Quadratic Presentations and Nilpotent Kähler Groups

James A. Carlson and Domingo Toledo To appear in Journal of Geometric Analysis

1. Introduction

It has been known for at least thirty years that certain nilpotent groups cannot be Kähler groups, i.e., fundamental groups of compact Kähler manifolds. The best known examples are lattices in the three-dimensional real or complex Heisenberg groups. It is also known that lattices in certain other standard nilpotent Lie groups, e.g., the full group of upper triangular matrices and the free k-step nilpotent Lie groups, k > 1, are not Kähler. The Heisenberg case was known to J-P. Serre in the early 1960’s, and unified proofs of the above statements follow readily from Sullivan’s theory of minimal models [6],[15], [19], or from Chen’s theory of iterated integrals [4], [10], or from more recent developments such as [9].

We recall that in [6] compact Kähler manifolds are shown to be formal in the sense of rational homotopy theory. This implies formality of the one-minimal model of the manifold, meaning that this object can be constructed formally from H^{1} and the kernel of the cup-product map. Equivalently, the Lie algebra of the Malcev completion of the fundamental group is quadratically presented, i.e., is the quotient of the free Lie algebra on its abelianization by an ideal generated in degree two. It is not hard to see that the Lie algebras of the groups mentioned above do not admit quadratic presentations.

It seems to be less well known that many nilpotent Lie algebras have quadratic presentations, for instance the real or complex Heisenberg Lie algebras of dimension at least five. Thus the methods of rational homotopy theory do not exclude lattices in the corresponding Lie groups from being Kähler, contrary to an apparently commonly held impression that the only nilpotent Kähler groups are almost abelian. An exception is [12], where the problem of lattices in higer-dimensional Heisenberg groups being Kähler is explicitly posed.

This matter has finally been settled by the remarkably simple examples of nilpotent Kähler groups found by Campana [2]. Remarkable also is the fact that such examples were already in the literature [18], presumably with the connection with nilpotence passing unnoticed. Campana constructs Kähler lattices in all real Heisenberg groups of dimension at least nine. In our unpublished notes [3] we had proved that no lattice in the five or seven-dimensional Heisenberg group is Kähler, so this construction is sharp for the Heisenberg groups. It is also sharp with respect to the rank of the abelianization. Namely, [3] gives (and we prove this below) that the abelianization of a nilpotent, not almost abelian Kähler group must have rank at least eight. The fact that abelianization of rank four is impossible appears in [2] by a different argument, but the mention there of the impossibility of rank six is based on [3].

The general question raised by these results is: which finitely generated nilpotent groups with quadratically presented Malcev Lie algebras are Kähler groups? A less delicate question would be:

Research partially supported by the National Science Foundation for which quadratically presented nilpotent real Lie algebras (with rational structure constants) is there a lattice in the corresponding simply connected nilpotent Lie group which is a Kähler group? For the Heisenberg Lie algebras of dimension at least five this question is answered above.

This paper addresses two related questions. First, how wide is the class of quadratically presented Lie algebras? We give an infinite family of examples of quadratically presented three-step nilpotent Lie algebras, thus indicating that this class is quite wide. We also classify quadratically presented complex nilpotent Lie algebras with abelianization of dimension at most five. It turns out that there are only finitely many isomorphism classes, and they are at most three-step nilpotent. As part of work now in progress, S. Chen has produced examples of quadratically presented nilpotent Lie algebras of arbitrarily high class of nilpotency, beginning with a four-step nilpotent one with 6-dimensional abelianization. Thus this class of algebras is even wider than indicated by the results of this paper.

The second question we address is how restricted within the class of quadratically presented Lie algebras is the subclass of Malcev Lie algebras of nilpotent Kähler groups. Using Kähler geometry we deduce a lower bound on the rank of certain elements in the kernel of the cup-product map. This implies the lower bound on the rank of abelianization mentioned above. We also show that this excludes certain infinite families of quadratically presented Lie algebras, containing algebras with abelianization of arbitrarily large dimension, from being Malcev algebras of Kähler groups. This appears to be the only restriction beyond quadratic presentation that is known as of this writing on nilpotent Kähler groups.

These results raise two specific questions (also raised in [2]). First, are there nilpotent Kähler groups of class larger than two? Note that there are no known obstructions to many of our threestep nilpotent examples being Kähler. Second, can the kernel of the cup-product map of a Kähler manifold with nilpotent fundamental group have elements of type (2,0)? If the answer to this question were negative, then our main theorem would imply that there are no elements of rank less than eight in the kernel of the cup-product map of a nilpotent Kähler group.

In view of the preceding discussion it appears that the difficult problem of classifying Kähler groups is wide open even in the restricted context of nilpotent groups. It is interesting to note that the situation for polycyclic groups is better, in the sense that it has been reduced to nilpotent groups. Namely, Arapura and Nori prove in [1] that a polycyclic Kähler group must be almost nilpotent.

We thank W. Goldman, J. Kollár and J. Millson for helpful discussions, and S. Chen, P. Deligne and J-P. Serre for suggesting improvements to our original manuscript. This paper is based on [3] and on a lecture delivered at the joint meeting of the American Mathematical Society and the Sociedad Matemática Mexicana held in Mérida in December of 1993 to celebrate the 50^{th} anniversary of the SMM. We thank the organizers of this meeting for their hospitality.

2. Differential algebras and Lie algebras

As noted above, a compact Kähler manifold is formal, as is the 1-minimal model of a formal space. In general the 1-minimal model of a space X is a free, connected, minimal differential algebra M^{(1)}, generated in degree one, which maps to the de Rham algebra of X, induces an isomorphism on H^{1}, and induces an injection on H^{2}. The 1-minimal model M^{(1)} of X is said to be formal if it is the same as the 1-minimal model of the differential algebra H^{∗}(X) with zero differential. Equivalent to this is the existence of a map of differential algebras M^{(1)} ⟶ H^{∗}(X) inducing an isomorphism on H^{1} and an injection on H^{2}. In terms of M^{(1)} alone this means that there is a map of differential algebras M^{(1)} ⟶ H^{∗}(M^{(1)}) inducing an isomorphism on H^{1} and an injection on H^{2}.

We also recall from [6, p. 87] that the 1-minimal model is constructed inductively as an increasing union

M_{1}^{(1)} ⊂ M_{2}^{(1)} ⊂ ⋯ ⊂ M_{i}^{(1)} ⊂ ⋯ ⊂ M^{(1)},

where each M_{i+1}^{(1)} is a Hirsch extension of M_{i}^{(1)} defined precisely so as to kill the kernel of H^{2}(M_{i}^{(1)}) ⟶ H^{2}(X). More explicitly, if V = H^{1}(X), then M_{1}^{(1)} = ΛV, and M_{2}^{(1)} = Λ(V ⊕ C_{0}), where C_{0}, a space homogeneous of degree one, and d : C_{0} ⟶ Λ^{2}V are defined so that the sequence

0 ⟶ C_{0} ⟶ Λ^{2}V ⟶ H^{2}(X)  (2.1)

is exact, where the last map is the cup-product. Consequently C_{0} is isomorphic to the kernel of the cup-product map Λ^{2}H^{1}(X) ⟶ H^{2}(X). The differential on the free differential algebra M_{2}^{(1)} is determined by the requirement that it be zero on V and that on C_{0} it be as just defined. Thus d has degree one in the grading of Λ(V ⊕ C_{0}).

