j. differential geometry 47 (1997) 469-503

HARMONIC MAPS OF INFINITE ENERGY AND RIGIDITY RESULTS FOR REPRESENTATIONS OF FUNDAMENTAL GROUPS OF QUASIPROJECTIVE

VARIETIES

JÜRGEN JOST & KANG ZUO

Abstract

We show the existence of harmonic maps associated with reductive homomorphisms of the fundamental group of a quasiprojective variety into a linear algebraic group over an archimedean or p-adic field. The map we construct may have infinite energy, but it satisfies suitable estimates at infinity, and it is pluriharmonic. We use this map to complete a previous result of Jost-Yau [21] on strong rigidity of nonuniform lattices in Hermitian symmetric spaces, and to drop a topological restriction in our previous theory ([24], [25]) of representations of fundamental groups of quasiprojective varieties. AMS-Code: 58E20

Introduction

Hodge theory represents cohomology classes by harmonic forms, and it is a fundamental tool in Kähler geometry. Homology or cohomology groups, however, do not contain the complete topological information about a manifold X, and, in particular, the fundamental group may be much more complicated than the first homology group, its abelianized version, indicates. Via the Albanese period mapping, the fundamental group acts as a lattice on some vector space. This can be considered as an abelian representation of π_{1}(X), yielding the first homology group. It is therefore natural to study also nonabelian representations of π_{1}(X), in order to obtain further information about the topology of X. In the same way as the Albanese period map is harmonic (which is essentially the Hodge theorem for 1-forms), one tries to associate to a nonabelian representation a harmonic map u as well and to study the properties of u. This has been done in work of Al’ber [1], [2], Eells-Sampson [10], Siu [46], Jost-Yau [19], [22], [23], Sampson [44], Diederich-Ohsawa [9], Corlette [5], Carlson-Toledo [7], Hitchin [13], Simpson [42], Zuo [50], [51], Gromov-Schoen [11], Mok-Siu-Yeung [37], and many others, in the case of a compact Riemannian, Kählerian, or algebraic manifold X. In the noncompact case, Iitaka [14] extended the Albanese map to quasiprojective varieties. The harmonic map theory has been extended to the quasiprojective case by Jost-Yau [20], [21], and Jost-Zuo [24], [25]. All these papers, however, work in a situation where one can produce a harmonic map of finite energy, and this requires some additional assumptions about the representations of π_{1}(X) near infinity. Simpson [40] considered the case of complex dimension 1 and used harmonic maps with controlled behavior at infinity which he called “tame”. In dimension 1, however, many of the analytical difficulties that we have to encounter in the present paper are not present. (For the one-dimensional analysis, see also the papers [48] and [32] that will be discussed in more details below.)

Received May 3, 1995, and, in revised form, October 29, 1996.

The main technical achievement of the present paper is to remove the restriction that the associated harmonic map must have finite energy, while working in arbitrary dimension. In our Theorem 1.1, we show that for any reductive representation ρ of π_{1} of a smooth quasiprojective variety (or, more generally, for a Kähler manifold admitting a suitable compactification) into the isometry group of either a symmetric space of noncompact type or a locally compact Euclidean Tits building, we obtain a corresponding harmonic map u of possibly infinite energy that satisfies a precise growth estimate near infinity. (Here, a symmetric space is associated with a representation in a linear algebraic group defined over ℝ or ℂ, whereas the Tits buildings arise from representations over p-adic ground fields.) In fact, the map is even pluriharmonic, making the foliation technique first discovered in [19] applicable. (“Pluriharmonic” means that the restriction of u to any subvariety of X is again harmonic.)

The prototype of such an infinite energy harmonic map with the typical growth behavior near the puncture is the map

u_{0} : D^{∗} := {z ∈ ℂ : 0 < |z| ≤ 1} → S^{1}
re^{iϑ} ↦ ϑ.

harmonic maps of infinite energy and rigidity results 471

For the case of complex dimension 1, i.e., for holomorphic curves with cusps alias punctured Riemann surfaces, such a harmonic map was constructed by Wolf [48] and Lohkamp [32], controlling the growth of the energy density near the puncture through comparison with the prototype u_{0}. The same strategy also works in our situation, but the details become much harder.

