ASPHERICAL KÄHLER MANIFOLDS WITH SOLVABLE FUNDAMENTAL GROUP

OLIVER BAUES AND VICENTE CORTÉS

Für Frau Kähler anläßlich von Erich Kählers hundertstem Geburtstag

Abstract. We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of ℂ^{n} by a discrete group of complex isometries.

The purpose of this article is to give an account of recent developments concerning the classification of compact aspherical Kähler manifolds whose fundamental groups contain a solvable subgroup of finite index. This problem stirred some interest among differential geometers for quite some time. It traces back to the classification of Kähler Lie groups and became popular through a conjecture of Benson and Gordon on Kähler quotients of solvable Lie groups. Only recently satisfying answers to the original Benson-Gordon conjecture became available when methods from complex geometry were introduced into the subject. The story as it presents itself now may be seen as an interesting show case where research questions from differential geometry, complex geometry and topology meet.

Outline of the paper. The first section of the paper has a survey character. We introduce the Benson-Gordon conjecture and describe its relation to the classification of Kähler Lie groups, Kähler groups and Kähler manifolds. In the course we also sketch the proof of a theorem of Arapura and Nori on polycyclic Kähler groups. This leads us to the following rigidity result for Kähler infra-solvmanifolds, which, in particular, contains the solution to the conjecture of Benson and Gordon.

Theorem A. Let M be an infra-solvmanifold which admits a Kähler metric. Then M is diffeomorphic to a flat Riemannian manifold.

In section 2 we develop generalisations of the previously described results. These concern the classification of Kähler structures on general smooth aspherical manifolds with polycyclic fundamental group. One new key point is to establish that a Kähler manifold which is homotopy equivalent to a torus is also diffeomorphic to a torus. We prove the stronger result that a Kähler manifold which is a homotopy torus is biholomorphic to a complex torus. This result is based on a detailed study of the Albanese mapping of such Kähler manifolds, which is carried out in section 3. In section 2.2 we introduce our main result:

Date: June 8, 2006.

The first author gratefully acknowledges partial support by DFG Schwerpunkt 1154, Globale Differentialgeometrie. This work was finished at the ESI, Vienna, in October 2005. Both authors wish to thank ESI, and the organizers of the Special Research Semester on Geometry of pseudo-Riemannian manifolds for hospitality and support.

Theorem B. Let X be a compact complex manifold which is aspherical with virtually solvable fundamental group and assume X supports a Kähler metric. Then X is biholomorphic to a quotient of (the hermitian vector space) ℂ^{n} by a discrete group of complex isometries.

The theorem shows that the existence of a Kähler metric on a complex aspherical manifold with virtually solvable fundamental group forces not only the the fundamental group, but also the smooth structure and the complex structure to be as simple as possible.

In order to provide the reader with an overall view, we list the main arguments in the proof of Theorem B, which are essentially the following:

(i) A solvable group which is the fundamental group of a compact aspherical manifold is torsion-free and polycyclic (by results of Bieri mentioned in 1.5).

(ii) A polycyclic Kähler group is virtually nilpotent (by a theorem of Arapura

and Nori stated in 1.5 and explained in 1.6). (iii) A nilpotent aspherical Kähler group is abelian (by work of Benson and

Gordon, Hasegawa, see 1.3). (iv) A Kähler manifold which is homotopy equivalent to a complex torus is

biholomorphic to a complex torus (see Theorem 2 and Theorem 5 in 2.2, section 3 for the proofs).

Acknowledgement. We thank Jörg Winkelmann for discussions which were helpful for the proof of Theorem 5. We are grateful to Fréderic Campana for his detailed comments on the first draft of our paper. We also wish to thank Frank Herrlich for further discussions, and Vincent Koziarz for providing us reference [42].

1. From Kähler Lie groups to solvable Kähler groups

Recall the differential geometer’s definition of a Kähler manifold. A Kähler manifold is a Riemannian manifold endowed with a parallel skew-symmetric complex structure, see [41]. The closed non-degenerate two-form which is associated with such data is called the Kähler form. Notice that, by the Newlander-Nirenberg theorem, a Kähler manifold has an underlying structure of a complex manifold. Our paper is concerned with the existence problem for Kähler structures on solvmanifolds and related spaces. The story starts, maybe, with the study of homogeneous Kähler manifolds and, in particular, Kähler Lie groups.

1.1. Kähler Lie groups. A Lie group is called Kähler if it is equipped with a leftinvariant Kähler metric and a left-invariant Kähler form. The basic examples are flat Kähler Lie groups and solvable Kähler Lie groups acting simply transitively on a bounded domain in ℂ^{n} by biholomorphisms [29]. There is a fundamental conjecture concerning the classification of homogeneous Kähler manifolds up to biholomorphism [29], which was finally settled in [28]. As a reference for the rich structure of solvable Kähler Lie groups and their classification up to (holomorphic) isometry one may consult [52].

Typically, a solvable Lie group endowed with a left-invariant complex structure can admit many non-equivalent irreducible Kähler Lie group structures (contrary to what is claimed in [43]). However, a unimodular Kähler Lie group G is always flat, as proved by Hano [33]. In particular, the Lie group G is solvable of a very restricted type. This follows from the classification of flat Riemannian Lie groups which is given in [47]. As a particular consequence, a Kähler Lie group G which admits a lattice, that is, a discrete cocompact subgroup Γ ≤ G, is flat. Hence, any compact quotient space G/Γ, where G is a Kähler Lie group, is a compact flat Riemannian manifold, and, by Bieberbach’s theorem [64], it is finitely covered by a flat torus.