Now, if X is formal, then the construction of the remaining M_{i}^{(1)} is purely formal in the sense that it makes no further use of X. It can easily be checked that the construction of M^{(1)} described in [6] proceeds as follows in the formal case. Define C_{1} to be a space of homogeneous elements of degree one and define d : C_{1} ⟶ V ⊗ C_{0} so that the sequence

0 ⟶ C_{1} ⟶ V ⊗ C_{0} ⟶^{m} Λ^{3}V  (2.2)

is exact, where m is defined by m(v ⊗ c) = v ∧ dc for all v ∈ V and c ∈ C_{0}. For i ≥ 2, define a space C_{i} of homogeneous degree one and d : C_{i} ⟶ V ⊗ C_{i−1} inductively by the requirement that the sequence

0 ⟶ C_{i} ⟶ V ⊗ C_{i−1} ⟶^{m} Λ^{2}V ⊗ C_{i−2}  (2.3)

is exact, where m is defined to be the composition

V ⊗ C_{i−1} ⟶^{1⊗d} V ⊗ V ⊗ C_{i−2} ⟶^{∧⊗1} Λ^{2}V ⊗ C_{i−2}.

Then M_{i}^{(1)} is the free minimal differential algebra generated in degree one by the space

V ⊕ C_{0} ⊕ ⋯ ⊕ C_{i−2}.

It is therefore the exterior algebra

Λ(V ⊕ C_{0} ⊕ ⋯ ⊕ C_{i−2})  (2.4)

with differential determined by the conditions that on V, d = 0, and that on C_{i}, i ≥ 0, d is defined as above.

Observe that in [19, Theorem 12.1] it is proved that formality is independent of the ground field. For this reason we will sometimes be vague on the field of definition (ℚ, ℝ or ℂ).

Finally, we recall the relation of the 1-minimal model to the Malcev Lie algebra L of π_{1}(X). First, L is a tower of graded, nilpotent Lie algebras {L_{i}}:

⋯ ⟶ L_{3} ⟶ L_{2} ⟶ L_{1} ⟶ 0.

Each element of this tower L_{i} is dual to the differential algebra M_{i}^{(1)}. This means that

L_{i}^{∗} ≈ V ⊕ C_{0} ⊕ C_{1} ⊕ ⋯ ⊕ C_{i−2},

and that, under this isomorphism, the restriction to L_{i}^{∗} of the differential d defined above is the map d : L_{i}^{∗} ⟶ Λ^{2}L_{i}^{∗} dual to the Lie bracket Λ^{2}L_{i} ⟶ L_{i}. Formality of M^{(1)} is equivalent to quadratic presentation of L in the sense that the tower {L_{i}} is isomorphic to the tower of maximal i-step nilpotent quotients of F(V^{∗})/I, where F(V^{∗}) denotes the free Lie algebra [17] on V^{∗}, and I is a homogeneous ideal generated by I_{2}, its homogeneous component of degree 2. Moreover I_{2} ⊂ F_{2}(V^{∗}) ≈ Λ^{2}V^{∗} is the annihilator of C_{0} ⊂ Λ^{2}V. Equivalently, I_{2} is the image of the map H_{2}(X) ⟶ Λ^{2}V^{∗} = Λ^{2}H_{1}(X) dual to the cup-product map Λ^{2}H^{1}(X) ⟶ H^{2}(X). This is a special case of results in [15], and can be seen by a discussion similar to, but simpler than, that surrounding the proof of Theorem (9.4) of that paper, namely drop all discussions of bi-grading, and replace his generating space A by our simpler V^{∗}.

Since we are only interested in formal spaces X with π_{1}(X) nilpotent, we will only be interested in the case in which the tower L is finite. Equivalently M^{(1)} = M_{j}^{(1)} for some j, or C_{i} = 0 for some i. We collect this summary of known facts in the following proposition:

2.1. Proposition: There is a one-to-one correspondence between nilpotent, quadratically presented Lie algebras and finite dimensional, formal, free minimal differential algebras generated in degree one. This correspondence assigns to a Lie algebra L = F(V^{∗})/I, where F(V^{∗}) is the free Lie algebra on V^{∗} and I is a homogeneous ideal generated by I_{2}, its homogeneous component of degree two, the differential algebra (2.4). The space d(C_{0}) ⊂ Λ^{2}V is the annihilator of I_{2} and C_{i} is defined as above for i ≥ 1. If C_{j} ≠ 0 and C_{j+1} = 0, the class of nilpotency of L is j + 2. The Malcev Lie algebra of π_{1}(X), where X is a formal space with π_{1}(X) nilpotent, is obtained in this way with V = H^{1}(X) and d(C_{0}) the kernel of the cup-product map Λ^{2}V ⟶ H^{2}(X).

Using this proposition, we will sometimes describe a Lie algebra by describing the corresponding differential algebra. This correspondence also gives us a very useful invariant of a quadratically presented Lie algebra, which we now define.

2.2. Definition: If L is a nilpotent Lie algebra as in the proposition (or a quadratically presented tower as explained above), its characteristic subspace C ⊂ Λ^{2}V is the annihilator of I_{2} ⊂ Λ^{2}V^{∗} ≈ F_{2}(V^{∗}). The characteristic classes of L are the elements of C.

Remark: If L is the Malcev Lie algebra of π_{1}(X), then its characteristic subspace C ⊂ Λ^{2}V = Λ^{2}H^{1}(X) is the kernel of the cup-product map. For formal X we have seen that C determines the 1-minimal model of X and hence the Malcev Lie algebra L.

We illustrate this proposition and the concept of the characteristic subspace with the simplest examples of non-abelian, quadratically presented nilpotent Lie algebras.

