invariant form on G/L, then ω^{∗} = p^{∗}(ω) is an element of g^{∗} such that:

(a) ω^{∗}(l) = 0,

(b) ad(x)ω^{∗} = ω^{∗} for all x ∈ l.

2. Conversely from the above statements, any form in g^{∗} satisfying conditions a) and b) is the image under p^{∗} of a G-invariant form on G/L.

3. If ω is a G-invariant form on G/L with ω^{∗} = p^{∗}ω, then (dω)^{k} ≠ 0 if and only if (dω^{∗})^{k} ≠ 0.

Let (,) denote the Killing-Cartan form of g. Then there exists a unique Z ∈ g such that ω^{∗}(X) = (Z, X). Set c(Z) = {X ∈ g : [X, Z] = 0}. For any X, Y ∈ g,

dω^{∗}(X, Y) = −\frac{1}{2}ω^{∗}([X, Y])
= −\frac{1}{2}(Z, [X, Y])
= −\frac{1}{2}([Z, X], Y).

So, ι(X)dω^{∗} = 0 if and only if [Z, X] = 0 (since (,) is nondegenerate). Hence, c(Z) = {X ∈ g : ι(X)dω^{∗} = 0}. Furthermore, l ⊂ c(Z).

Set N = dimCG so that dimCL = N − 3. We know that dω^{∗} ≠ 0 and that (dω^{∗})^{2} = 0 by statement b) above. So, with respect to some dual basis {e_{1}^{∗}, …, e_{N}^{∗}} of g^{∗}, dω^{∗} = e_{1}^{∗} ∧ e_{2}^{∗}, which means that c(Z) = ⟨e_{3}, …, e_{N}⟩C. i.e. dimCc(Z) = dimCG − 2. Therefore, c(Z) = l ⊕ ⟨Z⟩C.

With respect to the nilpotent subalgebra ⟨Z⟩C ⊂ g, g has the root decomposition

g = \underbrace{⟨Z⟩C ⊕ l}_{=g_{0}} ⊕g_{λ} ⊕ g_{−λ},

for nonzero root λ ∈ (⟨Z⟩C)^{∗}, where g_{λ} and g_{−λ} are the one-dimensional subspaces of g satisfying:

1. [g_{0}, g_{λ}] ⊂ g_{λ},

2. [g_{0}, g_{−λ}] ⊂ g_{−λ},

3. [Z, l ] = 0, 4. [g_{λ}, g_{−λ}] ∩ ⟨Z⟩ ≠ (0).

Since both of the nonzero root spaces are one-dimensional, we can set g_{λ} = ⟨X_{1}⟩ and g_{−λ} = ⟨X_{2}⟩. Then, for some a ≠ 0 and W ∈ l,

[X_{1}, X_{2}] = aZ + W
[X_{1}, Z] = λ(Z)X_{1}
[X_{2}, Z] = −λ(Z)X_{2}.

Since [g_{0}, g_{±λ}] ⊂ g_{±λ}, we also know that there are some numbers μ_{1} and μ_{2} such that

[X_{1}, W] = μ_{1}X_{1},
[X_{2}, W] = μ_{2}X_{2}.

However, an easy application of the Jacobi identity tells us that μ_{2} = −μ_{1}. Thus, by adjusting and renaming the coefficients properly, we have

[X_{1}, Z] = λX_{1}, [X_{2}, Z] = −λX_{2},
[X_{1}, X_{2}] = Z + W, [Z, W] = 0,
[X_{1}, W] = μX_{1}, [X_{2}, W] = −μX_{2}.

In particular, ⟨X_{1}, X_{2}, Z, W ⟩ is a Lie subalgebra of g, and we can thus assume that G is 4-dimensional with gas described above.

