Non-Kählerianity of the fundamental group of some non-positively curved 3-manifolds

Luis Hernández-Lamoneda ^{∗} Centro de Investigación en Matemáticas, A.C. (CIMAT) Guanajuato, México

February 10, 2000

Abstract

In this note a simple rank estimate for harmonic maps, with domain a Kähler manifold and target a compact 3-manifold, is proved. As an application we give new examples of fundamental groups of compact manifolds which are not Kähler groups.

Keywords: Kähler group, harmonic map, curvature. AMS Subject classification number:32.

1 Introduction

A Kähler group is a group that appears as the fundamental group of a compact Kähler manifold. Even though every finitely presented group is the fundamental group of a compact complex and symplectic manifold, there are obstructions for it to be a Kähler group. For example, it is well known (see [3]) that the fundamental group of a compact hyperbolic manifold of dimension at least three is non Kähler. The same thing holds for the fundamental group of manifolds whose curvature is sufficiently close to −1, although for general non-positively curved manifolds both Kähler and non-Kähler groups arise. The main reference for learning about Kähler groups is [1].

In this note I will look at harmonic maps from a Kähler manifold into certain non-positively curved compact 3-manifolds. Proposition 1 is the main tool, it is used to prove the following result.

Theorem 1 Let M be a compact riemannian 3-manifold of non-positive sectional curvature, such that the set of points where the curvature is strictly negative is dense.

Then π_{1}(M) is non-Kähler.

∗The author acknowledges support from grant 28491-E of CONACyT

Remark. Theorem 1 also holds under the more technical (though weaker) condition that the rank of the, negative semi-definite, curvature operator be at least 2 on a dense subset of M.

2 Some definitions

Let (M, ⟨, ⟩) be an arbitrary riemannian manifold, and pick some p ∈ M. Complexify the tangent space at p, T_{p}^{ℂ}M = T_{p}M ⊗ ℂ, and extend ⟨, ⟩ to a complex bilinear form on T_{p}^{ℂ}M. Let R denote the Riemann curvature tensor, thought of as a symmetric linear operator on 2-vectors, R : Λ^{2}T_{p}M → Λ^{2}T_{p}M. It can be also extended to a complex linear operator R^{ℂ} on Λ^{2}T_{p}^{ℂ}M.

Definition 1 The manifold (M, ⟨, ⟩) is said to have non-positive complex curvature, K_{M}^{ℂ} ≤ 0, if

⟨R^{ℂ}(u ∧ v), ū ∧ v̄⟩ ≤ 0

for every u, v ∈ T_{p}^{ℂ}M and every p ∈ M.

Now we recall a theorem of J.H. Sampson, [5].

Sampson’s Theorem. Let X be a compact Kähler manifold, M a compact riemannian manifold with K_{M}^{ℂ} ≤ 0. Let f : X → M be a harmonic map and W_{x} = df_{x}^{ℂ}(T_{x}^{1,0}X) ⊂ T_{f(x)}^{ℂ}M, where df_{x}^{ℂ} stands for the complex linear extension of df_{x}.

Then, for every u, v ∈ W_{x}, ⟨R^{ℂ}(u ∧ v), ū ∧ v̄⟩ = 0.

3 Proof of Theorem 1

Proposition 1 Let X be a compact Kähler manifold, M a 3-manifold with sectional curvature K ≤ 0, and f : X → M a harmonic map.

Let U = {x ∈ X : rank (df_{x}) = 3}, then f(U) is an open subset of M contained in the set {p ∈ M : rank(R_{p}) ≤ 1}.

Proof. f(U) is clearly open, so it remains to be shown that for p ∈ f(U), R_{p} has rank at most 1.

On a 3-manifold K ≤ 0 implies K^{ℂ} ≤ 0 (Proposition 2.6 of [4]), in fact, it implies that R is a negative semidefinite operator. Hence, Sampson’s theorem applies and we get that for every x ∈ X and u, v ∈ W_{x} = df_{x}^{ℂ}(T_{x}^{1,0}X)

⟨R^{ℂ}(u ∧ v), ū ∧ v̄⟩ = 0.  (1)

Pick an x ∈ U, then W_{x} + \overline{W}_{x} = T_{f(x)}^{ℂ}M and so dim_{ℂ}W_{x} ≥ 2. Therefore W_{x} ⋂ \overline{W}_{x} has real dimension at least 2. Thus W_{x} ⋂ \overline{W}_{x} contains a real vector e_{1}, say, of length 1.

