Dolbeault cohomology and deformations of nilmanifolds

Sergio Console

Abstract: In these notes I review some classes of invariant complex structures on nilmanifolds for which the Dolbeault cohomology can be computed by means of invariant forms, in the spirit of Nomizu's theorem for de Rham cohomology.

Moreover, deformations of complex structures are discussed. Small deformations remain in some cases invariant, so that, by Kodaira-Spencer theory, Dolbeault cohomology can be still computed using invariant forms.

2000 Mathematics Subject Classification. 53B05, 53C05

Partially supported by GNSAGA of INdAM and MIUR (Italy)

1. Introduction

A nilmanifold is a compact quotient , where is a real simply-connected nilpotent Lie group and is a discrete cocompact subgroup of . The homogeneous space is complex if it is endowed with an invariant complex structure .

Nilmanifolds provide examples of symplectic manifolds with no Kähler structure (see for instance [OT]). Indeed Benson and Gordon [BG] proved that a complex nilmanifold is Kähler if and only if it is a torus (see also [H]).

This is one reason for the interest in computing Dolbeault cohomology. Since nilmanifolds are homogeneous spaces, it is natural to expect this to be possible using invariant differential forms. However, whether this is true in general is still an open problem.

In these notes I shall review some classes of complex structures whose Dolbeault cohomology is computed by means of the complex of forms of type on the (complexified) Lie algebra, i.e., , where is the Lie algebra of . Special attention will be devoted to the class of abelian complex structures [BDM, CF, CFP, CFGU, MPPS]. On one hand, this is because proving that is simpler when dealing with abelian complex structures, and suggests how to tackle other situations. On the other hand, and more importantly, abelian complex structures have a very strong property related to deformations: they admit a locally complete family of deformations consisting (solely) of invariant complex structures. Using Kodaira-Spencer theory, one can show that still holds for any small deformation of an abelian complex structure (cf. Theorem 4.2).

2. Real nilmanifolds

Recall that is (-step) nilpotent if and only if there exists a (minimal) integer such that where

If is nilpotent with rational structure constants or, equivalently, has a rational structure (i.e., there exists a rational such that ) then a result of Mal'čev [M] asserts that there exists a discrete subgroup such that is compact.

The computation of de Rham cohomology can be carried out by means of invariant forms. This is a result of Nomizu going back to the fifties [N]. We will outline the main ideas, since some of them can be used to understand Dolbeault cohomology.

Let us begin remarking that the complex of differential forms on can be regarded as that of -invariant forms on . On the other hand, the complex of -invariant forms on can be identified with the exterior algebra . By this, the differential of is given by

The differential complex

is usually called the Chevalley-Eilenberg complex.

Theorem 2.1 (Nomizu [N]). , for any , where is the cohomology of the Chevalley-Eilenberg complex.

Sketch of the proof: Set where is the last non-zero term in the lower central series. Observe that is central, in particular abelian. Let be the connected Lie subgroup of with Lie algebra . Since is simply-connected and is connected, is simply-connected as well. Hence is diffeomorphic to .

By standard results on discrete subgroups [R] and are discrete cocompact subgroups. Thus, is a compact nilmanifold with dimension smaller than and is a torus. Moreover, is a fibration.

Next, consider the Leray-Serre spectral sequence associated with the above fibration. One has

The idea is to construct a second spectral sequence which is the Leray-Serre spectral sequence relative to the complex of -invariant forms (i.e., the Chevalley-Eilenberg complex). Since is subcomplex of , and

Now, since is a nilmanifold of lower dimension than , an inductive argument on the dimension shows that for any . Thus and one has the equality . Therefore, for any . □

3. Complex structures

We now assume that has an invariant complex structure , so that comes from a (left-invariant) complex structure on . Recall that determines a decomposition of the complexification of and of its dual

Moreover, is integrable. This means that the Nijenhuis tensor

vanishes or, equivalently,

Here is the space of -forms on and the differential is extended to by -linearity.

3.1. Dolbeault cohomology. Observe that, by the above relation, is complex (integrable) if and only if . This allows to define a Dolbeault complex for , i.e., for the invariant forms of type on . We denote by the cohomology of this complex. On the other hand, is the cohomology of the Dolbeault complex of -invariant forms on .