Let G be a solvable Lie group which admits a lattice Γ. Independently of the existence of a Kähler Lie group structure on G, the question arises whether the compact smooth manifold G/Γ carries a Kähler structure. Benson and Gordon initiated the study of this question in [10, 11]. Their conjecture, which we explain in the next paragraph, suggested an answer to this question for quotient spaces of completely solvable Lie groups.

1.2. A conjecture of Benson and Gordon. A solvable Lie group is called completely solvable if its adjoint representation has only real eigenvalues. In particular, nilpotent Lie groups are completely solvable. Notice that a completely solvable unimodular Kähler Lie group is abelian in virtue of Hano’s result and the classification of flat Lie groups as contained in [47].

Lattices in completely solvable groups satisfy strong rigidity properties, analogous to the Malcev rigidity [44] of lattices in nilpotent Lie groups. This was first proved by Saito, see [54]. In particular, the embeddings of Γ in G are conjugate by an automorphism of G, and the completely solvable Lie group G is determined by Γ up to isomorphism. These results show a particular strong link between the properties of the fundamental group of the space G/Γ and the group G. This may serve as a motivation for the following

Benson-Gordon conjecture [10, 11]: Let M = G/Γ be a compact quotient space of a completely solvable Lie group G by a discrete subgroup Γ. If M admits a Kähler metric then M is diffeomorphic to a standard 2n-torus T^{2n} = S^{1} × ⋯ × S^{1}.

1.3. Kähler nilmanifolds. A nilmanifold N/Γ is a quotient space of a simply connected nilpotent Lie group N by a discrete cocompact subgroup Γ ≤ N. Nilmanifolds satisfy the Benson-Gordon conjecture. This means that any compact nilmanifold which is Kähler is diffeomorphic to a torus.

The reasons for this fact stem from the Hodge theory of Kähler manifolds. The de Rham cohomology ring of a compact Kähler manifold satisfies strong extra conditions, which arise from the existence of the Kähler class and from Hodge theory. Elementary restrictions are that the second Betti-number is positive and odd Betti numbers are even. See [32, 60, 62] for basic reference on this subject, and [26] for the formality of the minimal model of a Kähler manifold.

Now by a result of Nomizu [51], the de Rham cohomology of a nilmanifold N/Γ is computed by the finite-dimensional Koszul complex for the Lie algebra cohomology of N. It was observed in [10] that if the Lie algebra cohomology of N satisfies the strong Lefschetz property, which holds for the cohomology ring of a Kähler manifold, then N is abelian. Also the Koszul complex of N is a minimal model (in the sense of Sullivan, see [26]) of the de Rham complex for N/Γ. The minimal model of a Kähler manifold is formal [26]. As observed in [35], if the Lie algebra cohomology of N is formal, N must be abelian. Another approach, announced in [25], uses the fact that Massey products in Kähler manifolds vanish [26].

However, these techniques did not suffice to prove the full conjecture of Benson and Gordon. (Note that the proof given in [59] contains a mistaken argument.) Moreover, there do exist compact quotient spaces of completely solvable Lie groups with non virtually nilpotent fundamental group, and which have the cohomology ring of a Kähler manifold, see [11, 53].

1.4. Aspherical Kähler groups. In general, it is a delicate problem to understand which finitely presented groups can occur as the fundamental group of a compact Kähler manifold. See [1] for some results and techniques. Traditionally, the fundamental group of a compact Kähler manifold is called a Kähler group. A manifold is called aspherical if its universal covering space is contractible. We call a group an aspherical Kähler group if it is the fundamental group of a compact aspherical Kähler manifold.

The homotopy properties of an aspherical manifold, its homotopy type, and, in particular, the cohomology ring and minimal model depend only on its fundamental group. Recall that every finitely generated torsion-free nilpotent group occurs as the fundamental group of a compact nilmanifold [44]. Thus, the results of Benson and Gordon [10], Hasegawa [35] show that a finitely generated torsion-free nilpotent group is not an aspherical Kähler group, unless Γ is abelian.

Note that this fact, rephrased in topological terms, also shows that a compact aspherical Kähler manifold with nilpotent fundamental group is homotopy equivalent to a torus T^{2n}, for some n.

Let us further remark that, although there are no non-abelian nilpotent aspherical Kähler groups, there are (as shown by Campana [20]) examples of torsionfree two-step nilpotent (in particular not virtually-abelian) Kähler groups. Despite these pioneering examples, nilpotent Kähler groups remain a mystery. Note, however, that formality of Kähler manifolds forces a nilpotent Kähler group to be quadratically presentable, see [22].

1.5. Polycyclic Kähler groups. A polycyclic group is a group which admits a finite normal series with cyclic quotients. See [55] for a comprehensive account on the theory of these groups. Note that, in particular, finitely generated nilpotent groups are polycyclic. But the class of polycyclic groups is much larger. By a result of Mostow [50], the fundamental group of a quotient space of a solvable Lie group is a polycyclic group. More generally, a result of Bieri [12, 13] characterizes torsion-free polycyclic groups as solvable Poincaré duality groups. This implies that a solvable group which is the fundamental group of a compact aspherical manifold is a torsion-free polycyclic group.

Let P be a property of groups. Recall that a group is said to be virtually P if it has a finite index subgroup which satisfies P. Concerning polycyclic Kähler groups there is the following remarkable result which is proved in [4]:

Theorem (Arapura-Nori). Let Γ be a (virtually) polycyclic Kähler group then Γ is virtually nilpotent.

