Proceedings Ist International Meeting on Geometry and Topology Braga (Portugal)

Public. Centro de Matemática da Universidade do Minho p. 207–221, 1998

The holomorphic fiber bundle structure of some compact complex nilmanifolds

Luis A. Cordero, Marisa Fernández, Alfred Gray and \underline{Luis Ugarte}

Abstract:

Strongly non-nilpotent complex structures on compact complex nilmanifolds Γ\G are defined and characterized. As an application, a compact complex nilmanifold Γ\G with a non-nilpotent complex structure which is not strongly nonnilpotent is described as the total space at the top in a tower of holomorphic principal torus bundles. This description extends the one given in [CFGU1] for the case of Γ\G possessing a nilpotent complex structure. The problem of computing the Dolbeault cohomology of Γ\G is also discussed.

1. Introduction

A compact nilmanifold is a homogeneous space Γ\G, where G is a simply– connected connected nilpotent Lie group and Γ a lattice of G of maximal rank. Nomizu [N] has proved that the qth de Rham cohomology group H^{q}(Γ\G) of Γ\G is canonically isomorphic to the Chevalley–Eilenberg cohomology group H^{q}(g), g denoting the Lie algebra of G. A key ingredient in Nomizu’s proof is the fact that Γ\G can be seen as the total space of a principal torus bundle over another compact nilmanifold.

In [CFG] the ascending central series {g_{l}}_{l≥0} of g has been used to describe Γ\G as the total space at the top in a tower of principal torus bundles. Let us recall that each term in {g_{l}}_{l≥0} is an ideal of g defined by

g_{0} = {0}, g_{l} = {X ∈ g | [X, g] ⊆ g_{l−1}}, l ≥ 1. (1)

This work has been partially supported through grants DGICYT (Spain), Projects PB-94-0633-C02-01 and PB-94-0633-C02-02, and through grants UPV, Projects 127.310-EC 248/96 and 127.310-EA 043/97

208 L. A. Cordero, M.Fernández, A. Gray and L. Ugarte

Thus g_{1} is the center of g, and if g is s-step nilpotent then g_{0} ≠ g_{1} ≠ ⋯ ≠ g_{s−1} ≠ g_{s} = g_{l} = g, for all l ≥ s. The tower of principal fiberings obtained in [CFG] has s − 1 steps.

Since Γ\G is a parallelizable manifold, it is possible to define geometric structures on it in a natural way. For example, if the real dimension of G is even an almost complex structure J on Γ\G can be defined by passing to the quotient a left invariant almost complex structure on G. We say that Γ\G is a compact complex nilmanifold if it possesses an integrable almost complex structure J defined in this way. (It is worthy to remark here that there are even dimensional nilpotent Lie groups with no left invariant complex structures [FGG], and hence the compact manifolds Γ\G constructed from them are never compact complex nilmanifolds.)

Such a complex structure J on Γ\G can be thought as an endomorphism of g satisfying J^{2} = −I and the integrability condition given by the vanishing of its Nijenhuis tensor. But if g is not a complex Lie algebra then J may not leave invariant some term neither of the descending central series nor of the ascending central series of g. Therefore, if G is not a complex Lie group these central series cannot be used in general to obtain a description of Γ\G as a total space of a holomorphic principal torus bundle.

Such a fiber structure of Γ\G has been obtained by Sakane for the case of G being a complex Lie group [S]. (In this case Γ\G is complex parallelizable in the sense of Wang [W].) Actually, Sakane has exploited this fact to prove a result similar to Nomizu’s theorem for the Dolbeault cohomology H∗,∗ ; ¯∂(Γ\G) of Γ\G. Recently, a variation of (1) adapted to the complex structure J of G has been introduced (see [CFGU1]) in order to get a description of a larger class of compact complex nilmanifolds Γ\G, namely the class of compact complex nilmanifolds whose complex structure is nilpotent (see Definition 1), as the total space at the top in a tower of holomorphic principal torus bundles. As an application, this description of Γ\G is used in [CFGU1] to extend Sakane’s result for H∗,∗ ; ¯∂(Γ\G) to any nilpotent complex structure J on a compact nilmanifold Γ\G.

In the present paper we discuss the situation not considered in [CFGU1], that is the case of J being non-nilpotent. The concept of strongly non-nilpotent complex structure on Γ\G is introduced (see Definition 2), and the following result is proved (Theorem 9): if Γ\G is a compact complex nilmanifold whose complex structure J stems from a non-nilpotent left invariant complex structure which is not strongly non-nilpotent, then it can be realized as the total space at the top in a tower of holomorphic principal bundles whose structure groups are all complex tori; moreover, each base space in this tower is a compact nilmanifold endowed with a non-nilpotent complex structure, and the complex structure on the base space at the bottom of the tower is always strongly non-nilpotent.

The holomorphic fiber bundle structure 209

The paper is structured as follows. In Section 2 we show some properties of left invariant complex structures J on nilpotent Lie groups and, in Theorem 4, a result of [CFGU2] concerning with a necessary and sufficient condition in dimension 6 for J to be nilpotent is extended to any dimension 2n. Section 3 is devoted to the construction of examples of compact complex nilmanifolds with non-nilpotent complex structure; moreover, these examples show that the hypothesis in Theorem 4 cannot be weakened. In Section 4, following the steps of [CFGU1], we obtain the desired fiber structure for any compact complex nilmanifold whose complex structure is not strongly non-nilpotent. As an application, the general problem of computing a model for the Dolbeault complex of any compact complex nilmanifold is reduced to solve the problem for the particular class of strongly non-nilpotent complex structures.