2.3. Example: The higher-dimensional Heisenberg algebras. Let V^{∗} be a symplectic vector space of dimension at least four with symplectic form ω = x_{1} ∧ y_{1} + ⋯ x_{n} ∧ y_{n}, n ≥ 2. Let C ⊂ Λ^{2}V be the one-dimensional space with basis ω. The kernel of the map m in (2.2) is {v⊗ω : v∧ω = 0} = 0, (because n ≥ 2), and so this map is injective. Thus, if we let C_{0} be a one-dimensional space with basis z of degree one, and let d : C_{0} ⟶ C defined by dz = ω, then the process gives C_{1} = 0. Consequently the minimal differential algebra

Λ(x_{1}, y_{1} ⋯, x_{n}, y_{n}, z), dz = x_{1} ∧ y_{1} + ⋯ x_{n} ∧ y_{n}  (2.5)

is formal and corresponds to a two-step nilpotent, quadratically presented Lie algebra, the 2n + 1dimensional Heisenberg Lie algebra, which we denote by H(n). This is the Lie algebra of the Heisenberg group H(n) of matrices of the form

g = ( \begin{matrix}1 & x & z \\ 0 & I & y \\ 0 & 0 & 1 \end{matrix} )  (2.6)

where I is the n × n unit matrix, x is an n-dimensional row vector, y is an n-dimensional column vector, and z is a scalar. A quadratic presentation of H(n) is obtained by taking the dual basis {X_{1}, Y_{1}, ⋯, X_{n}, Y_{n}} of V^{∗} and writing down the annihilator of ω. Doing so, we obtain the relations

[X_{i}, X_{j}] = 0, [Y_{i}, Y_{j}] = 0, [X_{i}, Y_{j}] = 0, i ≠ j

and

[X_{i}, Y_{i}] = [X_{j}, Y_{j}].

From these and the Jacobi identity one easily finds that Z = [X_{1}, Y_{1}] = ⋯ = [X_{n}, Y_{n}] is central and that the algebra is two step nilpotent. (The assumption n ≥ 2 is critical here.)

Remark If in the above example we set n = 1 we obtain the three-dimensional Heisenberg Lie algebra H(1) which is well known not to be quadratically presented. One proof is as follows. Let x and y be a basis for V, and let X, Y be the dual basis. Since [X, Y] is nonzero, the image of x ∧ y is nonzero. Therefore C_{0} is the one-dimensional space with basis x∧y. But the image of x⊗(x∧y) and y ⊗ (x ∧ y) under the map m in (2.2) is zero, so C_{1} is nonzero. Thus, if H(1) were formal, it would have index of nilpotence greater than two, a contradiction. Otherwise said, the formal 1-minimal model determined by the same cup-product relations as H(1) is infinite-dimensional: for each i ≥ 1 the elements (z_{j_{1}} ⊗⋯⊗z_{j_{i}})⊗(x∧y), where each z equals either x or y, give linearly independent elements of C_{i}. Thus for all i, C_{i} ≠ 0, and so the dual Lie algebra is not nilpotent. This way of thinking about H(1) leads to a very useful lemma:

2.4. Lemma: Let C ⊂ Λ^{2}V be the characteristic subspace of a nilpotent, quadratically presented Lie algebra L = F(V^{∗})/I, and let v ∈ C ⊗ ℂ, v ≠ 0. Then v is not decomposable.

Proof: Suppose, on the contrary, that L is quadratically presented and v = x ∧ y ∈ C ⊗ ℂ. Apply the last argument of the remark to contradict the nilpotency of L ⊗ ℂ. An alternative argument would be to complete x, y to a basis of V ⊗ ℂ, let X, Y ∈ V^{∗} ⊗ ℂ be dual to x, y, and observe that the elements of I_{2} contain no occurence of the monomial [X, Y], so that the free Lie algebra F(X, Y) has zero intersection with I, hence injects into L ⊗ ℂ, contradicting the nilpotency of L. In this connection, compare with Theorem 4.1 of [4] .

2.5. Corollary: The full algebra of strictly upper-triangular (n + 1) × (n + 1) matrices, n > 1, is not quadratically presented.

Proof: This Lie algebra is generated by the space V^{∗} consisting of matrices with only the entries directly above the main diagonal not zero, which has basis X_{1}, ⋯, X_{n} where X_{i} has all entries zero except for position (i, i + 1) where it has a one. Thus we can write the algebra as F(V^{∗})/I, and it is easy to see that the homogeneous component of degree two, I_{2}, has basis

[X_{i}, X_{j}], j − i > 1.

Therefore, if x_{1}, ⋯, x_{n} denotes the dual basis of V, then the characteristic space C has basis

x_{i} ∧ x_{i+1}, i = 1, ⋯, n − 1,

thus, by the lemma, I_{2} cannot generate I.

2.6. Example: Three-step nilpotent algebras: We now produce, for each integer m of the form m = k(k+3)/2, k ≥ 2 a three-step nilpotent quadratically presented Lie algebra with abelianization of dimension m. Let V be a vector space of dimension m, with a basis which we divide in two subsets:

{x_{1}, ⋯ x_{k}}, {y_{ij} : i, j = 1, ⋯, k; y_{ij} = y_{ji}}.

Let c_{i} ∈ Λ^{2}V be defined by

c_{i} = ∑_{j} x_{j} ∧ y_{ij},

and let C ⊂ Λ^{2}V be the subspace with basis c_{1}, ⋯, c_{k}. Set C_{0} = C, with the homogeneous degree shifted down by one, and define d : C_{0} ⟶ C to be the identity map. For i ≥ 1 let C_{i} be defined by the inductive process described above. Form the corresponding differential algebra, let L(k) denote the corresponding Lie algebra, nilpotent according to Proposition 2.7, and let L(k) denote the corresponding simply connected nilpotent Lie group. Since the structure constants of L(k) are rational, this group contains lattices [13, Therorm 7].

2.7. Proposition: The Lie algebra L(k) just defined for each k ≥ 2 is a quadratically presented three-step nilpotent Lie algebra of dimension (k^{2} + 5k + 2)/2. Its abelianization has dimension k(k + 3)/2 and its center is one-dimensional.

Proof: By construction, L(k) is quadratically presented and its abelianization is the vector space V of the asserted dimension. To prove the remaining assertions we must show that dim (C_{1}) = 1 and that C_{2} = 0. Recall that C_{1} is defined to be the kernel of the restriction to V ⊗ C_{0} of the wedge product map V ⊗ Λ^{2}V ⟶ Λ^{3}V.

We claim that C_{1} is one-dimensional with basis b = x_{1} ⊗ c_{1} + ⋯ + x_{k} ⊗ c_{k}. First, note that b ∈ C_{1}. Next, suppose a ∈ C_{1}. Since c_{1}, ⋯, c_{k} is a basis for C_{0}, there exist v_{1}, ⋯, v_{k} ∈ V such that a = v_{1} ⊗ c_{1} + ⋯ + v_{k} ⊗ c_{k}. Write v_{i} = ∑_{l} α_{il}x_{l}. Then a ∈ C_{1} means that

∑_{i,j,l} α_{il} x_{l} ∧ x_{j} ∧ y_{ij} = 0.

If i ≠ l, the coefficient of x_{l} ∧ x_{i} ∧ y_{ii} in this expression is α_{il}, so that α_{il} = 0 for i ≠ l. Moreover, if i ≠ j, the coefficient of x_{i} ∧ x_{j} ∧ y_{ij} is α_{ii} − α_{jj}, hence α_{ii} = α_{jj}. Thus there is a scalar λ so that v_{i} = λx_{i} for i = 1, ⋯, k, i.e., a = λb, and C_{1} is one-dimensional with basis b, as asserted.

If x ∈ C_{2}, then there exists a v ∈ V such that x = v ⊗ b and ∑(v ∧ x_{i}) ⊗ c_{i} = 0. It follows that v ∧ x_{i} = 0 for i = 1, ⋯, k, hence v = 0 since k ≥ 2. The proof of the proposition is complete.