We shall explain the proof of this key result further below. The result of Arapura and Nori as stated in [4] is more general and concerns also quasi-Kähler groups. It was shown by Campana [21] that the assumption “polycyclic” in the above theorem can be generalised to “linear solvable”. The general case seems to have been settled by Delzant in a recent preprint [27] implying that every solvable Kähler group is virtually nilpotent.

Combining the theorem with the results mentioned in 1.4 and 1.5 yields:

Corollary. Every aspherical solvable Kähler group Γ contains a finitely generated abelian subgroup of finite index. In particular, Γ is a Bieberbach group.

Recall that, by definition, a Bieberbach group is a finitely generated torsion-free group which contains an abelian subgroup of finite index. A version of the corollary is also mentioned in [45].

The Benson-Gordon conjecture is actually a straightforward consequence of the corollary. We shall explain this in 1.7. The relevance of the Theorem of Arapura and Nori for the Benson-Gordon conjecture was first recognized by Arapura in [3]. His main result is concerned with the biholomorphic properties of Kähler solvmanifolds and a conjecture of Hasegawa [36, 38]. (Note, however, that the proof contains a gap. See MR 2038772 (2005a:53123).)

Before continuing with the Benson-Gordon conjecture we shall briefly describe the techniques which are behind the proof of the above theorem of Arapura and Nori. The proof is based on ideas and results of Beauville [8, 9] on the character varieties of Kähler groups.

1.6. The character variety of a polycyclic Kähler group. Let Γ be a finitely generated group, and let be k the number of elements of a fixed generating set. Then, as is easy to see, the set of all homomorphisms from Γ to ℂ^{∗} forms an affine algebraic variety, the character variety X_{Γ} ⊂ (ℂ^{∗})^{k} of Γ. Each ρ ∈ X_{Γ} defines a one-dimensional Γ-module ℂ_{ρ}. Let X_{Γ}^{1} denote the subset of characters of Γ which satisfy dim H^{1}(Γ, ℂ_{ρ}) > 0.

Now let Γ be a Kähler group. Note that, for each ρ ∈ X_{Γ}, the Γ-module ℂ_{ρ} defines a flat holomorphic line bundle L_{ρ} over any Kähler manifold X with π_{1}(X) = Γ. Moreover, H^{1}(Γ, ℂ_{ρ}) = H^{1}(X, L_{ρ}). Using this correspondence, geometric arguments and (non-abelian) Hodge theory on X provide information about the set X_{Γ}^{1}. According to Beauville’s main result [9, Corollaire 3.6], X_{Γ}^{1} is a finite union of translates of connected subgroups of X_{Γ} (which arise from homomorphisms of Γ to surface groups) and a finite number of unitary characters. In particular, isolated points of X_{Γ}^{1} are unitary characters. Moreover, it was conjectured in [9] that the isolated points of X_{Γ}^{1} are characters of finite order. This conjecture was proven for the fundamental groups of smooth projective algebraic varieties by Simpson [56] and for general Kähler groups by Campana [21]. For polycyclic Kähler groups the conjecture follows from a direct argument which is based on the next lemma.

Lemma. Let Γ be a finitely generated group with DΓ/D^{2}Γ finitely generated. Then X_{Γ}^{1} is a finite set. If, in addition, Γ is a Kähler group, then the characters of X_{Γ}^{1} are of finite order.

Here, D^{i}Γ denotes the derived series of Γ. We briefly recall the proof of this Lemma, which is given in [9].

Proof. The formula H^{1}(Γ, ℂ_{ρ}) = Hom_{Γ^{ab}}(DΓ/D^{2}Γ, ℂ_{ρ}), where Γ^{ab} = Γ/DΓ denotes the abelianisation, shows that the set X_{Γ}^{1} is finite. Moreover, the values of any ρ ∈ X_{Γ}^{1} on Γ are algebraic integers, since, as the formula shows, ρ occurs as a weight of the adjoint representation of Γ on DΓ/D^{2}Γ ⊗ ℂ. If Γ is Kähler then, by the structure result on X_{Γ}^{1}, ρ is unitary. Therefore, these algebraic integers are of absolute value one, and also all of their algebraic conjugates are. Hence, by Kronecker’s Lemma, these algebraic integers are roots of unity. It follows that ρ is of finite order. □

A polycyclic group satisfies the assumption of the lemma. In fact, every subgroup of Γ is finitely generated, see [55]. Hence, for a polycyclic Kähler group, the elements of X_{Γ}^{1} are of finite order. As observed in [4], we furthermore have:

Lemma. Let Γ be a polycyclic group with X_{Γ}^{1} consisting of characters of finite order, then Γ is virtually nilpotent.

A direct proof for this remark (following the lines of [4]) can be given, by considering the embedding of Γ into its algebraic hull H_{Γ}. The hull H_{Γ} is a linear algebraic group which contains Γ as a Zariski-dense subgroup, and furthermore it has the property that the centraliser of the unipotent radical U of H_{Γ} is contained in U. (See [17], for the general concept of a linear algebraic group, and [49], [4] and also [7, Appendix A] for details on the construction of H_{Γ} and further applications.)