2. Nilpotent complex structures

In this section we study necessary and sufficient conditions for a left invariant complex structure on a s-step nilpotent Lie group G to be nilpotent. In particular, we extend to dimension 2n a result given in [CFGU2] for dimension 6.

Let g be the Lie algebra of G, and J a left invariant complex structure on G. In [CFGU1] it is introduced the ascending series {a_{l}(J)}_{l≥0} of g compatible with J, whose terms a_{l}(J) are defined by:

\{ \begin{matrix}a_{0}(J) & = \{0\} , \\ a_{l}(J) & = \{X ∈ g | [X, g] ⊆ a_{l−1}(J) and [JX, g] ⊆ a_{l−1}(J)\} , l ≥ 1. \end{matrix}

Notice that each term a_{l}(J) in this series is a J-invariant ideal of g. In particular, a_{1}(J) is the largest subspace of the center g_{1} of g which is invariant under J.

Definition 1. The complex structure J is called nilpotent if a_{t}(J) = g for some integer t > 0.

Suppose that J is nilpotent, and let t denote the first positive integer for which a_{t}(J) = g. Then it is easy to see that {a_{l}(J)}_{l≥0} satisfies

0 ≠ a_{1}(J) ≠ ⋯ ≠ a_{t−1}(J) ≠ a_{t}(J) = a_{l}(J) = g,

for all l ≥ t. A characterization of a nilpotent complex structure in terms of the structure equations of the Lie group is given in the following

Theorem 1. [CFGU1] Let G be a nilpotent Lie group of real dimension 2n. A left invariant complex structure J on G is nilpotent if and only if there exists an ordered basis \{ω_{i}, \overline{ω}_{i}; 1 ≤ i ≤ n\} of left invariant complex 1-forms on G satisfying

dω_{i} = ∑_{j<k<i} A_{ijk} ω_{j} ∧ ω_{k} + ∑_{j,k<i} B_{ijk} ω_{j} ∧ \overline{ω}_{k} ,

for 1 ≤ i ≤ n, where each ω_{i} is of type (1, 0) with respect to J.

The only 2-dimensional nilpotent Lie group is, up to isomorphism, the Abelian Lie group IR^{2}, and then any left invariant complex structure on it is nilpotent. If a s-step nilpotent Lie group G of real dimension 4 has a left invariant complex structure then s must be ≤ 2 [FGG]. Moreover, there are two nilpotent Lie groups of dimension 4 admitting left invariant complex structures: IR^{4} and H ×IR, where H is the Heisenberg group.

Theorem 2. [CFGU2] Let G be a simply–connected Lie group, of real dimension 4, nilpotent in step s ≤ 2. Then, any left invariant complex structure on G is nilpotent.

As the examples in Section 3 show, Theorem 2 does not hold in dimension 2n > 4.

Let us now suppose that a_{q}(J) ≠ g for some q. Since a_{q}(J) is an ideal of g, then g̃_{q} = g/a_{q}(J) is a nilpotent Lie algebra. Let G̃_{q} be the simply–connected connected nilpotent Lie group whose Lie algebra is g̃_{q}. Since a_{q}(J) is invariant under J, then there is an induced left invariant almost complex structure J̃_{q} on G̃_{q} defined by: J̃_{q}(X̃) = J̃X, (1)

for X̃ ∈ g̃_{q}, where X̃ and J̃X denote the class of X and JX, respectively, in the quotient g̃_{q}. Notice that J̃_{q} is integrable, and hence it defines a complex structure on G̃_{q}, because NJ̃_{q} = Ñ_{J} and J is integrable, NJ̃_{q} and N_{J} denoting the Nijenhuis tensors of J̃_{q} and J, respectively.

Lemma 3. With the notation given above, let {a_{l}(J̃_{q})}_{l≥0} be the ascending series of g̃_{q} compatible with J̃_{q}. Then

a_{l}(J̃_{q}) ≅ \frac{a_{q+l}(J)}{a_{q}(J)} , for all l ≥ 0. (2)

Proof.

Since (2) is obvious if a_{q}(J) = 0, assume that 0 ≠ a_{q}(J) ≠ g. Then, (2) being trivial for l = 0, we proceed by induction on l. Thus, from a_{l−1}(J̃_{q}) ≅ a_{q+l−1}(J)/a_{q}(J), (1) and the definition of a_{l}(J̃_{q}), we get

a_{l}(J̃_{q}) = {X̃ ∈ g̃_{q} | [X̃, g̃_{q}] ⊆ a_{l−1}(J̃_{q}) and [J̃_{q}(X̃), g̃_{q}] ⊆ a_{l−1}(J̃_{q})}
= {X̃ ∈ g̃_{q} | [X, g] ⊆ a_{q+l−1}(J) and [JX, g] ⊆ a_{q+l−1}(J)}
≅ a_{q+l}(J)/a_{q}(J),

and, therefore, (2) holds for all l ≥ 0. □

Theorem 4. Let Γ\G be a compact complex nilmanifold, of real dimension 2n > 4, with complex structure J. If a_{n−3}(J) ≠ a_{n−2}(J) then J is nilpotent. Moreover, the converse holds if G is s-step nilpotent with s > n − 3.