In the next Sections, we will discuss the following

Problem. For which complex structure does the isomorphism

hold?

In general, we have the following

Lemma 3.1 ([CF]). The inclusion induces an injective morphism

Proof. If is endowed with an invariant Hermitian metric, and its adjoint preserve -invariant forms but also the forms orthogonal to the -invariant ones. Thus, if , for some , we may assume with lying in the orthogonal complement of the -invariant forms. Therefore, is a -invariant form. Since

must be zero, which implies in . □

We let denote the cohomology of the Dolbeault complex of -invariant forms orthogonal to the -invariant ones [CF].

The isomorphism can be rephrased as

We begin examining some important subclasses of complex structures for which the isomorphism (or equivalently ) holds.

3.2. Complex parallelizable nilmanifolds. A nilmanifold is called complex parallelizable if is a complex Lie group, or equivalently

It is known that for complex parallelizable nilmanifolds the isomorphism (*) holds. Actually one can say more.

Theorem 3.2 (Sakane [S]). If is a complex parallelizable nilmanifold

where is the cohomology of the complex

In particular,

We shall not prove this, focusing instead on another class of complex structures (the so-called abelian ones). We briefly explain why the Dolbeault cohomology is not simply equal to the cohomology of invariant forms but the stronger result of Theorem 3.2 holds. Since , , so the Dolbeault complex splits as

In other words, , with and . Hence, the cohomology of the Dolbeault complex is the same as the cohomology of , which equals .

3.3. Abelian complex structures. A complex structure is called abelian [BDM] if is abelian. This means

or equivalently

Nilmanifolds with abelian complex structures are therefore dual to complex parallelizable ones, in some sense. The following result should thus be read as the counterpart to Theorem 3.2.

Theorem 3.3 ([CF, CFP, CFGU]). If is a nilmanifold endowed with an abelian complex structure

Moreover,

where is the structure sheaf of .

Proof. We use a method very similar to the proof of Nomizu's Theorem. We consider the upper central series given by , , , .

If is abelian (observe that this is not true for the lower central series).

Moreover, is central, in particular abelian, and we have the following exact sequence of Lie algebras

Let be the connected Lie subgroup of with Lie algebra ( is actually diffeomorphic to , since it is abelian and simply-connected), and let be the simply-connected nilpotent Lie group with Lie algebra .

Given the uniform discrete subgroup , since is a rational subalgebra of , is uniform in (cf. [R]). So we have the holomorphic fibration

whose fibre is the torus .

The main tool to get cohomological information about the total space of this bundle is Borel's spectral sequence [Hi, Appendix II by A. Borel, Theorem 2.1]: Let be a holomorphic fibre bundle, with compact connected fibre , and and connected. If is Kähler there is a spectral sequence , () with the following properties:

1. is 4-graded by fibre-degree, base-degree and type;
2. If , then ;
3. .

As for the proof of Nomizu's Theorem, one can use a first Borel spectral sequence for the complex of forms invariant by the lattice and a second one, denoted by , for the forms invariant by the group action (the forms on the Lie algebra).

We proceed by induction on the dimension. We have

Now, is a nilmanifold with an abelian complex structure of dimension lower than , so . Thus and . This implies that .

We want to show that . Similarly to the complex parallelizable case, we have that , for any , since , and the Dolbeault complex is the zero complex. □

One can also compute the Dolbeault cohomology with coefficients in the tangent sheaf of . This was done in [MPPS] for 2-step nilmanifolds with an abelian complex structure and in [CFP] in the general case.

Theorem 3.4 ([CFP, MPPS]). Let be nilmanifold with an abelian complex structure and denote by the holomorphic tangent sheaf of . Then is canonically isomorphic with the cohomology of a complex constructed using invariant forms and invariant vectors.

3.4. Nilpotent complex structures. An important property of abelian complex structures (used in 3.3) is that the complex structure preserves the elements of the upper central series , i.e., .

Given a complex structure on a -dimensional nilpotent Lie algebra , one may modify the upper central series so that it is preserved by . For this purpose, consider the series defined inductively by and

A complex structure is said to be nilpotent [CFGU] if the series satisfies for some positive integer .