Proof. By replacing Γ with a finite index subgroup we can assume that H_{Γ} is connected. By standard theory of linear algebraic groups [17], the group H_{Γ} has an algebraic splitting H_{Γ} = T U, where T is an algebraic torus, that is, T is a connected abelian group of semi-simple elements. Note that DΓ ⊂ U, and N = DH_{Γ} is the Zariski-closure of DΓ. Since the centraliser of U is contained in U, the abelian reductive group T is faithfully represented on N, and therefore (as holds for reductive actions on unipotent groups) also on N/DN = DΓ/D^{2}Γ ⊗ ℂ. The weights of T on N/DN restrict to elements of X_{Γ}^{1}. Since Γ is Zariski-dense, also the weights of T are of finite order. Since T is connected, they are trivial. Therefore, T centralises U. This implies that H_{Γ} = U is a nilpotent group, and so is Γ. □

1.7. Proof of the Benson-Gordon conjecture. The above corollary to the theorem of Arapura and Nori easily implies the original Benson-Gordon conjecture. In fact, let G be simply connected and completely solvable, and let Γ ≤ G a lattice. By the corollary, the fundamental group Γ of a compact Kähler manifold G/Γ is virtually abelian. The Saito rigidity for lattices in completely solvable Lie groups then implies that G and, therefore, also Γ are abelian, and moreover, G/Γ is diffeomorphic to the torus T^{2n}.

1.8. Kähler infra-solvmanifolds. In the application of the Corollary in 1.5, it is not necessary to restrict oneself to the case of quotient spaces of completely solvable Lie groups. The quotient space G/H of a solvable Lie group G by a closed subgroup H is called a solvmanifold. More generally, we can consider infrasolvmanifolds. Infra-solvmanifolds are constructed using isometric affine actions on solvable Lie groups, see [7, 63] for precise definition and further reference. This class of manifolds contains, in particular, solvmanifolds and flat Riemannian manifolds.

As is well known, every Bieberbach group is the fundamental group of a compact flat Riemannian manifold, see [64]. (Also the converse is true.) Moreover, see [7], every torsion-free virtually polycyclic group occurs as the fundamental group of a compact infra-solvmanifold.

Flat Riemannian manifolds with isomorphic fundamental groups are diffeomorphic, by Bieberbach’s theorem [14]. The same is true for compact solvmanifolds, by Mostow’s famous result [48]. Only recently (see [7]), it was established that infrasolvmanifolds are smoothly rigid, that is, these manifolds are determined by their fundamental group up to diffeomorphism. Therefore, infra-solvmanifolds constitute natural smooth model spaces for compact aspherical manifolds with a torsion-free virtually polycyclic fundamental group. The smooth rigidity of infra-solvmanifolds and the Corollary in 1.5 imply the following — generalised — version of the Benson-Gordon conjecture:

Corollary. Let M be an (infra-) solvmanifold which admits a Kähler metric. Then M is diffeomorphic to a flat Riemannian manifold.

But, of course, the story does not end here.

2. Classification of aspherical Kähler manifolds

We start this section with a short digression on the existence of exotic smooth structures on aspherical manifolds. Then we discuss the classification of aspherical Kähler manifolds with a virtually polycyclic fundamental group up to diffeomorphism and up to biholomorphic equivalence. By the previously discussed results on polycyclic Kähler groups, these classifications are reduced to the classification of Kähler structures on homotopy tori and their quotient spaces. The latter classification is obtained by studying the Albanese mapping of such a Kähler manifold.

2.1. Diffeomorphism classification.

Fake tori. Let T^{n} = S^{1} × ⋯ × S^{1} denote the n-torus with its natural differentiable structure. A compact topological manifold is called a homotopy torus if it is homotopy equivalent to T^{n}. For the classification of homotopy tori up to piece-wise linear equivalence, see [39]. Every homotopy torus is smoothable, but not uniquely. By the work of Browder [18] and Wall [61, §15 A, B], Hsiang and Shaneson [39] there do exist n-dimensional smooth homotopy-tori, n ≥ 5, which are not diffeomorphic to T^{n}. One way to obtain such fake tori is to glue with exotic spheres [18]. By the classification of PL-structures on T^{n}, see [39], there exist even more fake tori. Furthermore, all fake tori admit a finite smooth covering by a standard torus [61, §15 A, B], and are homeomorphic to T^{n} [40].

Recall from 1.6 that, by Mostow’s theorem [48], the natural smooth structure on a solvmanifold M = G/H, which is inherited from G and H, is independent of the choice of G and H, and it is determined by the fundamental group of G/H only. As already mentioned, the analogous fact also holds for the larger class of infra-solvmanifolds. The results cited above show that, surprisingly, there do exist smooth manifolds, even finite smooth quotients of a standard torus T^{n}, which are homeomorphic but not diffeomorphic to an (infra-) solvmanifold.

Aspherical Kähler manifolds with solvable fundamental group. In the light of these facts, it seems natural to ask for the diffeomorphism classification of all compact aspherical Kähler manifolds with virtually solvable fundamental group, instead of restricting our interest to quotient manifolds of solvable Lie groups only. The answer to this question is contained in the following generalisation of the result in 1.8:

Theorem 1. Let M be a compact aspherical Kähler manifold with virtually solvable fundamental group. Then M is diffeomorphic to a flat Riemannian manifold.

This result is implied by the classification of Kähler structures up to biholomorphism which is our next topic.

2.2. Classification up to biholomorphism. Recall that a Kähler manifold is equipped with a unique underlying complex manifold structure. Obvious obstructions for a complex structure to be Kähler arise from Hodge-theory, see [60]. Closely related, to the problem of classifying Kähler manifolds up to diffeomorphism, is the finer classification problem for the corresponding underlying complex manifolds.

A complex torus is a complex manifold ℂ^{n}/ Γ, where Γ ≅ ℤ^{2n} is a lattice in ℂ^{n}. Evidently, complex tori support a flat Kähler metric which is inherited from the flat Kähler metric on ℂ^{n}.