Proof.

First, we note that a_{t}(J) = a_{t+1}(J) for some t ≥ 0 implies that a_{l}(J) = a_{t}(J) for all l ≥ t. Therefore, since a_{n−3}(J) ≠ a_{n−2}(J) then

0 ≠ a_{1}(J) ≠ ⋯ ≠ a_{n−3}(J) ≠ a_{n−2}(J). (3)

Next, since a_{l}(J) is invariant under J then the real dimension of a_{l}(J) is even. Hence, from (3) it follows that dim a_{l}(J) ≥ 2l, for l = 0, …, n − 2. In particular, dim a_{n−2}(J) ≥ 2n − 4, the quotient Lie algebra g̃_{n−2} = g/a_{n−2}(J) has real dimension ≤ 2n − (2n − 4) = 4, and the induced almost complex structure J̃_{n−2} given by (1) is integrable on its corresponding Lie group G̃_{n−2}. Thus, there are two possibilities:

dim g̃_{n−2} = 4 or dim g̃_{n−2} = 2.

If dim g̃_{n−2} = 4, there are again two possibilities:

G̃_{n−2} ≅ IR^{4} or G̃_{n−2} ≅ H × IR,

where H is the Heisenberg group. Therefore, from Theorem 2 it follows that any left invariant complex structure on G̃_{n−2} is nilpotent. In particular, using Lemma 3 we get a_{n−1}(J)/a_{n−2}(J) ≅ a_{1}(J̃_{n−2}) ≠ 0, that is, a_{n−2}(J) ≠ a_{n−1}(J). If a_{n−1}(J) = g then J is nilpotent on G. Otherwise, g̃_{n−1} is a Lie algebra of real dimension 2, and the induced almost complex structure J̃_{n−1} is integrable on the Lie group G̃_{n−1} ≅ IR^{2}. Since J̃_{n−1} is nilpotent we have from Lemma 3 that a_{n}(J)/a_{n−1}(J) ≅ a_{1}(J̃_{n−1}) ≠ 0, that is, a_{n−1}(J) ≠ a_{n}(J) = g because dim a_{n}(J) ≥ 2n = dim g. Therefore, the complex structure J is nilpotent on G.

If dim g̃_{n−2} = 2, then G̃_{n−2} ≅ IR^{2}. Since J̃_{n−2} is nilpotent, taking into account once more Lemma 3, it follows that a_{n−1}(J)/a_{n−2}(J) ≅ a_{1}(J̃_{n−2}) ≠ 0. Thus, by obvious dimensional reasons, a_{n−2}(J) ≠ a_{n−1}(J) = g and, therefore, the complex structure J is still nilpotent on G.

Let us suppose now that J is a nilpotent complex structure on a s-step nilpotent Lie group G, with s > n − 3. Since g is not nilpotent in step ≤ n − 3 then g ≠ g_{n−3}, where g_{n−3} is the (n − 3)-th term in the ascending central series {g_{l}} of g. Moreover, a_{n−3}(J) ⊆ g_{n−3} and therefore a_{n−3}(J) ≠ g. The nilpotency of J insures the existence of an integer t ≥ n − 2 such that a_{t}(J) = g, i.e. a_{n−3}(J) ≠ a_{t}(J) = g. In particular, a_{n−3}(J) ≠ a_{n−2}(J). □

As a direct consequence of this theorem we get a simple necessary and sufficient condition for J to be nilpotent in (real) dimension 6 .

Corollary 5. [CFGU2] Let M^{6} = Γ\G be a compact complex nilmanifold, of (real) dimension 6, with complex structure J. Then, J is nilpotent if and only if a_{1}(J) ≠ 0.

3. Non-nilpotent complex structures

In this section we construct, for each n ≥ 3, a nilpotent Lie group K^{2n} endowed with a non-nilpotent (left invariant) complex structure J satisfying

0 ≠ a_{1}(J) ≠ a_{2}(J) ≠ ⋯ ≠ a_{n−4}(J) ≠ a_{n−3}(J) = a_{l}(J) ≠ g, (1)

for all l ≥ n − 3. In particular, this family of examples shows that Theorem 4 cannot be extended to step n − 4.

Example for n = 3 [AGS]:

Let K^{6} be the simply–connected connected nilpotent Lie group of real dimension 6 determined by the nilpotent Lie algebra K generated by X_{1}, Y_{1}, X_{2}, Y_{2}, X_{3}, Y_{3}, for which the nonzero defining bracket relations are the following:

[X_{1}, X_{2}] = −2X_{3}, [Y_{1}, X_{2}] = −2Y_{3}, [X_{1}, X_{3}] = [Y_{1}, Y_{3}] = −2Y_{2}.