2.8. Example: We write down in detail the Lie algebra L(2). Let x_{1}, ⋯, x_{5} be the basis for V, and let

c_{1} = x_{1} ∧ x_{3} + x_{2} ∧ x_{4}, c_{2} = x_{1} ∧ x_{4} + x_{2} ∧ x_{5}.

Let X_{1}, ⋯, X_{5} be the dual basis of V^{∗}. Then a basis for the space I_{2} of quadratic relations is given by

[X_{1}, X_{2}], [X_{1}, X_{5}], [X_{2}, X_{3}], [X_{3}, X_{4}], [X_{3}, X_{5}], [X_{4}, X_{5}]

and

[X_{1}, X_{3}] − [X_{2}, X_{4}], [X_{1}, X_{4}] − [X_{2}, X_{5}].

There is a single non-vanishing triple commutator formed with the basis elements, namely

[X_{1}, [X_{1}, X_{3}]] = [X_{1}, [X_{2}, X_{4}]] = [X_{2}, [X_{1}, X_{4}]] = [X_{2}, [X_{2}, X_{5}]] mod(I)

which generates the center of L(2), and all four-fold commutators are in I.

3. Classification for V of small dimension

The purpose of this section is to classify nilpotent quadratically presented Lie algebras L over ℂ such that the dimension of the abelianized algebra L_{ab} is at most five. We take them to be of the form L = F(V^{∗})/I, where the ideal I is generated by I_{2} ⊂ F_{2}(V^{∗}) ≈ Λ^{2}V^{∗}, its homogeneous component of degree two, and where dim (V) ≤ 5. Recall that L is determined by its characteristic subspace C ⊂ Λ^{2}V, the annihilator of I_{2}.

If V is a vector space of dimension at most 3, then every element of Λ^{2}V is decomposable, so by Lemma 2.4, it follows that C = 0, and so L is abelian. If n ≥ 4, the lines determined by decomposables constitute the Plücker embedding of the Grassmann manifold G(2, V) of 2dimensional subspaces of V in the projective space of Λ^{2}V. The former has dimension 2(n − 2), where n = dim (V) while the latter has dimension n(n − 1)/2 − 1.

If dim (V) = 4, then the Grassmannian has dimension four and the projective space has dimension five, so if dim C > 1, then C would contain a decomposable element, contrary to Lemma (2.4). Consequently, if L is finite-dimensional, then dim C = 0 or dim C = 1. In the first case L is abelian. In the second case there is a single characteristic class ω which is not decomposable and so is nondegenerate, i.e., symplectic. Thus L is the five-dimensional Heisenberg Lie algebra H(2).

If dim (V) = 5, then the Grassmannian has dimension six and the projective space dimension nine. If dim C > 3, then C would contain a decomposable element, contrary to Lemma 2.4. Thus we must have dim C ≤ 3. If dim C = 0, then L is abelian, and if dim C = 1, then L = H(2) ⊕ ℂ, the direct sum of the five-dimensional Heisenberg algebra and a one-dimensional abelian algebra. It remains to analize the cases dim C = 2 and dim C = 3. This will be done in the following three lemmas.

3.1. Lemma: Let V be a 5-dimensional vector space over ℂ, and let C ⊂ Λ^{2}V be a two-dimensional subspace which contains no decomposables. Then

a) The map S^{2}C ⟶ Λ^{4}V defined by c_{1}c_{2} ⟶ c_{1} ∧ c_{2} is injective.

b) For any basis c_{1}, c_{2} of C, there exists a basis x_{1}, ⋯, x_{5} of V so that

c_{1} = x_{1} ∧ x_{3} + x_{2} ∧ x_{4}, c_{2} = x_{1} ∧ x_{4} + x_{2} ∧ x_{5}

is a basis for C. In particular, all such spaces are equivalent under automorphisms of V.

Proof: Recall that c ∈ Λ^{2}V is decomposable if and only if c ∧ c = 0. Thus c ∧ c ≠ 0 for all nonzero c in C. Define the carrier of an element of Λ^{2}V to be the vector space obtained by contracting it with the dual of V. For any nonzero element c ∈ C, the carrier is four-dimensional and c ∧ c is a volume element for it. Consider a basis c_{1}, c_{2} of C and suppose for a moment that the volume forms of c_{1} and c_{2} are proportional. Then the carriers of these elements coincide and so constitute a single four-dimensional vector space U. But then C is two-dimensional subspace of Λ^{2}U, and so must contain decomposables, a contradiction. Now suppose that c_{1} ∧ c_{2} = 0. Then (c_{1} + c_{2}) ∧ (c_{1} + c_{2}) = (c_{1} − c_{2}) ∧ (c_{1} − c_{2}). But { c_{1} − c_{2}, c_{1} + c_{2} } is also a basis for C and our previous argument applies to exclude proportionality of the associated volume forms. It follows that c_{1} ∧ c_{2} ≠ 0 for any pair of non-zero elements c_{1}, c_{2} ∈ C, and this proves the first assertion.

For the second assertion, let c_{1}, c_{2} be a basis of C, and choose a non-zero element w ∈ Λ^{5}V. Then there is a unique z ∈ V^{∗} so that c_{1} ∧ c_{2} = ι(z)w, where ι denotes interior product. Since ι is a derivation, (ι(z)c_{1}) ∧ c_{2} + c_{1} ∧ ι(z)c_{2} = 0. Thus, if x_{1} = ι(z)c_{2} and x_{2} = ι(z)c_{1}, then

x_{1} ∧ c_{1} + x_{2} ∧ c_{2} = 0.

It follows that x_{1} and x_{2} are independent (otherwise a linear combination of c_{1} and c_{2} would be decomposable). Then it is well known and easy to check that there exist elements x_{3}, x_{4}, x_{5} ∈ V so that

c_{1} = x_{1} ∧ x_{3} + x_{2} ∧ x_{4}, c_{2} = x_{1} ∧ x_{4} + x_{2} ∧ x_{5}.

If the set x_{1}, ⋯, x_{5} were dependent, C would be a subspace of two-vectors in a vector space of dimension ≤ 4, thus it would contain decomposables. Therefore x_{1}, ⋯, x_{5} forms a basis for V, and the proof is complete.

3.2. Corollary: Let L be a nilpotent, quadratically presented Lie algebra over ℂ with dim L_{ab} = 5 and dim (C) = 2. Then L is isomorphic to the three-step nilpotent Lie algebra L(2) of Example 2.8.

It remains to consider the case of dim (C) = 3. To this end, we introduce the following:

3.3. Lemma: Let V be a five-dimensional complex vector space, and let C ⊂ Λ^{2}V be a threedimensional subspace which contains no decomposables. Then there is a basis x_{1}, ⋯, x_{5} for V so that

c_{1} = x_{1} ∧ x_{3} + x_{2} ∧ x_{4}, c_{2} = x_{1} ∧ x_{4} + x_{2} ∧ x_{5}, c_{3} = x_{2} ∧ x_{3} + x_{5} ∧ x_{4}

is a basis for C. In particular, all such spaces C are equivalent under automorphisms of V.