It is conceivable that our existence result holds in more generality. Namely, one would like the target to be a more general space of nonpositive curvature. In our argument, besides the nonpositivity of the curvature, however, we need a decay condition for Jacobi fields in order to handle parabolic or quasihyperbolic elements in the image of ρ. (Wolf [48] and Lohkamp [32] only consider the case of hyperbolic (or elliptic) elements which suffices for their purposes.) Finally one would like to extend the result to nonlocally compact images, as in [15], [16], [18], [28].

For our present purposes, however, our existence result suffices, and we can essentially complete the nonabelian Hodge theory for quasiprojective manifolds.

Some partial existence results for harmonic maps on noncompact Kähler manifolds have recently obtained by J. Li [31]. It is not clear, however, whether the maps produced by his method can be shown to be pluriharmonic.

In §2, we complete strong rigidity for nonuniform lattices in bounded symmetric domains D in the holomorphic context. The result is that if a quasiprojective manifold X is homotopically equivalent to a quotient of an irreducible bounded symmetric domain by a lattice, then X is already biholomorphically equivalent to this quotient (with the standard exception where D is the hyperbolic plane). This is the holomorphic version of Margulis’ rigidity theorem. In the compact case, it was shown by Siu [46], and in the noncompact case by Jost-Yau [20], [21] with an additional technical restriction on the compactification of X.

Our second application, presented in §3, extends our previous work [24], [25]. (In a subsequent paper, Katzarkov [27] stated results similar to those of [25], together with some applications to the Shafarevich conjecture.)

After recalling some preliminary constructions, we shall show two main results. The first one concerns Zariski dense representations ρ of π_{1}(X) in a simple algebraic group G defined over ℂ, with fixed characteristic polynomials at infinity. If such a representation is not rigid, then it factors through a morphism f : X → X′, with dim X′ ≤ rk_{ℂ} G.

In other words, any representation that does not factor through one obeying this rank restriction is rigid. The second result concerns the case where G is defined over a complete field K with a discrete valuation v. (We assume that the associated Tits building Δ(G(K_{v})) is locally compact.) If ρ is unbounded with respect to this valuation, i.e., if ρ(π_{1}(X)) is not contained in a compact subgroup of G(K_{v}), i.e., in the stabilizer of a vertex of Δ(G(K_{v})), then ρ factors as before.

As explained in more detail in the introduction of [25], these results have far reaching consequences for archimedean and p-adic representations of fundamental groups of quasiprojective manifolds.

The paper partly grew out of a discussion with Peter Kronheimer, and we thank him for his suggestions.

We thank the referee for finding an error in the first version of our paper, and we should point out that the first Lemma in our announcement [26] is incorrect in the generality stated.

Our work was supported by SFB 237 and the Leibniz program of the DFG. The first author also thanks Harvard University and ETH Zürich for hospitality.

1. The existence of pluriharmonic maps with controlled growth at infinity

1.1. Statement of the result

Theorem 1.1. Let X be a Kähler manifold admitting a compactification X̄ as a compact Kähler manifold for which D := X̄ \ X is a divisor with simple normal crossings. Let Y be either

a) a symmetric space of noncompact type

or

b) a locally compact Euclidean Tits building (with isometry group operating transitively on the set of vertices).

Let

ρ : π_{1}(X) → I(Y)

be a homomorphism from the fundamental group of X into the isometry group of Y. Assume that ρ is reductive (see Def. below). Then there exists a ρ-equivariant pluriharmonic

u : X̃ → Y

from the universal cover of X to Y satisfying the following estimate:

For every noncompact holomorphic curve Σ in X, we represent a neighborhood of a node conformally as a punctured disc

D^{∗} := {z ∈ ℂ : 0 < |z| < 1} ⊂ ℂ.

With respect to the (noncomplete) Euclidean metric on D^{∗} and the natural metric of Y, we then have for the norm of the derivative of u

∥du(z)∥^{2} ≤ \frac{const}{|z|^{2}}.