The ascending central series {K_{l}} of K is then

K_{1} = {Y_{2}}, K_{2} = {Y_{2}, X_{3}, Y_{3}}, K_{3} = K,

that is, K is 3-step nilpotent. Let J be the left invariant almost complex structure on K^{6} defined by JX_{1} = Y_{1}, JX_{2} = Y_{2}, JX_{3} = Y_{3}. (2)

It is easy to check that the Nijenhuis tensor of J vanishes, and therefore (2) defines a left invariant complex structure on K^{6}. Since a_{1}(J) = 0, from Corollary 5 it follows that J is non-nilpotent.

The Lie algebra K can be also described by using the exterior differential d on the dual space K^{∗}. Since dα(X, Y) = −α([X, Y]), for α ∈ K^{∗} and X, Y ∈ K, in terms of the basis {α_{1}, β_{1}, α_{2}, β_{2}, α_{3}, β_{3}} of K^{∗} dual to the basis {X_{1}, Y_{1}, X_{2}, Y_{2}, X_{3}, Y_{3}} of K the structure equations of K^{6} are:

\{ \begin{matrix}dα_{1} = dβ_{1} = dα_{2} = 0, \\ dβ_{2} = 2α_{1} ∧ α_{3} + 2β_{1} ∧ β_{3}, \\ dα_{3} = 2α_{1} ∧ α_{2}, \\ dβ_{3} = 2β_{1} ∧ α_{2}. \end{matrix} (3)

It is worthy to remark that, since the center K_{1} of K is 1-dimensional and the term a_{1}(J) being always the largest J-invariant subspace of K_{1}, we have a_{1}(J) = 0 for any left invariant complex structure J on K^{6} and, therefore, any such J is non-nilpotent.

Example for n = 4:

Let K^{8} be the simply–connected nilpotent Lie group of real dimension 8 determined by the structure equations (3) together with the equations

\{ \begin{matrix}dα_{4} = α_{1} ∧ α_{3} − β_{1} ∧ β_{3}, \\ dβ_{4} = β_{1} ∧ α_{3} + α_{1} ∧ β_{3}. \end{matrix} (4)

If X_{4} and Y_{4} denote the dual elements of α_{4} and β_{4}, respectively, then the Lie algebra of K^{8} is generated by X_{1}, Y_{1}, X_{2}, Y_{2}, X_{3}, Y_{3}, X_{4}, Y_{4}, and the structure equations of the Lie group in terms of this basis are:

\begin{matrix}[X_{1}, X_{2}] = −2X_{3}, [Y_{1}, X_{2}] = −2Y_{3}, [X_{1}, Y_{3}] = [Y_{1}, X_{3}] = −Y_{4}, \\ [X_{1}, X_{3}] = −X_{4} − 2Y_{2}, [Y_{1}, Y_{3}] = X_{4} − 2Y_{2}. \end{matrix}

Now it is easy to check that the left invariant almost complex structure J defined by

JX_{1} = Y_{1}, JX_{2} = Y_{2}, JX_{3} = Y_{3}, JX_{4} = Y_{4}

is integrable on K^{8}. Moreover, 0 ≠ a_{1}(J) = {X_{4}, Y_{4}} = a_{l}(J) ≠ K for all l ≥ 2. Therefore J is a non-nilpotent complex structure on K^{8} satisfying (1) for n = 4.

Example for n ≥ 5:

The previous example for n = 4 can be generalized to any n ≥ 5 in the following way: the Lie group K^{8} has been obtained by adding the equations (4) to the equations (3), that is, to the structure equations of K^{6} we have added the equations

\{ \begin{matrix}dα_{n} = α_{1} ∧ α_{n−1} − β_{1} ∧ β_{n−1}, \\ dβ_{n} = β_{1} ∧ α_{n−1} + α_{1} ∧ β_{n−1}, \end{matrix} (5)

with n = 4. In general, for an arbitrary n ≥ 5 we construct the nilpotent Lie group K^{2n} of real dimension 2n by adding the equations (5) to the structure equations of the nilpotent Lie group K^{2(n−1)}.

Then, if {X_{1}, Y_{1}, …, X_{n}, Y_{n}} is the basis for the Lie algebra of K^{2n} dual to the basis {α_{1}, β_{1}, …, α_{n}, β_{n}}, the almost complex structure J defined by JX_{i} = Y_{i}, for i = 1, …, n, is integrable on K^{2n}. Moreover,

0 ≠ a_{1}(J) = {X_{n}, Y_{n}} ≠ a_{2}(J) = {X_{n−1}, Y_{n−1}, X_{n}, Y_{n}} ≠ ⋯
⋯ ≠ a_{r}(J) = {X_{n−r+1}, Y_{n−r+1}, …, X_{n}, Y_{n}} ≠ ⋯
⋯ ≠ a_{n−3}(J) = {X_{4}, Y_{4}, …, X_{n}, Y_{n}} = a_{l}(J) ≠ K,

for all l ≥ n − 2. Therefore J is a non-nilpotent complex structure satisfying (1).

The complex structure J on the Lie group K^{2n} can be easily described in terms of the complex structure equations of K^{2n} as follows. Denote by ω_{i} the left invariant complex 1-form on K^{2n} of type (1, 0) with respect to J defined by

ω_{i} = α_{i} + \sqrt{−1} β_{i}, 1 ≤ i ≤ n.