Proof: Let G denote the Grassmann manifold of three-dimensional subspaces of Λ^{2}V, which is a manifold of dimension 21. We compute the dimension of the SL(V) orbit of C in G. To this end consider the map ϕ : S^{2}C ⟶ Λ^{4}V given by ϕ(c_{1}c_{2}) = c_{1} ∧ c_{2}. By the first assertion of Lemma 3.1, for any two-dimensional subspace C′ ⊂ C, the restriction of this map to S^{2}C′ is injective. Otherwise said, the kernel of ϕ contains no non-zero symmetric tensors of rank ≤ 2. Now the subvariety of symmetric tensors of rank ≤ 2 is the cone on the secant variety of the Veronese variety and so has codimension one. If the kernel of ϕ is of dimension greater than one it must intersect the Veronese variety, a contradiction. We conclude that ϕ has one-dimensional kernel and hence is surjective. The kernel is generated by an element q of rank three which we take to be of the form q = c_{1}^{2} + 2c_{2}c_{3} for some basis c_{1}, c_{2}, c_{3} of C.

Let H ⊂ SL(V) denote the identity component of the isotropy group of C. Then the action of H on C preserves q up to scalars. On the other hand, the action of H on C contains no homotheties by λ, where λ^{2} ≠ 1. Indeed, such a homothety would map by the surjection ϕ : S^{2}C ⟶ Λ^{4}V to a homothety by λ^{2} on Λ^{4}V. Since SL(V) acts on unimodularly on Λ^{4}V, this is impossible. It follows that the action of H on C preserves q. Since the group that preserves a quadratic tensor in dimension three is three-dimensional, dim (H) ≤ 3. Since SL(V) has dimension 24 and G has dimension 21, the orbit of C in G has dimension 21, hence is open. Moreover, it follows that H is isomorphic to an orthogonal group in three variables, hence operates transitively on the set of homothety classes of bases for C satisfying the relation c_{1}^{2} + 2c_{1}c_{2} = 0. Consequently all such spaces C and all such choices of basis c_{1}, c_{2}, c_{3} are equivalent (up to homothety) under the action of SL(V).

If x_{1}, ⋯, x_{5} is a basis for V and c_{1}, c_{2}, c_{3} are defined as in the statement of the lemma, then it is easy to check that the space C = span{ c_{1}, c_{2}, c_{3} } contains no decomposables and that the relation c_{1}^{2} + 2c_{2}c_{3} = 0 is satisfied. This completes the proof of the lemma.

3.4. Lemma: Let V be a five-dimensional complex vector space with basis x_{1}, ⋯, x_{5}, let C ⊂ Λ^{2}V be the three-dimensional subspace with basis

c_{1} = x_{1} ∧ x_{3} + x_{2} ∧ x_{4}, c_{2} = x_{1} ∧ x_{4} + x_{2} ∧ x_{5}, c_{3} = x_{2} ∧ x_{3} + x_{5} ∧ x_{4},

and let C′ ⊂ V ⊗ C be the five-dimensional subspace with basis

x_{1} ⊗ c_{1} + x_{2} ⊗ c_{2}, x_{3} ⊗ c_{1} + x_{4} ⊗ c_{3}, x_{4} ⊗ c_{2} − x_{2} ⊗ c_{3},
x_{2} ⊗ c_{1} + x_{5} ⊗ c_{2} + x_{1} ⊗ c_{3}, x_{4} ⊗ c_{1} + x_{3} ⊗ c_{2} + x_{5} ⊗ c_{3}.

Then the sequences

0 ⟶ C′ ⟶ V ⊗ C ⟶ Λ^{3}V ⟶ 0,  (3.1)

and

0 ⟶ V ⊗ C′ ⟶ Λ^{2}V ⊗ C  (3.2)

are exact.

Proof: To see that the last map in (3.1) is surjective, observe that if x_{i} ∧ x_{j} ∧ x_{k}, i < j < k, is a basis element for Λ^{3}V other than x_{1} ∧ x_{3} ∧ x_{5}, then it is of the form ±x_{l} ∧ c_{m} for some l, m. This is because c_{1}, c_{2}, c_{3} are non-degenerate two-vectors in the four-dimensional subspaces of V obtained by deleting the basis vector x_{5}, x_{3}, x_{1}, respectively. But x_{1} ∧ x_{3} ∧ x_{5} = c_{1} ∧ x_{5} + c_{2} ∧ x_{4}, so the surjectivity of this map follows.

By counting dimensions, we see that the kernel of this last map is five-dimensional. One checks easily that C′ is contained in this kernel, and so the exactness of (3.1) follows.

To prove the exactness of (3.2), let γ_{1}, ⋯, γ_{5} denote the basis elements of C′, written as listed in the lemma. We must show that if v_{1}, ⋯, v_{5} ∈ V and a = v_{1} ⊗ γ_{1} + ⋯ + v_{5} ⊗ γ_{5} is in the kernel of the map V ⊗ C′ ⟶ Λ^{2}V ⊗ C, then v_{1} = ⋯ = v_{5} = 0. Writing explicitly the coefficients of c_{1}, c_{2}, c_{3} in the image of a under this map, we obtain the system of equations

v_{1} ∧ x_{1} + v_{2} ∧ x_{3} + v_{4} ∧ x_{2} + v_{5} ∧ x_{4} = 0
v_{1} ∧ x_{2} + v_{3} ∧ x_{4} + v_{4} ∧ x_{5} + v_{5} ∧ x_{3} = 0
v_{2} ∧ x_{4} − v_{3} ∧ x_{2} + v_{4} ∧ x_{1} + v_{5} ∧ x_{5} = 0

for v_{1}, ⋯, v_{5}. At this point it is convenient to relabel the unknowns as w_{1}, ⋯, w_{5}, where w_{1} = v_{2}, w_{2} = v_{5}, w_{3} = v_{1}, w_{4} = v_{4}, w_{5} = v_{3}. The equations then become

\begin{matrix}w_{3} ∧ x_{1} + w_{1} ∧ x_{3} + w_{4} ∧ x_{2} + w_{2} ∧ x_{4} = 0 \\ w_{3} ∧ x_{2} + w_{5} ∧ x_{4} + w_{4} ∧ x_{5} + w_{2} ∧ x_{3} = 0 \\ w_{1} ∧ x_{4} − w_{5} ∧ x_{2} + w_{4} ∧ x_{1} + w_{2} ∧ x_{5} = 0 \end{matrix}  (3.3)

Write w_{j} = ∑_{i} a_{ij}x_{i}. Then, since the indices 5, 1, 3 do not appear in the first, respectively second, respectively third equation, the matrix a_{ij} must have the form