We also note that h = ⟨Z, W ⟩ is a nilpotent subalgebra of g, which induces a root space decomposition of g given by:

g = g_{0} ⊕ g_{ν} ⊕ g_{−ν},

where ν ∈ h^{∗} is given by the equations:

ν(Z) = λ, ν(W) = μ,

and g_{0} = h, g_{ν} = g_{λ}, g_{−ν} = g_{−λ}. Thus, h is a Cartan subalgebra of g. However, it is known that for a Cartan subalgebra h, the set of nonzero roots Δ of h spans h^{∗} (For reference, see [11]). In our case, the set of nonzero roots consists of {±ν}, so that h^{∗} is one-dimensional. Thus, L ⊂ G is a discrete subgroup, and the universal cover G of M is a three-dimensional semi-simple Lie group. Case 2. Suppose G is solvable, i.e. S = (0).

To prove the lemma for this case, we need to make use of the following proposition due to Winkelmann [18]. Proposition 3.4. Let G be a solvable, connected Lie group acting transitively on a complex manifold X with dimCX ≤ 3. Then there exists a solvable, connected complex Lie group K acting transitively on X with dimCK = dimCX.

Thus, in our situation, we know that M = K/Γ for some solvable complex Lie group K and some discrete subgroup Γ ⊂ K. The problem is that it is not clear from the proof of this proposition that the original complex contact structure is preserved by this group K. In fact, in general, it won’t be. In order to remedy this, we will construct a possibly new complex contact structure which is preserved by K.

We assume that M = K. At e ∈ M, we choose linearly independent vectors X, Y ∈ H_{e}^{1,0}. We then extend these to form linearly independent holomorphic vector fields in H^{1,0} near e. Also, let X′ and Y′ be the K-left-invariant holomorphic vector fields on K given by X′_{e} = X_{e}, Y′_{e} = Y_{e}. So, we have a K-left-invariant subbundle H′ = ⟨X′, Y′⟩ in T^{1,0}M.

Both H^{1,0} and H′ are 2-dimensional subspaces of a 3-dimensional vector space, so that H^{1,0} ∩ H′ ≠ (0) at each point on M. Thus, we can assume that locally H^{1,0} ∩ H′ = ⟨X⟩ , i.e. X = X′.

Since H^{1,0} is a complex contact structure, we know that [X, Y] = W ∉ H^{1,0}. So, {X, Y, W } is a local holomorphic basis of T^{1,0}M. Also, since we’re assuming X = X′, we can also assume that Y′ = Y + ϵW for some local holomorphic function ϵ with ϵ(e) = 0. Furthermore,

[X, Y′] − [X, Y] = [X, ϵW] = (Xϵ)W + ϵ[X, W].

In particular, [X, Y′]_{e} − [X, Y]_{e} = (Xϵ)(e)W_{e}. So, [X, Y′]_{e} ∈ ⟨W ⟩ . Since H′_{e} = H^{1,0}, we see that [X, Y′] ∉ H′. But [X, Y′] and H′ are both K-left-invariant structures on M. So, [X, Y′] ∉ H′ at all points on M. Thus, H′ is a K-left-invariant complex contact structure on M = K. Case 3. Suppose G is mixed, i.e. S ≠ (0) ≠ R.

By Winkelmann’s list in [19], we know that all three-dimensional complex homogeneous manifolds in this category are C, C^{∗}, or C^{2} bundles over complex homogeneous manifolds. Let π : M → N be the corresponding bundle map in our case with structure group K ⊂ G. Since this map π preserves the actions of G on M and N, we know that ker(π_{∗}) is G-invariant. Set H′ = H ∩ ker(π_{∗}). So, H′ is a K-invariant subbundle of M.

We have three possibilities:

1. H ∩| ker(π_{∗}),

2. H ⊂ ker(π_{∗}),

3. H ∩ ker(π_{∗}) ≠ ∅, but H ⊄ ker(π_{∗}).

If (1) were true, then M would be a complex Boothby-Wang fibration over a complex-homogeneous complex symplectic manifold, which as we have seen must be a complex torus. In particular, the universal cover of M is the complex Heisenberg group. If (2) were true, then by dimension-counting, H = ker(π_{∗}). This is clearly impossible, since the vertical subbundle of any submersion is integrable.

