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Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006


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 M = G ∕Γ , where G is a real simply-connected nilpotent Lie group and Γ is a discrete cocompact subgroup of G . The homogeneous space M is complex if it is endowed with an invariant complex structure J .

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 (p, q) on the (complexified) Lie algebra, i.e.,  *,* *,* ℂ H ∂- (M ) ~= H ∂-(𝔤 ) , where 𝔤 is the Lie algebra of G . Special attention will be devoted to the class of abelian complex structures [BDMCFCFPCFGUMPPS]. On one hand, this is because proving that H *,*(M ) ~= H *,*(𝔤ℂ) ∂ ∂ 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 H *,*(M, J) ~= H *,*(𝔤ℂ) ∂ ∂ still holds for any small deformation J of an abelian complex structure (cf. Theorem 4.2).

2. Real nilmanifolds

Recall that G is (k -step) nilpotent if and only if there exists a (minimal) integer k such that 0 ⁄= 𝔤k- 1 ⊃ 𝔤k = {0} where

𝔤0 = 𝔤, 𝔤i = [𝔤i-1,𝔤 ]. (lower central series)

If G is nilpotent with rational structure constants or, equivalently, 𝔤 has a rational structure (i.e., there exists a rational 𝔤ℚ such that 𝔤 = 𝔤ℚ ⊗ ℝ ) then a result of Mal'čev [M] asserts that there exists a discrete subgroup Γ such that M = G∕ Γ 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 ∧ *M of differential forms on M can be regarded as that of Γ -invariant forms on G . On the other hand, the complex of G -invariant forms on G can be identified with the exterior algebra ∧ * * 𝔤 . By this, the differential of  ∧k α ∈ 𝔤 * is given by

 ∑ i+j dα (x1, ...,xk+1) = (- 1) α ([xi,xj],x1,...ˆxi,..., ˆxj,...,xk+1 ) i<j

The differential complex

 ∧k - 1 d ∧k d ∧k+1 ⋅⋅⋅ → 𝔤*→ 𝔤*→ 𝔤* → ...

is usually called the Chevalley-Eilenberg complex.

Theorem 2.1 (Nomizu [N]). HkdR(M ) ~= Hk (𝔤) , for any k , where Hk (𝔤) is the cohomology of the Chevalley-Eilenberg complex.

Sketch of the proof: Set  k-1 𝔥 := 𝔤 where  k-1 𝔤 is the last non-zero term in the lower central series. Observe that 𝔥 is central, in particular abelian. Let H be the connected Lie subgroup of G with Lie algebra 𝔥 . Since G is simply-connected and H is connected, H is simply-connected as well. Hence H is diffeomorphic to ℝn .

By standard results on discrete subgroups [R] H Γ ∕H and H ∩ Γ are discrete cocompact subgroups. Thus, --- M := G ∕H Γ is a compact nilmanifold with dimension smaller than M and 𝕋n := H∕H ∩ Γ is a torus. Moreover,  π --- 𝕋n `→ M → M is a fibration.

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

 --- --- ∧ Ep,2q = HpdR(M ,HqdR (𝕋n)) ~= HpdR(M ) ⊗ q ℝn , Ep,∞q ⇒ Hp+dqR (M ).

The idea is to construct a second spectral sequence E˜p,*q which is the Leray-Serre spectral sequence relative to the complex of G -invariant forms (i.e., the Chevalley-Eilenberg complex). Since ∧ * * 𝔤 is subcomplex of ∧ * M ,  ˜p,q p,q E * ⊆ E* and

 ∧ ˜Ep,2q = Hp (𝔤∕𝔥) ⊗ qℝn , ˜Ep,q ⇒ Hp+q (𝔤) . ∞

Now, since --- M is a nilmanifold of lower dimension than M , an inductive argument on the dimension shows that Hp (M--) ~= Hp (𝔤 ∕𝔥) dR for any p . Thus E = E˜ 2 2 and one has the equality  ˜ E ∞ = E∞ . Therefore,  k ~ k H dR (M ) = H (𝔤) for any k . □

3. Complex structures

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

{ ℂ 1,0 0,1 𝔤 = 𝔤∧1,0⊕ 𝔤 ∧0,1 (𝔤ℂ)* = (𝔤 ℂ)* ⊕ (𝔤ℂ )* = 𝔤*(1,0) ⊕ 𝔤*(0,1)

Moreover, J is integrable. This means that the Nijenhuis tensor

NJ (Z,W ) = [Z, W ] + J [J Z,W ] + J [Z, JW ] - [JZ, JW ], Z,W ∈ 𝔤

vanishes or, equivalently,

d𝔤*(1,0) ⊂ 𝔤*(1,1) ⊕ 𝔤*(2,0) .