( \begin{matrix}a_{11} & 0 & 0 & 0 & 0 \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ 0 & 0 & a_{33} & 0 & 0 \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ 0 & 0 & 0 & 0 & a_{55} \end{matrix} ) .  (3.4)

Recall the fact that, if y_{1}, ⋯ y_{5} ∈ V and y_{1} ∧ x_{1} + ⋯ + y_{5} ∧ x_{5} = 0, then the matrix expressing the y_{j} in terms of the basis x_{i} is symmetric. Thus each equation imposes some condition on an appropiately rearranged 4 × 4 sub-matrix of a_{ij}. Explicitly, the first equation implies that the matrix

( \begin{matrix}0 & 0 & a_{11} & 0 \\ a_{23} & a_{24} & a_{21} & a_{22} \\ a_{33} & 0 & 0 & 0 \\ a_{43} & a_{44} & a_{41} & a_{42} \end{matrix} )

is symmetric. Thus we must have a_{23} = a_{21} = a_{43} = a_{41} = 0. Set these coefficients equal to zero in the matrix (3.4), and then write down the condition imposed by the second equation, namely that the matrix

( \begin{matrix}0 & a_{22} & a_{25} & a_{24} \\ a_{33} & 0 & 0 & 0 \\ 0 & a_{42} & a_{45} & a_{44} \\ 0 & 0 & a_{55} & 0 \end{matrix} )

is symmetric, hence a_{25} = a_{24} = a_{42} = 0. Set these coefficients also equal to zero in the matrix (3.4), and write down the condition resulting from the third equation, namely that the matrix

( \begin{matrix}0 & 0 & a_{11} & 0 \\ 0 & 0 & 0 & a_{22} \\ a_{44} & −a_{45} & 0 & 0 \\ 0 & −a_{55} & 0 & 0 \end{matrix} )

is symmetric, hence a_{45} = 0. These leaves the coefficients a_{11}, a_{22}, a_{33}, a_{44}, a_{55}. From the symmetry of the first two sub-matrices we see that all these coefficients are equal, and from the symmetry of the third, that a_{22} = −a_{55}. Thus all coefficients also vanish, and the proof is complete.

3.5. Example: Let V be a 5-dimensional vector space over any field (say ℚ, ℝ, ℂ) with basis x_{1}, ⋯ x_{5}, and let C, and C′ be the spaces defined by the bases in the last lemma. Let C_{0} = C except that C_{0} is declared to have homogeneous degree one, let d : C_{0} ⟶ C be the identity, and for i ≥ 1, define C_{i} by the process described in section 2. Then the lemma yields C_{1} ≅ C′ and C_{2} = 0. Let K denote the corresponding nilpotent Lie algebra and K the corresponding simply connected nilpotent Lie group. Then K is a three-step quadratically presented nilpotent Lie algebra of dimension 13 with five-dimensional abelianization and five-dimensional center. Note that the orthogonal group of the form q defined in the proof of Lemma 3.1 acts on K by Lie algebra automorphisms. Note also that K contains lattices since K has rational structure constants [13, Thm 7].

3.6. Corollary: Let L be a nilpotent, quadratically presented Lie algebra over ℂ with dim L_{ab} = 5 and dim (C) = 3. Then L is isomorphic to the three-step nilpotent Lie algebra K of Example 3.5.

We can summarize the discussion of this section as the following classification theorem.

3.7. Theorem: Let L be a nilpotent, quadratically presented Lie algebra over ℂ with dim L_{ab} ≤ 5. Then

a) If dim L_{ab} ≤ 3, then L is abelian.

b) If dim L_{ab} = 4, then L is either abelian or the 5-dimensional Heisenberg algebra H(2) of Example 2.3.

c) If dim L_{ab} = 5, then L is either abelian, or H(2) ⊕ ℂ, or the algebra L(2) of Example 2.8 or the algebra K of Example 3.5.

In particular, these algebras are classified up to isomorphism by the dimensions of L_{ab} and of the characteristic space C.

4. Restrictions on nilpotent Kähler groups

In this section we derive restrictions on nilpotent Kähler groups that use Kähler geometry and go beyond the conclusions of rational homotopy theory. We work first in the context of a formal topological space X with nilpotent fundamental group Φ. By passing to a finite cover of X we may assume that Φ is torsion free, in fact Φ is a lattice in the simply connected real nilpotent Lie group L with Lie algebra L the Malcev completion of Φ [13].

Let T be a torus of dimension rank(Φ_{ab}) and let f : X ⟶ T be a continuous map that induces an isomorphism Φ_{ab} ⟶ π_{1}(T). Then f^{∗} : H^{1}(T, ℝ) ⟶ H^{1}(X, ℝ) = V is an isomorphism, where V is as in the notation of section 2. Because f^{∗} preserves the multiplicative structure of cohomology, the spaces ker (f^{∗}) ⊂ H^{2}(T, ℝ) ≅ Λ^{2}H^{1}(T, ℝ) and C ⊂ Λ^{2}V correspond.

Now suppose X is a compact Kähler manifold, which we now denote by M. Then the torus T can be given a complex structure as the Albanese torus of M, namely

Alb(M) = H^{1,0}(M)^{∗}/H_{1}(M, ℤ),

and the map f can be chosen to be holomorphic, namely the Albanese map. Then we have:

4.1. Lemma: Let M be a compact Kähler manifold, let T be its Albanese torus, let f : M ⟶ T be the Albanese map, and let L = F(V^{∗})/I be the real Malcev Lie algebra of π_{1}(M). Then the canonical isomorphism H^{2}(T, ℝ) ≈ Λ^{2}V takes ker(f^{∗}) to the characteristic subspace C. In particular, V has an integral structure and a Hodge structure that induce integral and Hodge structures on Λ^{2}V. With respect to these structures, C is a subspace of Λ^{2}V defined over the integers and has a Hodge structure.

Now consider a real (p, p) form α on a complex manifold M. At each point x of M one can find a coordinate system which diagonalizes α in the sense that α_{x} is a linear combination of basic monomials, i.e., of products of the forms

\sqrt{−1}dz_{k} ∧ dz̄_{k}

If the coefficients of the basic monomials are nonnegative we say that α is semipositive at x. If α is semipositive at all points of M and has at least one positive coefficient at one point, we call it quasipositive. The following lemma is basic:

4.2. Lemma: If a closed (p, p) form α is quasipositive and M is Kähler, then α is nonzero in cohomology.

For the proof, consider β = α ∧ ω^{m−p}, where m is the dimension of M and ω is its Kähler form. Then β is a quasipositive (m, m) form, hence a form with positive integral. Therefore it, and so α as well, is nonzero in cohomology. The next result is an immediate consequence of the preceding two lemmas:

4.3. Corollary: Let ω be a characteristic class of a Kähler group. Then the (1, 1) component of ω is not (positive or negative) definite.