Finally, we come to case (3). There exists a K-connection D of the fibration M → N such that H′′ = D ∩ H ≠ (0). Then H = H′ ⊕ H′′. Let X ∈ H′, Y ∈ H′′ be local K-invariant vector fields. Then π_{∗}(X) = 0, and π_{∗}(Y) is a well-defined vector field on N. Thus, [π_{∗}(X), π_{∗}(Y)] = π_{∗}[X, Y] = 0. So, [X, Y] ∈ ker(π_{∗}). This means that ker(π_{∗}) is 2-dimensional. By Winkelmann’s list, then, we know that M is the C^{2}-bundle over CP^{1} given by the transition functions:

w_{1} = −w_{0} (\frac{z_{0}}{z_{1}} )^{n}

v_{1} = v_{0} (\frac{z_{0}}{z_{1}} )^{np+n−2} − w_{0}^{p+1} (\frac{z_{0}}{z_{1}} )^{np+n−1} .

Furthermore, although the subbundle V = ⟨U⟩ may not live in ker(π_{∗}), we can set pr : T M → ker(π_{∗}) to be the holomorphic projection with respect to the holomorphic splitting T M = ker(π_{∗}) ⊕ D. Then we will have that U′ = pr(U) is a nowhere-zero holomorphic vector field in ker(π_{∗}), which can be taken as a nowhere-zero holomorphic section of M as a C^{2}-bundle over CP^{1}. This is a contradiction, since it is known that this bundle has no holomorphic sections. Thus, we have proven our lemma.

4 Final remarks

Given a 3-dimensional complex contact manifold M with local complex contact forms {ω}, we can define a C^{∗}-bundle B over M locally by

B = {fω : f ∈ C^{∗}} ⊂ Λ^{1,0}M.

Let π : B → M be the projection. It is well-known that there exists a globally defined 1-form Ω on B such that

1. (dΩ)^{2} ≠ 0.

2. Ω = 0 on vectors tangent to the fibres created by the fibration B → M.

3. (R_{z})^{∗}Ω = zΩ for all z ∈ C^{∗}, where R_{z} is the right-handed C^{∗}-action on B.

Finally, it also well-known that any contactomorphism of M determines a unique biholomorphism of B which (1) commutes with right translations, and (2) preserves Ω. Conversely, any biholomorphism of B satisfying these conditions determines a unique contactomorphism of M. For a given x ∈ M and b ∈ π^{−1}(x), set L = {g ∈ G : g(x) = x} and L_{1} = {g ∈ G : g(b) = b} . Then L_{1} ⊂ L; and either L = L_{1} or L \ L_{1} = C^{∗}. All of these facts are documented in [4]. It is clear that M has a global complex contact form if and only if B is holomorphically trivial.

Using this terminology, our main theorem gives us the following corollary.

Corollary 4.1 Let M be a three-dimensional complex contact manifold with transitive group of holomorphic contactomorphisms G. If B is trivial, then (1) G preserves each global complex contact form and hence (2) the subgroup of G preserving a fibre of B over M acts trivially on that fibre.

This corollary allows us to extend one of Boothby’s results, using a similar proof as he does in [4].

Theorem 4.2 Let M be a three-dimensional complex homogeneous complex contact manifold with transitive group of holomorphic contactomorphisms G. Then there are two possible cases:

1. The following equivalent statments hold:

(a) G is transitive on B;

(b) B is non-trivial as a bundle over M;

(c) L \ L_{1} = C^{∗}; 2. The following equivalent statments hold:

(a) The subset of G preserving a fibre of B acts as the identity on that fibre;

(b) B is trivial as a bundle over M;

(c) L = L_{1}.

Boothby’s original theorem assumed that M was compact and simply-connected and also that G was semi-simple. Furthermore, in [5], Boothby showed that under these same circumstances, the second category in his theorem is null. It is unknown to the author whether there are any known examples of manifolds in category (1) which are not homogeneous twistor spaces.

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Case Western Reserve University ^{1}

1Research supported by NSF grant # DMS95-20481