Here  *(p,q) ∧p,q ℂ * 𝔤 = (𝔤 ) is the space of (p,q) -forms on𝔤 and the differential d is extended to 𝔤 ℂ by ℂ -linearity.

3.1. Dolbeault cohomology. Observe that, by the above relation, J is complex (integrable) if and only if ∂2 = 0 . This allows to define a Dolbeault complex for ∧p,q ℂ * *(p,q) (𝔤 ) = 𝔤 , i.e., for the G - invariant forms of type (p,q) on M . We denote by  *,* ℂ H ∂- (𝔤 ) the cohomology of this complex. On the other hand, H *,*(M ) ∂ is the cohomology of the Dolbeault complex  ∧ -- ( p,q(M ),∂) of Γ -invariant forms on M .

In the next Sections, we will discuss the following

Problem. For which complex structure J does the isomorphism

 *,* ~ *,* ℂ (*) H ∂ (M )= H ∂ (𝔤 )


In general, we have the following

Lemma 3.1 ([CF]). The inclusion ∧ ∧ p,q(𝔤 ℂ) `→ p,q(M ) induces an injective morphism

 *,* j *,* H ∂- (𝔤ℂ)`→ H∂- (M ).

Proof. If M = G∕ Γ is endowed with an invariant Hermitian metric, -- ∂ and its adjoint -- ∂ * preserve G -invariant forms but also the forms orthogonal to the G -invariant ones. Thus, if j[ω ] = 0 , for some [ω] ∈ Hp,q (𝔤 ℂ) ∂ , we may assume j(ω) = ∂-φ with φ lying in the orthogonal complement of the G -invariant forms. Therefore, -*-- ∂ ∂ φ is a G -invariant form. Since

 ---- -- -- 0 = ⟨φ, ∂*∂φ ⟩ = ⟨∂φ,∂ φ⟩,

-- ∂φ must be zero, which implies[ω] = 0 in  -p,q ℂ H∂ (𝔤 ) .

We let  ⊥*,* H∂- (M ) denote the cohomology of the Dolbeault complex of Γ -invariant forms orthogonal to the G -invariant ones [CF].

The isomorphism H *,*(M ) ~= H-*,*(𝔤ℂ) ∂ ∂ can be rephrased as

 ⊥*,* (**) dim H ∂ (M ) = 0 .

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

3.2. Complex parallelizable nilmanifolds. A nilmanifold M = G ∕Γ is called complex parallelizable if G is a complex Lie group, or equivalently

 *(1,0) *(2,0) d𝔤 ⊂ 𝔤

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

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

 ∧p Hp,q(M ) ~= 𝔤*(1,0) ⊗ Hq (𝔤*(0,1)) , ∂

where  q *(0,1) H (𝔤 ) is the cohomology of the complex

 ∧q - ∧q+1 - ⋅⋅⋅ → 𝔤 *(0,1) = 𝔤*(0,q) ∂→ 𝔤*(0,1) = 𝔤*(0,q+1) ∂→ ...

In particular,

 p,0 ~ ∧p *(1,0) *(p,0) H ∂- (M )= 𝔤 = 𝔤 .