Then, in terms of the complex 1-forms ω_{1}, ω_{2}, ω_{3} and their conjugates, the structure equations of K^{6} become

\{ \begin{matrix}dω_{1} = 0, \\ dω_{2} = \sqrt{−1} (ω_{1} ∧ \overline{ω}_{3} − ω_{3} ∧ \overline{ω}_{1}), \\ dω_{3} = ω_{1} ∧ ω_{2} + ω_{1} ∧ \overline{ω}_{2}. \end{matrix} (6)

For the Lie group K^{8} the complex structure equations are then given by (6) together with the equation dω_{4} = ω_{1} ∧ ω_{3}. In general, the complex structure equations of K^{2n} will be obtained by adding the equation

dω_{n} = ω_{1} ∧ ω_{n−1} (7)

to the complex structure equations of K^{2(n−1)}, for n ≥ 4.

Since the complex structure on K^{2n} is non-nilpotent we know, from Theorem 1, that there does not exist any ordered basis \{ω_{1}, …, ω_{n}, \overline{ω}_{1}, …, \overline{ω}_{n}\} of left invariant complex 1-forms on K^{2n} satisfying that dω_{i} ∈ Λ^{∗,∗}(ω_{1}, …, ω_{i−1}, \overline{ω}_{1}, …, \overline{ω}_{i−1}), for 1 ≤ i ≤ n. However, (7) shows that there is “some order” in the structure equations of our examples for n ≥ 4. This is directly related to the fact that the non-nilpotent complex structure J on K^{2n} satisfies (1) for any n ≥ 4. In fact, proceeding in an analogous way as in the proof of Theorem 1 (see [CFGU1] for details) it can be proved the following result.

Theorem 6. Let G be a simply-connected connected nilpotent Lie group of (real) dimension 2n with a left invariant complex structure J, and denote by g the Lie algebra of G. Then, a_{q}(J) ≠ a_{q+1}(J) for some q ≥ 0 if and only if there exists an ordered basis \{ω_{1}, …, ω_{n}, \overline{ω}_{1}, …, \overline{ω}_{n}\} of left invariant complex 1-forms on G satisfying

dω_{i} = ∑_{j<k≤n−n_{l+1}} A_{ijk} ω_{j} ∧ ω_{k} + ∑_{j,k≤n−n_{l+1}} B_{ijk} ω_{j} ∧ \overline{ω}_{k},

for n − n_{l+1} + 1 ≤ i ≤ n − n_{l}, and for all 0 ≤ l ≤ q, where 2n_{l} = dim a_{l}(J).

Notice that the only left invariant complex structures J which are not covered by Theorem 6 are those satisfying a_{l}(J) = 0, for all l ≥ 0. This fact motivates the following Definition 2. A left invariant complex structure J on a nilpotent Lie group G is called strongly non-nilpotent if a_{1}(J) = 0.

Equivalently, J is strongly non-nilpotent if and only if {0} is the only Jinvariant subspace of g which is contained in its center. Moreover, from Corollary 5 we have:

Corollary 7. Any non-nilpotent complex structure on a compact nilmanifold Γ\G of (real) dimension 6 is strongly non-nilpotent.

Remark 8. We refer the reader to [CFGU2] for the construction of a nilpotent Lie group of real dimension 10 with a left invariant complex structure J which is strongly non-nilpotent.

Finally, we must note that since the coefficients in the structure equations (3)– (5) are all rational numbers, a theorem of Mal’cev [M] guarantees the existence of a lattice Γ^{2n} of K^{2n} of maximal rank, for each n ≥ 3. Therefore, for each n ≥ 3, Γ^{2n}\K^{2n} is a compact complex nilmanifold whose complex structure is nonnilpotent, that is, it comes from a non-nilpotent left invariant complex structure on the Lie group K^{2n} by passing to the quotient.

4. On the holomorphic fiber bundle structure of Γ\G

A compact complex nilmanifold Γ\G with nilpotent complex structure J has been described in [CFGU1] as the total space at the top in a tower of holomorphic principal torus bundles with a complex torus as the base space at the bottom of the tower. In this section we extend this study to the case of J being a nonnilpotent complex structure on Γ\G, but restricting our attention to those J’s which are not strongly non-nilpotent.

Let Γ\G be a compact complex nilmanifold with non-nilpotent complex structure J satisfying a_{1}(J) ≠ 0, and let g be the Lie algebra of G. Since J is not strongly non-nilpotent, there is an integer t > 0 such that

0 ≠ a_{1}(J) ≠ a_{2}(J) ≠ ⋯ ≠ a_{t−1}(J) ≠ a_{t}(J) = a_{l}(J) ≠ g, (1)

for all l ≥ t. Therefore, the series {a_{l}(J)} induces the sequence of quotient Lie algebras

g ⟶ g/a_{1}(J) ⟶ ⋯ ⟶ g/a_{q}(J) ⟶^{π_{q+1}} g/a_{q+1}(J) ⟶ ⋯ ⟶ g/a_{t}(J), (2)

where π_{q+1} is the natural surjective homomorphism, and ker π_{q+1} = a_{q+1}(J)/a_{q}(J). In fact π_{q+1}: g̃_{q} ⟶ g̃_{q+1} and ker π_{q+1} = a_{1}(J̃_{q}), according to the notation and results in Section 2. Therefore, (2) gives the following tower of quotient Lie algebras

a_{1}(J) = a_{1}(J̃_{0}) ↪ g̃_{0} = g ; ↓ ; a_{1}(J̃_{1}) ↪ g̃_{1} ; ↓ ; ⋮ ; ↓ ; a_{1}(J̃_{t−1}) ↪ g̃_{t−1} ; ↓ ; g̃_{t} (3)

where J̃_{0} = J. Since a_{1}(J̃_{q}) is contained in the center of g̃_{q}, it is an Abelian Lie algebra for all 0 ≤ q ≤ t − 1.