Our principal result in this section asserts that complex tori are the only examples of complex manifold structures on homotopy tori which can be Kähler:

Theorem 2. Let X be a compact complex manifold which is Kähler and homotopy equivalent to a torus T^{2n}. Then X is biholomorphic to a complex torus.

The theorem is a direct consequence of Theorem 5 below. As F. Catanese kindly informed us, Theorem 2 may also be deduced as a special case of [23, Proposition 4.8].

Together with the results of Bieri [12], Benson-Gordon [10], Hasegawa [35], Arapura-Nori [4], Theorem 2 implies:

Corollary 3. Let X be a compact complex manifold which is aspherical with virtually solvable fundamental group and assume X supports a Kähler metric. Then X is biholomorphic to a finite quotient of a complex torus.

We obtain the classification of Kähler aspherical manifolds with solvable fundamental group up to biholomorphism, as formulated in Theorem B in the introduction:

Corollary 4. Let X be a compact complex manifold which is aspherical with virtually solvable fundamental group and assume X supports a Kähler metric. Then X is biholomorphic to a quotient of ℂ^{n} by a discrete group of complex isometries.

Proof. By Corollary 3, X is finitely covered by a complex torus, and the deck transformation group is a finite group of holomorphic maps of a complex torus. Every biholomorphic map of a complex torus is induced by a complex affine transformation of the vector space ℂ^{n}, see [32]. Thus it follows that X is a quotient space of ℂ^{n} by a discrete group Γ of complex affine transformations. Since the translations in Γ form a subgroup of finite index, Γ preserves an invariant scalar product on ℂ^{n}. □

Since the compact complex manifolds occuring in Corollary 4 are quotient spaces of ℂ^{n} by affine isometries, these manifolds also carry a flat Riemannian metric. Therefore, Corollary 4 implies Theorem 1.

For a discussion of the classification of discrete groups of complex isometries on ℂ^{n}, as appearing in Corollary 4, in further detail, see also [45].

The Albanese torus of a Kähler manifold. We shall invoke now a natural construction from complex geometry which is the main tool in our proof of Theorem 2. Let us denote with X a complex manifold, and let us put dim X for its complex dimension. The Albanese torus Alb(X) of X is a complex torus which is naturally associated to any compact Kähler manifold X. Its complex dimension is dim Alb(X) = n, where 2n = dim H^{1}(X, ℂ). See Section 2.1, and [15, 20, 60], or [1, Chapter 1] for further notes on this topic.

Roughly, the Albanese torus is constructed as follows: Let H^{0}(X, Ω_{X}^{1}) denote the complex vector space of holomorphic one-forms on X. Since X is Kähler, Hodge theory implies that Alb(X) = H^{0}(X, Ω_{X}^{1})^{∗}/H_{1}(X, ℤ) is a complex torus. There is a natural holomorphic map, the Albanese morphism, alb_{X} : X → Alb(X), where alb_{X}(x) = ∫_{x_{0}}^{x} associates to x the integral along any path from (a fixed) x_{0} to x, acting on holomorphic one-forms.

Theorem 2 is a consequence of the following rigidity result for Kähler manifolds. Roughly speaking, the result asserts that a Kähler manifold which has the integral cohomology ring of a torus T^{2n} must be biholomorphic to a complex torus:

Theorem 5. Let X be a compact complex manifold, n = dim X, which is Kähler and satisfies 2 dim X = dim H^{1}(X, ℂ). If furthermore dim H^{2}(X, ℂ) ≤ \binom{2n}{2} and H^{2n}(X, ℤ) is generated by integral classes of degree 1, then the Albanese morphism alb_{X} : X → Alb(X) is a biholomorphic map. In particular, X is biholomorphic to a complex torus.

A version of this theorem under the (stronger) assumption that X is diffeomorphic to a 2n-torus is stated in [3] without proof.

Projective examples, abelian varieties. Variants of our results also hold in the algebraic setting. A smooth projective variety is a compact submanifold of complex projective space which is defined by algebraic equations. Since projective space is Kähler, smooth projective varieties are naturally Kähler manifolds as well. An abelian variety is an irreducible complex projective variety with an abelian group structure. One can show (cf. [60, §12.1.3]), if X is projective, then Alb(X) is projective and an abelian variety. Furthermore, the map alb_{X} is a morphism of algebraic varieties.

Corollary 6. Let X be a smooth projective variety which is homotopy-equivalent to a torus T^{2n} then X is an abelian variety.

Proof. By the above, alb_{X} is a biholomorphic morphism X → Alb(X) of projective varieties. By the GAGA theorem, alb_{X} is actually an algebraic isomorphism, that is, the holomorphic inverse of alb_{X} is a morphism of algebraic varieties, cf. [34, Appendix B, Exerc. 6.6]. □

Remark that, in complex dimensions n = 1 and n = 2, all complex structures on T^{2n} are Kähler. In dimension n = 1, this fact is elementary, for n = 2, it may be deduced from the classification of surfaces, as described in [6, VI 1.]. In particular, by Corollary 4, any complex surface X homotopy equivalent to T^{4} is biholomorphic to a complex torus. However, the analogue of this result fails in higher dimensions. In fact, Sommese [57, Remark III, E)] observed the existence of exotic non-Kähler complex structures on the 6-dimensional real torus. These examples are also discussed in Catanese’s paper [23], where the classification problem for complex structures on K(π, 1) compact manifolds is formulated in a spirit similar to the point of view of our paper.