Recall that if ω ∈ Λ^{2}V, then the rank of ω (necessarily even) is the dimension of the smallest subspace W ⊂ V such that ω ∈ Λ^{2}W. Observe that rank(ω) = 2 if and only if ω is decomposable. With these preliminaries we can state our main theorem:

4.4. Theorem: Let M be a compact Kähler manifold with nilpotent fundamental group Φ which is not almost abelian, let C and V be as above, and let ω ∈ C ⊂ Λ^{2}V, ω ≠ 0 be integral and of type (1,1) (in the sense of Lemma 4.1). Then rank(ω) ≥ 8.

Proof: Let ω be as in the hypothesis. We may view ω as an integer-valued skew form on the real vector space V^{∗} = H_{1}(T, ℝ), the universal cover of T. Let J be the complex structure on V^{∗} which is the pullback of the complex structure on T. Since ω is of type (1,1), it is invariant under J, so its null-space is J-invariant, i.e., a complex subspace, and defined over ℤ. It therefore covers a complex subtorus T′ of T and determines an exact sequence

0 ⟶ T′ ⟶ T ⟶^{q} T′′ ⟶ 0

of complex tori, and so a holomorphic map g = q ∘ f of M to T′′. Moreover, dim_{ℝ}(T′′) = rank(ω), and ω is the pullback of a an integral (1, 1) class ω′′ on T′′ of maximum rank.

We already know from Lemma 2.4 that rank(ω) ≥ 4. Thus, two prove the theorem we only need consider two cases:

Case 1: rank(ω) = 4. Then the map g : M ⟶ T′′ has real rank 2 or 4. In the latter case it is surjective, g^{∗}ω^{′′2} is quasipositive, hence nonzero in cohomology, which contradicts g^{∗}ω′′ = 0 in H^{2}(M). In the former case the image of g is a (possibly singular) Riemann surface Y. Since g^{∗} factors through the 1-dimensional space H^{2}(Y), its kernel has codimension one and hence contains decomposables, in contradiction to Lemma 2.4.

Case 2: rank(ω ) = 6. Then the map g : M ⟶ T′′ has real rank 2, 4, or 6, and the case of rank 2 can be disposed of as above. The case of maximum rank 6 is also as above: g^{∗}ω^{′′3} is quasipositive, hence nonzero in cohomology, which contradicts g^{∗}ω′′ = 0 in H^{2}(M).

It remains to treat the case in which the real rank of g is 4. Then Y = g(M) is a possibly singular algebraic surface and T′′ is an algebraic torus. Consider first the case in which Y is ample and smooth. Factor g as j ∘ k where j : Y ⟶ T′′ is the inclusion. By the Lefschetz hyperplane theorem [14, p. 41], j^{∗} : H^{2}(T′′) ⟶ H^{2}(Y) is injective, so that j^{∗}ω′′ is nonzero. Let η be a class in H^{2}(Y) such that j^{∗}ω′′ ∪ η is a positive multiple of the fundamental class. Then k^{∗}(j^{∗}ω′′ ∪ η) is quasipositive of rank 2 and so is nonzero in cohomology. But k^{∗}(j^{∗}ω′′ ∪ η) = g^{∗}ω′′ ∪ k^{∗}η, so g^{∗}ω′′ ≠ 0, and so we arrive at the by now familiar contradiction.

For the singular case a slightly more elaborate argument is required, but the conclusion is the same: g^{∗}η ≠ 0. To see this, we begin with the observation that the Lefschetz theorem [14, p. 41] still applies to give j^{∗}η ≠ 0. Next, consider a resolution of singularities p : Ŷ ⟶ Y. By a mixed Hodge-theoretic argument [5, Proposition 8.2.7] we know that the kernel of p^{∗}j^{∗} is the same as the kernel of j^{∗}. Therefore p^{∗}j^{∗}η ≠ 0. Now let M̂ be a desingularization of the fiber product Ŷ ×_{Y} M, and let k̂ : M̂ ⟶ Ŷ, q : M̂ ⟶ M be the natural maps. Since p^{∗}j^{∗}η ≠ 0, there is a class α such that p^{∗}j^{∗}η ∪ α represents a multiple of the fundamental class. Therefore k̂^{∗}(p^{∗}j^{∗}η ∪ α) ≠ 0, and so k̂^{∗}p^{∗}j^{∗}η ≠ 0. But then g^{∗}η = q^{∗}k^{∗}j^{∗}η ≠ 0, which is what was claimed.

Let us now consider the case in which Y is not ample. Then T′′ is reducible, and Y is the pullback of a divisor on a quotient torus [16, pp 25–26]. In addition there is a finite unramified cover of T′′ which splits as E × A, where E is an elliptic curve and A is an abelian variety of dimension 2. Replace M and T′′ by the corresponding unramified covers so as to reduce the problem to one in which T′′ splits. Then the divisor Y has the form E × C, where C is a genus two curve, or E × E′, or { p } × A, where p is a point. The second and third cases are excluded, for otherwise g_{∗} would not be surjective on H_{1}. The first case is also excluded for it yields a nonconstant map of M to a Riemann surface, thus contradicting Lemma 2.4, as in Case 1. The proof of the Theorem is complete.

4.5. Corollary: Let Φ be a nilpotent, not almost abelian, Kähler group. Then rank(Φ_{ab}) ≥ 8.

Proof: We only need to show that, if Φ were a nilpotent Kähler group with rank(Φ_{ab}) ≤ 6 and ω ∈ C, ω ≠ 0, then ω has type (1,1). But if ω had a non-zero (2,0)-component ω^{(2,0)}, then ω^{(2,0)} ∈ Λ^{2}H^{2,0}(T). Since dim H^{2,0}(T) ≤ 3, ω^{(2,0)} would be decomposable, in contradiction to Lemma 2.4. Thus ω is of type (1,1), in contradiction to the Theorem, and the proof is complete.

4.6. Corollary: Let Φ be a nilpotent Kähler group which is not almost abelian, and let C be as above. Then there exists x ∈ C with rank(x) ≥ 8. If dim C ≤ 2, then rank(x) ≥ 8 for all x ∈ C, x ≠ 0.

Proof: If Φ is not almost abelian, then C ≠ 0. If C is of type (1,1), i.e., C ⊗ ℂ = C^{1,1}, then the theorem gives us that every non-zero element of C has rank at least 8. Otherwise, C^{2,0} ≠ 0, and if α is a nonzero element of C^{2,0}, then by Lemma 2.4, rank(α) ≥ 4. Thus there are complex linear coordinates z_{1}, ⋯, z_{n} for V so that

α = dz_{1} ∧ dz_{2} + ⋯ + dz_{2k−1} ∧ dz_{2k}, k ≥ 2.

It is then easy to check that any non-trivial linear combination of the real and imaginary parts of α has rank 4k ≥ 8. Thus there are elements of C of rank ≥ 8, and the first assertion follows. For the second assertion, if dim C = 1, then C is of type (1,1), and if dim C = 2, then the only possibilities for C is that it be either all of type (1, 1) or all of type (2, 0) + (0, 2). In either case, we have seen that all its elements have to have rank at least 8.