Let G̃_{q} denote the simply–connected connected nilpotent Lie group defined by g̃_{q}. From Lemma 3 and (1) it follows that

a_{t−q−1}(J̃_{q}) ≠ a_{t−q}(J̃_{q}) = a_{t−q+l}(J̃_{q}) ≠ g̃_{q}, (4)

for all l ≥ 0 and 0 ≤ q ≤ t − 1. Therefore, the complex structure J̃_{q} on G̃_{q} is non-nilpotent. Moreover, from (4) we have in particular that a_{1}(J̃_{q}) ≠ 0, that is, J̃_{q} is not strongly non-nilpotent for 0 ≤ q ≤ t − 1. However, using again Lemma 3 and (1), we get a_{1}(J̃_{t}) ≅ a_{t+1}(J)/a_{t}(J) = 0, and therefore J̃_{t} is a strongly non-nilpotent complex structure on G̃_{t}.

The Abelian Lie algebra a_{1}(J̃_{q}) with the restriction of the complex structure J̃_{q} becomes a complex Abelian Lie algebra for all 0 ≤ q ≤ t − 1. Therefore, it defines a simply–connected connected Abelian complex Lie group A_{q} ≅ C^{n_{q+1}−n_{q}}, where 2n_{l} = dim a_{l}(J). Notice that a_{1}(J̃_{q}) is an ideal in g̃_{q}, hence A_{q} is a closed normal Lie subgroup of G̃_{q} and G̃_{q+1} ≅ G̃_{q}/A_{q}, for 0 ≤ q ≤ t − 1.

Therefore, the tower (3) induces by integration a new tower at the level of Lie groups:

C^{n_{1}} ↪ G̃_{0} = G ; ↓ ; C^{n_{2}−n_{1}} ↪ G̃_{1} ; ↓ ; ⋮ ; ↓ ; C^{n_{t}−n_{t−1}} ↪ G̃_{t−1} ; ↓ ; G̃_{t} (5)

each π_{q+1}: G̃_{q} ⟶ G̃_{q+1} in the tower being a holomorphic map; furthermore, since [a_{1}(J̃_{q}), g̃_{q}] = 0 there is an induced holomorphic free action of C^{n_{q+1}−n_{q}} on G̃_{q} for 0 ≤ q ≤ t − 1. Thus, the Lie group G appears as the total space at the top in this tower of holomorphic principal bundles, whose structure groups are all Abelian, where the complex structure J̃_{q} on G̃_{q}, although non-nilpotent, is not strongly non-nilpotent for 0 ≤ q ≤ t − 1, and the complex structure J̃_{t} on the base space G̃_{t} at the bottom of (5) being always strongly non-nilpotent.

Moreover, if the real dimension of G is 2n, then Theorem 4 implies that the number of steps in this tower is always ≤ n − 3.

Theorem 9. Let Γ\G be a compact complex nilmanifold whose complex structure stems from a non-nilpotent (left invariant) complex structure on the Lie group G. If J is not strongly non-nilpotent then Γ\G can be described as the total space at the top in a tower of holomorphic principal bundles

𝕋^{n_{1}} ↪ Γ̃_{0}\G̃_{0} = Γ\G ; ↓ ; 𝕋^{n_{2}−n_{1}} ↪ Γ̃_{1}\G̃_{1} ; ↓ ; ⋮ ; ↓ ; 𝕋^{n_{t−1}−n_{t−2}} ↪ Γ̃_{t−2}\G̃_{t−2} ; ↓ ; 𝕋^{n_{t}−n_{t−1}} ↪ Γ̃_{t−1}\G̃_{t−1} ; ↓ ; Γ̃_{t}\G̃_{t} (6)

whose structure groups are all complex tori 𝕋^{n_{l+1}−n_{l}} (0 ≤ l ≤ t − 1), where 2n_{l} = dim a_{l}(J). The complex structure on each compact complex nilmanifold Γ̃_{l}\G̃_{l} is that obtained by passing to the quotient the non-nilpotent complex structure J̃_{l} on G̃_{l} given by (1), for 0 ≤ l ≤ t−1. Moreover, the complex structure on the base space Γ̃_{t}\G̃_{t} at the bottom of the tower is strongly non-nilpotent. Finally, if the (real) dimension of Γ\G is 2n then the number of steps in the tower is ≤ n − 3.

Proof.