We conclude our discussion with yet another research question. Though, as is clear now, nil- and solvmanifolds are not Kähler, unless they are (finitely covered by) tori, many of them admit a complex manifold structure, see, for example, [16, 30, 37, 24, 58]. It seems tempting to assert that the existence of a complex manifold structure on a smooth manifold which is homotopy equivalent to a solvmanifold forces the differentiable structure to be as nice as possible, that is, such a manifold should be diffeomorphic to a solvmanifold. Hence, is a homotopy torus with complex manifold structure diffeomorphic to a standard torus? More generally, we ask: Let X be a compact complex manifold, aspherical with virtually solvable fundamental group. Is X diffeomorphic to an infra-solvmanifold?

The rest of our paper is devoted to give a proof of Theorem 5.

3. The Albanese morphism of a Kähler manifold

We start by defining the Albanese variety Alb(X) of a Kähler manifold X and deduce some elementary properties of the Albanese morphism alb_{X}. For further reference on Alb(X) one may consult [15, 20], [1, Chapter 1] and [60, §12.1.3].

3.1. The Albanese morphism. Let X be compact Kähler and let H^{0}(X, Ω_{X}^{1}) denote the space of holomorphic one-forms on X. For every path γ : I → X, ω ↦ ∫_{γ} ω defines a linear map H^{0}(X, Ω_{X}^{1}) → ℂ. This correspondence induces a natural map

ι : H_{1}(X, ℤ) → H^{0}(X, Ω_{X}^{1})^{∗} .

Let us put Γ = ι(H_{1}(X, ℤ)). By Hodge-theory, Γ ≅ H_{1}(X, ℤ)/Torsion is a lattice in H^{0}(X, Ω_{X}^{1})^{∗}. Define the Albanese torus Alb(X) of X as

Alb(X) = H^{0}(X, Ω_{X}^{1})^{∗}/ Γ.

Now fix a base-point x_{0} ∈ X. Integration along paths defines a natural holomorphic map, the Albanese morphism alb_{X} : X → Alb(X), via

alb_{X} : x ↦ ∫_{x_{0}}^{x} .

A salient feature of the Albanese construction is that it reduces integration of holomorphic one-forms on X to integration of linear one-forms on Alb(X). In fact, let ω ∈ H^{0}(X, Ω_{X}^{1}) be a holomorphic one-form. By duality of vector spaces, ω defines a (linear) holomorphic one-form ω̄ ∈ H^{0}(Alb(X), Ω_{Alb(X)}^{1}). Then the identity

∫_{γ} alb_{X}^{∗} ω̄ = ∫_{γ} ω

is easily verified. The following lemma is a standard fact, and it is a direct consequence of the preceding remark:

Lemma 7. The Albanese morphism alb_{X} : X → Alb(X) induces an isomorphism of cohomology groups alb_{X}^{∗} : H^{1}(Alb(X), ℤ) → H^{1}(X, ℤ).

Proof. Consider Γ^{∗} ≤ H^{0}(X, Ω_{X}^{1}), the dual lattice for Γ. Then the corresponding linear forms ω̄, ω ∈ Γ^{∗}, define one-forms in Ω^{1}(Alb(X)) which represent the integral cohomology H^{1}(Alb(X), ℤ). Recall that H^{1}(X, ℤ) has no torsion. Then, by the definition of Γ, integration of forms on the cycles of X identifies Γ^{∗} with H^{1}(X, ℤ). Thus, the identity alb_{X}^{∗} ω̄ = ω proves the lemma. □

3.2. The mapping degree of alb_{X}. We employ here the differential topological notion of the mapping degree. Let f : M → N be a smooth map, where M and N are compact oriented smooth manifolds, k = dim M = dim N. Let y ∈ N be a regular value for f, then

deg f = ∑_{x∈f^{−1}(y)}sign det(df_{x})

is an integer which is independent of the choice of y, see [46]. The degree of f also coincides with the unique integer which is defined by the induced map f^{∗} : H^{k}(N, ℤ) → H^{k}(M, ℤ) relative to the orientation classes of M and N. Note the following particular useful consequence of the definition of degree: if deg f is nonzero then f must be surjective.

We shall apply this concept to holomorphic maps f : X → Y between compact complex manifolds X and Y. Here k = 2n, where n = dim X = dim Y. Both X and Y have natural orientations determined by their complex atlas. Moreover, with respect to these orientations, sign det(df_{x}) = 1, for all regular points x ∈ X. Thus if f is holomorphic, its degree deg f is a non-negative integer, and deg f counts precisely the number of elements in the fibers over the regular values of f.

Next we provide a criterion for finiteness of the fibers of a holomorphic map. A proper holomorphic map f : X → Y is called finite if for all y ∈ Y, f^{−1}(y) is a finite set.

Lemma 8. Assume that X is compact Kähler and that the Kähler class [ω] of X is contained in f^{∗}H^{2}(Y, ℝ) then f is finite.

Proof. It is enough to prove that the connected components of f^{−1}(y) are points. Let F, dim F = p be such a component, and consider ω_{p} = ω_{F}^{p} the restriction of the p-th power of the Kähler form ω on F. Then ω_{p} is a positive definite volume-form on the analytic subvariety F of X. The assumption on the Kähler class implies that ω_{p} is cohomologous to a pull-back form f^{∗}(β), where β ∈ Ω^{2p}(Y). The integral of forms on analytic varieties is well defined, and by a version Stoke’s theorem descends to cohomology classes, see [42] and [60, Theorem 11.21]. Since the pull-back f^{∗}(β) has zero integral on F, this also holds for ω_{p}. Hence, F must be a point. □

We are ready for the proof of our main result on the Albanese mapping.