(The constant depends only on the representation ρ, but not on the curve Σ; one may extract this from the subsequent proof, but we shall not go into the details. Note that in case b), u is only Lipschitz; the preceding estimate then holds a.e., and pluriharmonicity of u has to be interpreted in the sense of Gromov-Schoen [11].)

The following definition was suggested by Scot Adams.

Definition 1.1. Let Y be a complete, simply connected, locally compact space of nonpositive curvature with isometry group I(Y). Let Γ be a group, and ρ : Γ → I(Y) a homomorphism.

ρ is called reductive if there exists a complete totally geodesic subspace Z of Y stabilized by ρ(Γ) with the following property:

For every totally geodesic subspace Z′ of Z that has no Euclidean factor, ρ(Γ) does not fix any point in the sphere at infinity of Z′, i.e., is not contained in a parabolic subgroup of the isometry group of Z′.

In geometric terms, this means that there does not exist an unbounded sequence (y_{n})_{n∈ℕ} ⊂ Z′ with

dist (y_{n}, γ(y_{n})) ≤ c(γ)

for all γ ∈ ρ(Γ), with a constant c depending on γ, but not on n. Note that any semisimple representation is reductive.

For simplicity of notation, we shall consider u in the sequel as a map from X into

N := Y/ρ (π_{1} (X)) ,

although N may be singular. Sometimes, it will be implicitly understood (and not explicitly mentioned) that for the proper meaning of a statement about a harmonic map one has to lift the universal covers in a ρ-equivariant manner.

In particular, the energy of u can be computed on a fundamental region F_{X} for X in X̃:

E(u) = \frac{1}{2} ∫_{F_{X}} ∥du(x)∥^{2}dvol(x).

Here, we assume that X is equipped with some Kähler metric which then lifts to X̃ and induces a volume form dvol(x), and the norm of the differential du is computed with respect to this metric and the one on Y.

1.2. Harmonic and pluriharmonic maps

In this section, we collect some known results about harmonic and pluriharmonic that will be employed in the proof of Theorem 1.1.

(H1) Let M be a compact Riemannian manifold, possibly with nonempty boundary, and Y be either

a) a complete simply connected Riemannian manifold of nonpositive curvature

or

b) a locally compact Euclidean Tits buildings, with isometry group I(Y).

Let

ρ : π_{1}(M) → I(Y)

be a reductive representation. In case ∂M = ∅, let g be smooth ρ-equivariant boundary values. Then there exists a ρ-equivariant harmonic map

u : M̃ → Y,

with u|_{∂M̃} = g in case ∂M ≠ ∅. This was shown in [22] and [30] for a) building upon [1], [2], [10], [9], [8], [4], and for b) in [11]. Conversely, if such a ρ-equivariant harmonic map exists, then ρ is reductive; see [30]. Moreover, u minimizes energy in its class. In fact, instead of the compactness of M, one only needs to assume that there exists some finite energy map in the class under consideration.

(H2) If under the assumptions of (H1), u_{1} and u_{2} are ρ-equivariant harmonic maps, then there exists a family u_{t} : X̃ → Y of ρequivariant harmonic maps, 1 ≤ t ≤ 2, all of the same energy, and for each x ∈ X,

u_{t}(x) , 1 ≤ t ≤ 2,

is a geodesic arc from u_{1}(x) to u_{2}(x) parametrized proportionally to arclength, with length independent of x. This was shown in [1], [2], [12], [11].

(H3) Let u be a harmonic map as in (H1). Then u is smooth in case a) and Lipschitz in case b) as shown in [11]. Furthermore, it is shown in [11] that u has sufficient regularity properties to justify the application of Bochner type identities that were previously developed for the smooth case; see (H4). Also, on regions of controlled geometry, a harmonic map satisfies estimates depending only on its energy. See e.g. [17] and the references contained therein.

(H4) Suppose that M is a compact Kähler manifold and that Y is a symmetric space of noncompact type. Let u be a harmonic map as in (H1). It was shown, by Siu [46] in the Hermitian symmetric case and by Sampson [44] in general, that u is pluriharmonic meaning that its restriction to any complex submanifold of M is again harmonic. Since harmonicity of a map on a holomorphic curve Σ is a property that does not depend on the choice of a (conformal) metric on Σ, also pluriharmonicity of u is a property that does not depend on the Kähler metric, but only on the complex structure of M. As noted in (H3), this result was extended by Gromov-Schoen to Euclidean Tits buildings Y. Likewise, the pluriharmonicity was extended to the finite energy case by Jost-Yau [20], [21].