4.7. Corollary: Let Φ be a cartesian product of one of the following forms:

a) Φ = Φ_{1} × Φ_{2}, where Φ_{1} is a lattice in the the five or seven-dimensional real Heisenberg group H(2) or H(3), and Φ_{2} is a free abelian group.

b) Φ = Φ_{1} × Φ_{2} × Φ_{3} where Φ_{1} and Φ_{2} and are as in (a) and Φ_{3} is a lattice in a Heisenberg group H(k), k ≥ 2.

c) Φ = Φ_{1} × Φ_{2}, where Φ_{1} is a lattice in one of the real Lie groups L(3) or L(3) of Example 2.8 or the group K of Example 3.5, and Φ_{2} is free abelian.

Then the Malcev Lie algebra of Φ is quadratically presented, by suitable choice of Φ_{2} or Φ_{3} the rank of Φ_{ab} can be made even and arbitrarily large, yet Φ is not a Kähler group.

Proof: By construction, either dim C ≤ 2 and some characteristic class of Φ has rank at most 6, or, in the cases involving L(2) and K, dim C = 3 and all elements of C have rank at most six. Thus the last corollary excludes these as Kähler groups.

4.8. Remark: It is not known whether characteristic classes of type (2,0) can occur for nilpotent Kähler groups. If it were true that C always has type (1,1), then our theorem would give much stronger restrictions on nilpotent Kähler groups.

5. Remarks on the examples

Finally, we examine the characteristic classes of Campana’s examples of nilpotent Kähler groups [2]. For the construction it is helpful to consult section 9 of [7] and Part II of [9]. Let A and B be Abelian varieties of dimension n, and let f : A ⟶ ℙ^{n} and g : B ⟶ ℙ^{n} be finite maps. Set X = A × B and let h = f × g : X ⟶ ℙ^{n} × ℙ^{n}, which is also a finite map. Choose two copies Ā and B̄ of ℙ^{n} in ℙ^{2n+1} which are in general position, hence disjoint. Let V be the complement of these, and note that it fibers over Ā × B̄ with fiber ℂ^{∗}. Indeed, if x is a point of V, let a be the unique point of intersection of the (n + 1)-dimensional linear space spanned by x and B̄ with Ā. Reverse the roles of the two ℙ^{n}’s to construct a point b on B̄, and define p(x) = (a, b). The characteristic class of this principal ℂ^{∗} bundle is (ω, −ω), where ω is the class of the hyperplane section. Let X^{∗} be the pullback of V to A × B via h. Note that X^{∗} is a quasi-projective variety, and is a principal ℂ^{∗} bundle with characteristic class η = (f^{∗}ω, −g^{∗}ω). Let Φ = π_{1}(X^{∗}). The exact sequence of the fibration gives

0 ⟶ ℤ ⟶ Φ ⟶ ℤ^{2n} × ℤ^{2n} ⟶ 0,

thus Φ is the central extension of ℤ by ℤ^{4n} with characteristic class η.

Now let h̃ : X^{∗} ⟶ V be the map of ℂ^{∗} bundles covering h : X ⟶ ℙ^{n} × ℙ^{n}, and let L ⊂ ℙ^{2n+1} be a linear space of dimension 2 which is in general position with respect to Ā and B̄. It is contained in V and so we can form the projective variety Y = h̃^{−1}(L). The generalized Lefschetz theorem of [9, p. 195] implies that

π_{i}(X^{∗}, h̃^{−1}(L)) = 0

for i ≤ 2 provided that n ≥ 2. Thus

π_{1}(Y) ≅ π_{1}(X^{∗}) = Φ,

thus Φ is the fundamental group of the projective manifold Y provided n ≥ 2. This situation is similar in spirit to that of [20], [21] in which the fundamental groups of certain quasiprojective varieties turn out to be the fundamental groups of certain projective varieties, namely, of suitable linear sections.

Since η^{2n} ∈ H^{4n}(X, ℤ) is a non-zero multiple (namely, the degree of h) of the positive generator, η is a symplectic form on the vector space H_{1}(X, ℚ). This implies that the rational Malcev completion of Φ is the rational Heisenberg group H(2n) of Example 2.3, and that Φ is a subgroup of finite index in the integral Heisenberg group, namely the group of matrices (2.6) with integral entries. In fact, it is easy to see that, if d_{1}, ⋯, d_{2n} are the elementary divisors of η as a skew form on H_{1}(X, ℤ) ≈ ℤ^{4n}, then Φ is isomorphic to the subgroup of (2.6) where x_{i} is divisible by d_{i}, where x = (x_{1}, ⋯, x_{2n}).

Observe that Φ itself cannot be isomorphic to the full integral Heisenberg group. (If this were the case, all elementary divisors of f^{∗}ω and g^{∗}ω would be one, in which case the cohomology classes would define principal polarizations of A and B respectively. But the line bundle of a principal polarization has a one-dimensional space of sections, and so cannot be induced from a map to ℙ^{n}.) But Φ contains subgroups of finite index isomorphic to the integral Heisenberg group. In fact, P. Deligne pointed out to us that if U is a unipotent group over ℚ whose Lie algebra (as in our case) admits a positive grading, and if U_{1} and U_{2} are any two ℤ-forms of U, then a suitable power of the grading automorphism embeds U_{1} as a subgroup of finite index in U_{2}. Thus U_{1} is a Kähler group if and only if U_{2} is a Kähler group.

Concrete examples can be obtained as follows. Let E be an elliptic curve, let E ⟶ ℙ^{1} be the standard map of degree 2. Let E × E ⟶ ℙ^{1} × ℙ^{1} be the Cartesian product of two such maps, and f : E × E ⟶ ℙ^{2} the composition of this map with the standard degree two map ℙ^{1} × ℙ^{1} ⟶ ℙ^{2}. Let A = B = E ×E and f = g. Then f and g are of degree 8, consequently h is of degree 64. It can easily be checked that all the elementary divisors of η are 2, thus Φ is isomorphic to the subgroup of (2.6) with all x_{i} even, where x = (x_{1}, ⋯, x_{4}). The subgroup of Φ of index two where in addition z is required to be even is isomorphic to the integral Heisenberg group. Another example would be to take polarizations with elementary divisors 1, 5 on A and B and f and g generic projections to ℙ^{2} of their embeddings in ℙ^{4}, [11]. In this case f and g have degree 10 and Φ is isomorphic to the subugroup of (2.6) with x_{2} and x_{4} divisible by 5, and it contains, say, the subgroup of index 125 of matrices where x_{1}, x_{3} and z are also divisible by 5, which is isomorphic to the integral Heisenberg group.

Note that the characteristic class of each Φ in the examples just constructed is of type (1, 1). It is also indefinite (in accord with Corollary 4.3), and of rank at least eight, in accord with Theorem 4.4. It is not difficult to extend the construction just given to one for Kähler groups which are central extensions of ℤ^{k} by ℤ^{2m} for various m and k. (For instance, take more factors of ℙ^{n}, take factors of different dimensions, etc.) The characteristic classes are still of type (1,1), cf. Remark 4.8.

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