Complete to an orthonormal basis {e_{1}, e_{2}, e_{3}} of T_{f(x)}M. Since df_{x} is onto, there is a vector Y in T_{x}X, such that df_{x}(Y) = e_{2}. Denote by J the complex structure on X; then, since e_{1}, e_{2} − \sqrt{−1}df_{x}(JY) are in W_{x}, equation (1) can be applied as follows

\begin{matrix}0 = & ⟨R^{ℂ}(e_{1} ∧ (e_{2} − \sqrt{−1}df_{x}(JY))), e_{1} ∧ (e_{2} + \sqrt{−1}df_{x}(JY))⟩ \\ = & ⟨R(e_{1} ∧ e_{2}), e_{1} ∧ e_{2}⟩ + ⟨R(e_{1} ∧ df_{x}(JY)), e_{1} ∧ df_{x}(JY)⟩, \end{matrix}

and since each of the last two summands is non positive, ⟨R(e_{1}∧e_{2}), e_{1}∧e_{2}⟩ = 0; a similar argument shows that ⟨R(e_{1} ∧ e_{3}), e_{1} ∧ e_{3}⟩ = 0. Since the rank of R coincides with the dimension of the largest space where it is negative definite, and since dim Λ^{2}T_{f(x)}M = 3, we get the result.

Corollary 1 Let X be a compact Kähler manifold, M a compact 3-manifold with K ≤ 0, and such that the set of points where K < 0 is dense. Then if f : X → M is a harmonic map, rank (df_{x}) ≤ 2 at every x ∈ X.

PROOF OF THEOREM 1. Suppose false, and let X be a compact Kähler manifold with fundamental group isomorphic to Γ = π_{1}(M). Realize this isomorphism of fundamental groups via a harmonic map f : X → M.

The previous corollary implies that rank(df_{x}) ≤ 2 at every point x ∈ X. Now we can retrace the proof by Carslon-Toledo that the fundamental group of a hyperbolic manifold is non-Kähler (see Theorem 6.21 of [1] or Theorem 7.1 of [3]). Namely, the fact that the rank of df is bounded by two everywhere, implies that f = ψ ∘ ϕ where ϕ : X → S is some continuous map and S is a circle or a compact surface of genus at least 2. Now, ϕ_{∗}(Γ) is an isomorphic copy of Γ inside π_{1}(S), thus there is a cover Σ of S with fundamental group isomorphic to Γ. But then Σ and M are both K(Γ, 1)’s, therefore homotopically equivalent spaces, although H_{3}(M) ≃ ℤ and H_{3}(Σ) = 0. Contradiction.

4 Examples

The following construction is usually attributed to Heintze.

Given a graph G with finitely many vertices and edges, one can construct a real analytic 3-manifold with K < 0 except along finitely many totally geodesic embedded 2-dimensional tori (one for each edge) as follows. For each vertex pick a hyperbolic, finite volume 3-manifold with as many cusps as edges arrive to this vertex. For each edge we have two cusps, cut from each one of them an open neighborhood of infinity and glue them along the boundary tori. Do this for every edge of the graph. An smoothing operation converts this space into a (real analytic) 3- manifold of negative curvature except along the gluing tori.

By Theorem 1 the fundamental groups of these manifolds are not Kähler; it should be noted that these groups cannot be the fundamental group of a strictly negatively curved manifold nor of a locally symmetric one of rank> 1. To see this, note that these groups are of rank= 1, but they do contain ℤ^{2} as a subgroup (see [2]).

Other constructions can be found in [6] or in the references therein. Acknowledgments. I wish to thank both Prof. Maite Fernández and Mme. Juana Lamoneda for their helpful advice.

References

[1] J. Amorós, M. Burger, K. Corlette, D. Kotschick and D. Toledo, Fundamental Groups of Compact Kähler Manifolds, Mathematical Surveys and Monographs, 44, 1996.

[2] W. Ballmann and P. Eberlein, Fundamental groups of manifolds of nonpositive curvature, J. Diff. G. 25 (1987), 1-22.

[3] J. Carlson and D. Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces, IHES Publ. Math. 69 (1989), 173-201.

[4] L. Hernández-Lamoneda, Kähler manifolds and 1/4-pinching, Duke J. Math., 62 (1991), 601-611.

[5] J.H. Sampson, Applications of harmonic maps to Kähler geometry, Contemp. Math. 49 (1986), 125-133.

[6] V. Schroeder, Analytic manifolds of nonpositive curvature with higher rank subspaces, Arch. Math. 56 (1991), 81-85.