Proof of Theorem 5. Our strategy is to prove that the Albanese mapping alb_{X} : X → Alb(X) is a holomorphic branched covering (a so-called analytic covering, see [31, Ch. 7, §2] for the definition) using the assumptions on the cohomology. Since X and Alb(X) are compact complex manifolds, this amounts to showing that alb_{X} is surjective and has finite fibers, see [31, Ch. 9 §3.3].

Note that, by Lemma 7, the Albanese mapping alb_{X} induces an isomorphism H^{1}(Alb(X), ℤ) → H^{1}(X, ℤ). This and the assumption that H^{1}(X, ℤ) generates the top cohomology H^{2n}(X, ℤ) of X implies that alb_{X}^{∗} : H^{2n}(Alb(X), ℤ) → H^{2n}(X, ℤ) is an isomorphism on the integral top-cohomology groups. Thus deg alb_{X} = 1. In particular, alb_{X} is also a surjective map.

As a general fact, if X is compact Kähler and f : X → Y is a surjective holomorphic map then (using techniques similar to the proof of Lemma 8) the induced map f^{∗} : H^{∗}(Y, ℝ) → H^{∗}(X, ℝ) is injective, see [20, Lemme 2.4] or [60, Lemma 7.28]. In particular, alb_{X}^{∗} : H^{2}(Alb(X), ℂ) → H^{2}(X, ℂ) is injective and, hence, an isomorphism, since we have assumed that dim H^{2}(X, ℂ) ≤ dim H^{2}(Alb(X), ℂ). Applying Lemma 8 we can deduce that alb_{X} is a finite map and, hence, an analytic covering.

Since alb_{X} has degree 1, it is injective at regular points. But at the singular points the number of elements in the fibers cannot increase. For a proof of the latter fact, see for example [31, Chapter 8, §1]. Thus alb_{X} is bijective, hence a holomorphic homeomorphism.

The latter fact implies that alb_{X} is biholomorphic. This can be deduced as follows: The continuous map alb_{X}^{−1} : Alb(X) → X is holomorphic outside the set S_{Alb(X)} ⊂ Alb(X) of singular values of alb_{X}. The set of singular points S_{X} of alb_{X} is clearly an analytic subset of X and is mapped onto S_{Alb(X)} by the finite holomorphic map alb_{X}. This implies that S_{Alb(X)} ⊂ Alb(X) is an analytic subset, by [31, Proposition 4.1.6]. Since Alb(X) is a complex manifold, the Riemann extension theorem [31, Theorem 7.1.1] shows that alb_{X}^{−1} is holomorphic. Thus alb_{X} is biholomorphic. □

References

[1] J. Amors, M. Burger, K. Corlette, D. Kotschick, D. Toledo, Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, 44. American Mathematical Society, Providence, RI, 1996.

[2] D. Arapura, Geometry of cohomology support loci for local systems I, J. Algebraic Geom. 6 (1997), 563-597.

[3] D. Arapura, Kähler solvmanifolds, Int. Math. Res. Not. 3 (2004), 131-137.

[4] D. Arapura, M. Nori, Solvable fundamental groups of algebraic varieties and Kähler manifolds, Compositio Math. 116 (1999), no. 2, 173-188.

[5] L. Auslander, An exposition of the structure of solvmanifolds, I. Algebraic theory, Bull. Amer. Math. Soc. 79 (1973), no. 2, 227-261.

[6] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Bd. 4, Springer-Verlag, Berlin, 1984.

[7] O. Baues, Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology, 43 (2004), no. 4, 903-924.

[8] A. Beauville, Annulation du H^{1} et systèmes paracanonique sur les surfaces, J. reine angew. Math. 388 (1988), 149-157.

[9] A. Beauville, Annulation du H^{1} pour les fibrés en droites plats, Lecture Notes in Math. 1507, Springer-Verlag, 1992, 1-15.

[10] C. Benson, C.S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513-518.

[11] C. Benson, C.S. Gordon, Kähler structures on compact solvmanifolds, Proc. Amer. Math. Soc. 108 (1990), no. 4, 971-980.

[12] R. Bieri, Gruppen mit Poincaré-Dualität, Comment. Math. Helv. 47 (1972), 373-396.

[13] R. Bieri, Homological dimension of discrete groups, Queen Mary College Mathematical Notes (second edition), London, 1981. iv+198 pp.

[14] L. Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume II, Math. Ann. 72 (1912), 400-412.

[15] A. Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. (3) 73 (1956), 157-202.

[16] C. Borcea, Moduli for Kodaira surfaces, Compositio Math. 52 (1984), no. 3, 373-380.

[17] A. Borel, Linear algebraic groups, Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991.

[18] W. Browder, On the action of Θ^{n} (∂π), Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), 23-36, Princeton Univ. Press, Princeton, N.J., (1965).

[19] A. Brudnyi, Solvable quotients of Kähler groups, Michigan Math. J. 51 (2003), no. 3, 477-490.

[20] F. Campana, Remarques sur les groupes de Kähler nilpotents, Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 3, 307-316.

[21] F. Campana, Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler, J. Algebr. Geom. 10 (2001), no. 4, 599-622.

[22] J. Carlson, D. Toledo, Quadratic presentations and nilpotent Kähler groups, J. Geom. Anal. 5 (1995), no. 3, 359-377.