Since a_{1}(J̃_{q}) is contained in the center of g̃_{q} then the corresponding Lie subgroup A_{q} of G̃_{q} is also contained in the center of G̃_{q}, for 0 ≤ q ≤ t − 1. Therefore, if for some 0 ≤ q ≤ t − 1 we have a lattice Γ̃_{q} in G̃_{q} of maximal rank then A_{q} ∩ Γ̃_{q} is a lattice in A_{q}, and thus the compact nilmanifold (A_{q} ∩ Γ̃_{q})\A_{q} ≅ 𝕋^{n_{q+1}−n_{q}}; moreover, A_{q}Γ̃_{q} is closed in G̃_{q} and Γ̃_{q+1} = π_{q+1}(Γ̃_{q}) is a lattice in G̃_{q+1} (see, for example, [R]). Since (A_{q} ∩ Γ̃_{q})\A_{q} ≅ Γ̃_{q}\A_{q}Γ̃_{q}, we obtain the holomorphic

principal bundle 𝕋^{n_{q+1}−n_{q}} ↪ Γ̃_{q}\G̃_{q} ⟶^{π_{q+1}} Γ̃_{q+1}\G̃_{q+1}.

Taking into account this argumentation, if Γ is a lattice of G of maximal rank and we denote Γ̃_{0} ≡ Γ, then Γ̃_{q+1} = π_{q+1}(Γ̃_{q}) is a lattice in G̃_{q+1} for all 0 ≤ q ≤ t − 1. Therefore, replacing G̃_{q} by Γ̃_{q}\G̃_{q} (0 ≤ q ≤ t) in the tower (5), the desired tower of holomorphic principal bundles (6) arises. □

Remark 10. Note that the results about the fiber structure of compact complex nilmanifolds in Section 2 of [CFG] must be corrected accordingly. That is, they apply only to special nilpotent complex structures J, namely those J’s for which a_{l}(J) = g_{l} for all l ≥ 0.

It is well known that on an arbitrary complex manifold M there exists the so called Dolbeault complex of M,

⋯ ⟶ Λ^{p,q−1}(M) ⟶^{¯∂} Λ^{p,q}(M) ⟶^{¯∂} Λ^{p,q+1}(M) ⟶ ⋯, (7)

where Λ^{r,s}(M) denotes the space of forms of type (r, s) on M, and ¯∂ is the operator of type (0, 1) in the usual decomposition d = ∂ + ¯∂ of the exterior differential d. The cohomology groups of (7) are the Dolbeault cohomology groups of M.

In the particular case of M being a compact complex nilmanifold Γ\G, in addition to (7) there is also the complex

⋯ ⟶ Λ^{p,q−1}(g^{ℂ})^{∗} ⟶^{¯∂} Λ^{p,q}(g^{ℂ})^{∗} ⟶^{¯∂} Λ^{p,q+1}(g^{ℂ})^{∗} ⟶ ⋯, (8)

where g^{ℂ} denotes the complexification of the Lie algebra g of G, and ¯∂ is the corresponding differential of type (0, 1) in the canonical decomposition d = ∂ + ¯∂ of the Chevalley–Eilenberg differential d extended to Λ^{∗}(g^{ℂ})^{∗}. It is natural to ask if there is some relationship between the cohomology groups of (8) and the Dolbeault cohomology groups of M = Γ\G.

Although the relationship between (7) and (8) has not been completely investigated, as we have mentioned at the begining a result similar to Nomizu’s theorem has been obtained by Sakane [S] when the Lie group G is a complex Lie group. In [CFGU1] Sakane’s result has been extended to the class of compact complex nilmanifolds Γ\G whose complex structure J is nilpotent, by proving that there is a canonical isomorphism

Hp,q ; ¯∂(Γ\G) ≅ Hp,q ; ¯∂(g^{ℂ}),

between the cohomology groups Hp,q ; ¯∂(g^{ℂ}) of (8) and those of (7) for M = Γ\G. In fact, in [CFGU1] it is proved that the natural inclusion of differential bigraded algebras

i: (Λ^{∗,∗}(g^{ℂ})^{∗}, ¯∂) ⟶ (Λ^{∗,∗}(Γ\G), ¯∂)

induces an isomorphism at the cohomology level or, in other words, (Λ^{∗,∗}(g^{ℂ})^{∗}, ¯∂) is a minimal model (in the sense of [NT]) for the Dolbeault complex of Γ\G. This gives a partial answer to the following natural question:

Question:

Is (Λ^{∗,∗}(g^{ℂ})^{∗}, ¯∂) a model for the Dolbeault complex (Λ^{∗,∗}(Γ\G), ¯∂) of any compact complex nilmanifold Γ\G?

As far as we know, there is no complete answer to this question. The result in [CFGU1] gives a positive answer only for the class of compact complex nilmanifolds Γ\G with nilpotent complex structure J, but the question remains still unsolved for the class of non-nilpotent J’s. Nevertheless, the techniques developed in [CFGU1] together with the fiber structure obtained in Theorem 9 allow us to reduce the question to give an affirmative answer in the case of Γ\G being a compact complex nilmanifold with a strongly non-nilpotent complex structure.

Actually, let us suppose that Γ\G is a compact complex nilmanifold with nonnilpotent J. From Theorem 9 we have a description of Γ\G as the total space of the tower (6), where the complex structure on the space Γ̃_{t}\G̃_{t} at the bottom is strongly non-nilpotent. If (Λ^{∗,∗}(g̃ℂ_{t})^{∗}, ¯∂) were a model (not necessarily minimal) for the Dolbeault complex of Γ̃_{t}\G̃_{t}, then the iterative procedure from the bottom to the top of the tower used in Section 4.3 of [CFGU1] in the case of a nilpotent J can be easily adapted to the tower (6) in order to get that (Λ^{∗,∗}(g^{ℂ})^{∗}, ¯∂) is a model for the Dolbeault complex of Γ\G.