Note that an abelian complex structure is always nilpotent, with , for any .

Again, for nilpotent complex structures the isomorphism (*) holds.

Theorem 3.5 ([CFGU]). If is a nilmanifold endowed with a nilpotent complex structure

The proof relies on a similar technique to that of abelian complex structures.

3.5. Rational complex structures. A complex structure is called rational if it is compatible with the rational structure of the lattice , meaning . For this class as well we have

Theorem 3.6 ([CF]). If is a nilmanifold endowed with a rational complex structure

Sketch of the proof: One key point in the proof is to modify the lower central series so that it is preserved by . For each , one considers the smallest subalgebra containing which is invariant. More precisely, . One obtains a descending series in which each term fits into an exact sequence of Lie algebras

Here, is the smallest integer such that .

In general, the subalgebra is not a rational subalgebra of (in which case one would have trouble with discrete subgroups). But if is rational, any is rational in .

The passage to Lie groups and then to the quotient by uniform subgroups yields a set of holomorphic fibrations of nilmanifolds.

For these one considers a generalization of Borel's spectral sequence to the case of non-Kähler fibres [FW, L]. Once again, one considers two spectral sequences: one for the complex of forms invariant by the lattice and another for the forms invariant by the group action. An inductive argument (starting from and the "last fibration" whose fibres and base are tori) yields the proof. □

4. Deformations

Abelian complex structures behave well with respect to deformations. Indeed

Theorem 4.1 ([CFP, MPPS]). Any abelian invariant complex structure on a nilmanifold has a locally complete family of deformations (parametrized by a subset ) consisting entirely of invariant complex structures.

Thus, any deformation of an abelian invariant complex structure is necessarily equivalent to an invariant one, at least if the deformation is sufficiently small.

The main point for the proof is that the holomorphic tangent sheaf can be computed using invariant forms and invariant vectors (Theorem 3.4). This implies that there are harmonic representatives of , which are the main ingredient for Kuranishi deformations [K]. Moreover, this harmonic theory can be exploited on finite dimensional vector spaces of invariant forms and invariant vectors. Thus, Kuranishi's inductive process remains within invariant tensors.

4.1. Stability results. If we deform a complex structure of some type (abelian, nilpotent), the deformed structure can still be invariant but will not be of the same type in general. One calls a complex structure stable if it remains of the same type when deformed.

The deformation of abelian complex structures are not stable in this sense, beginning from real dimension six [MPPS].

However, on a nilmanifold with an abelian complex structure, a vector in the virtual parameter space is integrable to a 1-parameter family of abelian complex structures if and only if it lies in a linear subspace that defines the abelian condition infinitesimally (see [MPPS] for 2-step nilmanifolds and [CFP] in the general case).

In real dimension six, the deformation of abelian complex structures is stable within the class of nilpotent complex structures. There are examples showing that this phenomenon does not occur in higher dimension [CFP]. Consequently, neither nilpotent complex structures are stable.

Finally, one can see that rational complex structures are not stable under deformations, too [CF, Section 7, page 122].

4.2. Deformations and Dolbeault cohomology. Suppose we have a deformation () of a complex structure consisting (at least locally) of invariant complex structures.

Then we have (cf. [CF] in the case of rational complex structures)

Theorem 4.2. Suppose that holds that for a complex structure . Then this isomorphism still holds for any small deformation () of consisting of invariant complex structures.

Sketch of the proof: The idea goes as follows.

For we know that . Now, by Kodaira-Spencer theory [KS], is an upper semicontinuous function, so there exists a sufficiently small neighbourhood of such that

Thus in . □

We conclude with an application of this result. By Theorem 4.1, in fact, given an abelian complex structure , for any small deformation (), we have .

Acknowledgments: I am in great debt to Simon Chiossi, Anna Fino and Yat Sun Poon. I would also like to take this opportunity to thank the organizers for this wonderful conference.

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Sergio Console
Dipartimento di Matematica
Università di Torino
via Carlo Alberto 10
10123 Torino, Italy
sergio.console@unito.it

Recibido: 6 de octubre de 2005
Aceptado: 17 de septiembre de 2006