[23] F. Catanese, Deformation types of real and complex manifolds, Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000), 195-238, Nankai Tracts Math., 5, World Sci. Publishing, River Edge, NJ, 2002.

[24] S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), no. 2, 111-124.

[25] L. Cordero, M. Fernández, A. Gray, Minimal models in differential geometry, Proceedings of the Workshop on Recent Topics in Differential Geometry (Puerto de la Cruz, 1990), 31-41, Informes, 32, Univ. La Laguna, La Laguna, 1991.

[26] P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245-274.

[27] T. Delzant, L’invariant de Bieri Neumann Strebel des groupes fondamentaux des varietes kaehleriennes, arXiv:math.DG/0603038.

[28] J. Dorfmeister, K. Nakajima, The fundamental conjecture for homogeneous Kähler manifolds, Acta Math. 161 (1988), no. 1-2, 23-70.

[29] S.G. Gindikin, I I. Pjateckiĭ-Šapiro, E.B. Vinberg, Homogeneous Kähler manifolds 1968, Geometry of Homogeneous Bounded Domains (C.I.M.E., 3 Ciclo, Urbino, 1967), 3-87.

[30] G. Grantcharov, C. McLaughlin, H. Pedersen, Y.S. Poon, Deformations of Kodaira manifolds, Glasg. Math. J. 46 (2004), no. 2, 259-281.

[31] H. Grauert, R. Remmert, Coherent analytic sheaves, Grundlehren der mathematischen Wissenschaften, 265, Springer Verlag, 1984.

[32] P. Griffiths, J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience, New York, 1978.

[33] J. Hano, On Kählerian homogeneous spaces of unimodular Lie groups, Amer. J. Math. 79 (1957), 885-900.

[34] Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York, 1977.

[35] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65-71.

[36] K. Hasegawa, A class of compact Kählerian solvmanifolds and a general conjecture, Geom. Dedicata 78 (1999), no. 3, 253-258.

[37] K. Hasegawa, Complex and Kähler structures on compact solvmanifolds, Journal of symplectic geometry 3 (2005), no. 4, 749-767.

[38] K. Hasegawa, A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), 131-135.

[39] W.C. Hsiang, J.L. Shaneson, Fake tori, the annulus conjecture, and the conjectures of Kirby, Proc. N.A.S. 62 (1969), 687-691.

[40] W.C. Hsiang, C.T.C. Wall, On homotopy tori II, Bull. London Math. Soc. 1 (1969), 341-342.

[41] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969.

[42] P. Lelong, Intégration sur un ensemble analytique complexe, Bull. Soc. Math. France 85 (1957), 239-262.

[43] A. Lichnerowicz, Groupes kähleriens, C.R.Acad.Sci. Paris Sér. I Math. 310 (1990), no. 9, 671-676.

[44] A.I. Mal’cev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 39 (1951), 1-33.

[45] N. J. Michelacakis, On a special class of fibrations and Kähler rigidity, Pacific J. Math. 219 (2005), no. 1, 1-12.

[46] J. Milnor, Topology from the differentiable view point, Princeton University Press, Princeton, NJ, 1997.

[47] J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293-329.

[48] G.D. Mostow, Factor spaces of solvable groups, Ann. of Math. 60 (1954), 1-27.

[49] G.D. Mostow, Representative functions on discrete groups and solvable arithmetic subgroups, Amer. J. Math. 92 (1970).

[50] G.D. Mostow, On the fundamental group of a homogeneous space, Ann. of Math. (2) 66 (1957), 249-255.

[51] K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59, (1954), 531-538.

[52] I.I. Pyateskii-Shapiro, Automorphic functions and the geometry of classical domains, Mathematics and Its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris 1969.

[53] H. Sawai, T. Yamada, Lattices on Benson-Gordon type solvable Lie groups, Topology Appl. 149 (2005), no. 1-3, 85-95.

[54] M. Saito, Sur certains groupes de Lie résolubles II, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 7 (1957), 157-168.

[55] D. Segal, Polycyclic Groups, Cambridge University Press, London, 1983.

[56] C. Simpson, Subspaces of moduli spaces of rank one local systems, Ann. Sci. Éc. Norm. Supér., IV. Sér. 26 (1993), no. 3, 361-401.

[57] A. J. Sommese, Quaternionic manifolds, Math. Ann. 212 (1974/75), 191-214.

[58] A. Tralle, Symplectic manifolds with no Kähler structure, LNM 1661, Springer Verlag, Berlin-Heidelberg, 1997.

[59] A. Tralle, J. Kedra, Compact completely solvable Kähler solvmanifolds are tori, Internat. Math. Res. Notices 15 (1997), 727-732.

[60] C. Voisin, Hodge Theory and complex algebraic geometry I, Cambridge studies in advanced mathematics, 76, Cambridge University Press 2002.

[61] C.T.C. Wall, Surgery on compact manifolds, Math. Surveys and Monographs, 69, 1999 (second ed.), AMS, 1999.

[62] R.O. Wells, Differential analysis on complex manifolds, Graduate Texts in Mathematics, 65, Springer-Verlag, New York, 1980.

[63] B. Wilking, Rigidity of group actions on solvable Lie groups, Math. Ann. 317 (2000), no. 2, 195-237.

[64] J.A. Wolf, Spaces of constant curvature, Fifth edition, Publish or Perish, Inc., Houston, TX, 1984.

Mathematisches Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany

   E-mail address: baues@math.uni-karlsruhe.de

Institut Élie Cartan, Université Henri Poincaré - Nancy I, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France

   E-mail address: cortes@iecn.u-nancy.fr