The main tools to be used in such iterative procedure are a Hirsch-type Lemma and a suitable principal connection on each bundle

𝕋^{n_{q+1}−n_{q}} ↪ Γ̃_{q}\G̃_{q} ⟶^{π_{q+1}} Γ̃_{q+1}\G̃_{q+1}, for 0 ≤ q ≤ t − 1,

which allows to apply the mentioned Hirsch Lemma. Such suitable connection also exists in this case and is defined as follows. Since a_{q}(J) ≠ a_{q+1}(J), Theorem 6 implies in particular the existence of a basis {ω_{1}, …, ω_{n−n_{q}}} of left invariant complex 1-forms on G̃_{q} of type (1,0) such that

dω_{i} = ∑_{j<k≤n−n_{q+1}} A_{ijk} ω_{j} ∧ ω_{k} + ∑_{j,k≤n−n_{q+1}} B_{ijk} ω_{j} ∧ \overline{ω}_{k}, (9)

for n−n_{q+1} +1 ≤ i ≤ n−n_{q}. (For the sake of simplicity we denote with the same symbols the 1-forms ω_{1}, …, ω_{n−n_{q+1}} in g̃_{q} and those induced in g̃_{q+1}; moreover, ω_{1}, …, ω_{n−n_{l}} descend to Γ̃_{l}\G̃_{l}, l = q or q + 1, and we also denote with the same symbols the forms on the quotient.) If {Z_{1}, …, Z_{n_{q+1}−n_{q}}} is a basis of left invariant holomorphic vector fields of type (1,0) on 𝕋^{n_{q+1}−n_{q}}, then

ω = ∑_{i=1}^{n_{q+1}−n_{q}} ω_{n−n_{q+1}+i} ⊗ Z_{i}

defines a principal connection on the bundle. Notice that ω has type (1,0) and is compatible with the holomorphic structure in the bundle. Moreover, from (9) it follows that the curvature from Ω of ω on the base Γ̃_{q+1}\G̃_{q+1} is given by

Ω = ∑_{i=1}^{n_{q+1}−n_{q}} ( ∑_{j<k≤n−n_{q+1}} A_{n−n_{q+1}+i j k} ω_{j} ∧ ω_{k} + ∑_{j,k≤n−n_{q+1}} B_{n−n_{q+1}+i j k} ω_{j} ∧ \overline{ω}_{k})⊗Z_{i}.

References

[AGS] E. Abbena, S. Garbiero, S. Salamon, Private communication.

[CFG] L.A. Cordero, M. Fernández, A. Gray, La estructura fibrada de una nilvariedad compacta, Proc. Workshop Recent Topics in Differential Geometry (Ed. D. Chinea, J.M. Sierra), Puerto de la Cruz, Tenerife (Spain) 1990, Serie Informes 32, 43–64, Univ. La Laguna (Spain) 1991.

[CFGU1] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte, Compact nilmanifolds with nilpotent complex structure: Dolbeault cohomology, Preprint 1997.

[CFGU2] L.A. Cordero, M. Fernández, A. Gray, L. Ugarte, Nilpotent complex structures on compact nilmanifolds, Rendiconti Circ. Mat. Palermo, Serie II, Suppl. 49 (1997), 83–100.

[FGG] M. Fernández, M.J. Gotay, A. Gray, Four-dimensional compact parallelizable symplectic and complex manifolds, Proc. Amer. Math. Soc. 103 (1988), 1209–1212.

[M] I.A. Mal’cev, A class of homogeneous spaces, Amer. Math. Soc. Transl. No. 39 (1951).

[NT] J. Neisendorfer, L. Taylor, Dolbeault Homotopy Theory, Trans. Amer. Math. Soc. 245 (1978), 183–210.

[N] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. 59 (1954), 531-538.

[R] M.S. Raghunathan, Discrete subgroups of Lie groups, Springer–Verlag, Berlin, 1972.

[S] Y. Sakane, On compact parallelisable solvmanifolds, Osaka J. Math. 13 (1976), 187-212.

[W] H.C. Wang, Complex parallelisable manifolds, Proc. Amer. Math. Soc. 5 (1954), 771-776.

Luis A. Cordero Departamento de Geometría y Topología Facultad de Matemáticas Universidad de Santiago de Compostela 15705 Santiago de Compostela Spain

Email: cordero@zmat.usc.es

Marisa Fernández Departamento de Matemáticas Facultad de Ciencias Universidad del País Vasco Apartado 644, 48080 Bilbao Spain

Email: mtpferol@lg.ehu.es

\underline{Alfred Gray} Department of Mathematics University of Maryland College Park, Maryland 20742 USA

Email: gray@bianchi.umd.edu

\underline{Luis Ugarte} Departamento de Geometría y Topología Facultad de Ciencias Universidad de Zaragoza Campus Pl. San Francisco, 50009 Zaragoza Spain

Email: ugarte@posta.unizar.es