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 G- invariant forms but the stronger result of Theorem 3.2 holds. Since d𝔤*(1,0) ⊂ 𝔤*(2,0) , -- ∂𝔤*(p,0) = 0 , so the Dolbeault complex ∧ p,q(𝔤ℂ)* = 𝔤*(p,q) splits as

 𝔤*(p,q) = 𝔤*(p,0) ⊗ 𝔤*(0,q) | | 1 | |∂ | | *(p,q+1) *(p,0) *(0,q+1) 𝔤 = 𝔤 ⊗ 𝔤

In other words, -′ ′ -- ′ ∂ (ω ⊗ ¯ω ) = ω ⊗ ∂ω¯ , with  *(p,0) ω ∈ 𝔤 and  ′ *(0,q) ¯ω ∈ 𝔤 . Hence, the cohomology of the Dolbeault complex  ∧p,q ℂ * -- ( (𝔤 ) ,∂) is the same as the cohomology of  -- (𝔤*(p,0) ⊗ 𝔤*(0,q),∂′) , which equals ∧ p𝔤*(1,0) ⊗ Hq (𝔤*(0,1)) .

3.3. Abelian complex structures. A complex structure J is called abelian [BDM] if 𝔤1,0 is abelian. This means

[JX, J Y] = [X, Y], ∀X, Y ∈ 𝔤 ,

or equivalently

d𝔤*(1,0) ⊂ 𝔤*(1,1) .

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 ([CFCFPCFGU]). If M is a nilmanifold endowed with an abelian complex structure

 p,q ~ p,q ℂ H ∂ (M ) = H ∂ (𝔤 ).


 0,q q ~ ∧q *(0,1) *(0,q) H ∂-(M ) = H (M, OM ) = 𝔤 = 𝔤 ,

where OM is the structure sheaf of M .

Proof. We use a method very similar to the proof of Nomizu's Theorem. We consider the upper central series {𝔤ℓ} given by 𝔤0 = {0} , 𝔤1 = {X ∈ 𝔤 | [X, 𝔤] = 0 } , 𝔤ℓ = {X ∈ 𝔤 | [X, 𝔤] ⊂ 𝔤ℓ- 1} , 𝔤k = 𝔤 .

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

Moreover, 𝔤1 is central, in particular abelian, and we have the following exact sequence of Lie algebras

0 → 𝔤1 -→ 𝔤 -→ 𝔤 ∕𝔤1 → 0 . abelian

Let G1 be the connected Lie subgroup of G with Lie algebra 𝔤1 (G1 is actually diffeomorphic to  n ℝ , since it is abelian and simply-connected), and let  1 G be the simply-connected nilpotent Lie group with Lie algebra 𝔤∕𝔤1 .

Given the uniform discrete subgroup Γ ⊂ G , since 𝔤1 is a rational subalgebra of 𝔤 , p0(Γ ) is uniform in G1 (cf. [R]). So we have the holomorphic fibration

π0 : M = G ∕Γ → M 1 = G1 ∕p0(Γ )

whose fibre is the torus 𝕋n = G1 ∕Γ ∩ G1 .

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 p : P → B be a holomorphic fibre bundle, with compact connected fibre F , and P and B connected. If F is Kähler there is a spectral sequence (Er, dr) , (r ≥ 0 ) with the following properties:

  1. Er is 4-graded by fibre-degree, base-degree and type;
  2. If p + q = u + v , then  ∑ p,qEu,2v ~= k Hk,∂u- k(B) ⊗ Hp∂-k,q-u+k (F) ;
  3. p,qE*,*⇒ H *,*(P ) ∞ ∂ .

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

We proceed by induction on the dimension. We have

 u,v ∑ k,u- k ∧p -k,q- u+k p,qE 2 ~= k H ∂ (M 1) ⊗ (ℂn ), p,qE˜u,v ~= ∑ Hk,u- k((𝔤∕ 𝔤1)ℂ ) ⊗ ∧p -k,q- u+k(ℂn) 2 k ∂

Now, M 1 is a nilmanifold with an abelian complex structure of dimension lower than M , so Hk,u-k(M 1) ~= Hk,u-k((𝔤∕𝔤 )ℂ) ∂ ∂ 1 . Thus E = ˜E 2 2 and E = ˜E ∞ ∞ . This implies that  p,q ~ p,q ℂ H ∂-(M )= H ∂- (𝔤 ) .

We want to show that H0,q(M ) = Hq (M, O ) ~= ∧q(𝔤*(0,1)) ∂ M . Similarly to the complex parallelizable case, we have that -- *(0,q) ∂ 𝔤 = 0 , for any q , since d𝔤 *(1,0) ⊂ 𝔤*(1,1) , and the Dolbeault complex ∧0,* (𝔤 ℂ)* = 𝔤*(0,*) is the zero complex. □

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

Theorem 3.4 ([CFPMPPS]). Let M be nilmanifold with an abelian complex structure and denote by ΘM the holomorphic tangent sheaf of M . Then H *(M, ΘM ) 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 J preserves the elements of the upper central series { 𝔤ℓ} , i.e., J 𝔤ℓ ⊆ 𝔤ℓ .

Given a complex structure J on a 2n -dimensional nilpotent Lie algebra 𝔤 , one may modify the upper central series so that it is preserved by J . For this purpose, consider the series {𝔤Jℓ} defined inductively by 𝔤J0 = {0} and

 J J J 𝔤ℓ = {X ∈ 𝔤 : [X, 𝔤 ] ⊆ 𝔤ℓ-1, [J X, 𝔤] ⊆ 𝔤 ℓ-1}, ℓ ≥ 1.

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

Note that an abelian complex structure is always nilpotent, with 𝔤J = 𝔤ℓ ℓ , for any ℓ ≥ 0 .

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

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

Hp,q(M ) ~= Hp,q(𝔤ℂ ). ∂ ∂

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

3.5. Rational complex structures. A complex structure J is called rational if it is compatible with the rational structure of the lattice Γ , meaning J (𝔤 ℚ) ⊂ 𝔤ℚ . For this class as well we have

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

H *,*(M ) ~= H *,*(𝔤ℂ ). ∂ ∂

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

 p0 0 → 𝔤1J -→ 𝔤 -→ 𝔤∕𝔤1J ⁄= 0 → 0 abelian ... i i-1 pi-1 i-1 i 0 → 𝔤J -→ 𝔤J → 𝔤J ∕𝔤J → 0 abelian ... 0 → 𝔤s- → 𝔤s-1 p→s-1 𝔤s-1∕𝔤s → 0 J J J J abelian abelian

Here, s is the smallest integer such that 𝔤s+1 = {0 } J .

In general, the subalgebra  i 𝔤J is not a rational subalgebra of  i-1 𝔤J (in which case one would have trouble with discrete subgroups). But if J is rational, any  i 𝔤 J is rational in 𝔤iJ-1 .

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 (E ,d ) r r to the case of non-Kähler fibres [FWL]. 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 ℓ = s 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 ([CFPMPPS]). Any abelian invariant complex structure on a nilmanifold M = G ∕Γ has a locally complete family of deformations (parametrized by a subset  N Kur ⊂ ℂ ) 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 Θ M can be computed using invariant forms and invariant vectors (Theorem 3.4). This implies that there are harmonic representatives of  k H (M, ΘM ) , 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 J of some type (abelian, nilpotent), the deformed structure Jt 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 M with an abelian complex structure, a vector in the virtual parameter space  1 H (M, ΘM ) 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 Jt (t ∈ B ) of a complex structure J0 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  *,* ~ *,* ℂ H ∂- (M, J0)= H ∂-(𝔤 ) holds that for a complex structure J0 . Then this isomorphism still holds for any small deformation Jt (t ∈ B ) of J0 consisting of invariant complex structures.

Sketch of the proof: The idea goes as follows.

For J0 we know that dim H ⊥-*,*(M, J0) = 0 ∂ . Now, by Kodaira-Spencer theory [KS],  ⊥*,* dim H ∂- (M, Jt) is an upper semicontinuous function, so there exists a sufficiently small neighbourhood U0 of J0 such that

∀J ∈ U dim H ⊥*,*(M, J ) ≤ dim H ⊥*,*(M, J ) = 0 . t 0 ∂ t ∂ 0

Thus  *,* *,* H ∂-(M, Jt) ~= H ∂-(𝔤ℂ ) in U0 ⊂ B .

We conclude with an application of this result. By Theorem 4.1, in fact, given an abelian complex structure J , for any small deformation Jt (t ∈ U0 ⊆ Kur ), we have H *∂,*(M, Jt) ~= H *∂,*(𝔤 ℂ) .

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

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

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