1.3. Preliminary constructions

Let X̄ be a compactification for which D = X̄\X is a divisor with simple normal crossings as only possible singularities. We may also assume that each irreducible component D_{λ} of D is free from self intersections. Thus, at each intersection point, precisely two components of D meet. The irreducible components of D will be denoted by D_{1}, …, D_{l}.

For each irreducible component D_{λ} of D, we let σ_{λ} be a holomorphic section of O(X̄, [D_{λ}]) with a simple zero along D_{λ}. By multiplying σ_{λ} with a suitable constant, if necessary, we may assume that the derivative of σ_{λ} has no zeroes in the set |σ_{λ}| ≤ 1. We obtain a fibration of a neighborhood of D_{λ} in X̄ by holomorphic disks B = {z ∈ ℂ : |z| < 1} meeting D_{λ} transversally at 0 ∈ B by choosing local trivializations of the line bundle [D_{λ}], and on each such disk, σ_{λ} defines a local coordinate through the trivialization. Since in general, the normal bundle of D_{λ} is nontrivial, these local coordinates are not globally defined, but coordinate changes preserve these local disks. Thus, for each w ∈ D_{λ}, we obtain such a transversal disk B_{w}. The fibrations corresponding to two different components of D meet transversally near the intersection locus of these components.

Likewise, for sufficiently small δ > 0, |σ_{λ}| = δ defines the boundary Σ_{δ}^{λ} of such a tube around D_{λ}, fibered by circles. The intersection of two such boundaries Σ_{δ}^{λ_{1}}, Σ_{δ}^{λ_{2}} is fibered by tori.

1.4. Hyperbolic and elliptic representations

Let Y be a simply connected complete metric space with distance function d(·, ·) of nonpositive curvature in the sense of Alexandrov, with isometry group I(Y). We recall the following:

Definition 1.2. g ∈ I(Y) is called

- elliptic if inf_{y∈Y} d(y, gy) = 0 and there exists y_{0} ∈ Y with gy_{0} = y_{0},

- hyperbolic if inf_{y∈Y} d(y, gy) =: λ_{g} > 0 and there exists y_{0} ∈ Y with d(y_{0}, gy_{0}) = λ_{g},

- parabolic if inf_{y∈Y} d(y, gy) = 0, but the infimum is not achieved,

- quasihyperbolic if inf_{y∈Y} d(y, gy) > 0 and the infimum is not achieved.

Throughout this section, we assume:

(A) The representation

ρ : π_{1}(X) → I(Y)

has the property that the image of every small loop around D is hyperbolic or elliptic.

Actually, the case of elliptic elements can be handled by the analysis of our previous papers [24], [25], because it does not cause infinite energy. Elliptic elements, however, also easily succumb to the treatment of hyperbolic elements that we are going to present.

Each hyperbolic element in I(Y) can by represented by translation along some geodesic. In the quotient N = Y/ϱ(π_{1}(X)), this yields a closed geodesic c. Since each elliptic element has a fixed point, it corresponds to a point curve, i.e., a trivial closed geodesic in N.

We then map each circle in Σ_{δ}^{λ} onto the corresponding closed geodesic proportionally to arclength. This may be done smoothly on the family of circles defining Σ_{δ}^{λ}. On intersections Σ_{δ}^{λ_{1}} ∩ Σ_{δ}^{λ_{2}}, this may be performed in a compatible, i.e., continuous manner because the two elements of ϱ(π_{1}(X)) corresponding to the circles in Σ_{δ}^{λ_{1}} and Σ_{δ}^{λ_{2}} commute, since the two circles themselves commute as elements of π_{1}(X) near an intersection of D_{λ_{1}} and D_{λ_{2}}.

As explained above, we consider Σ_{δ}^{λ} as the boundary of a family of disks, S_{δ}^{λ} = ⋃_{w∈D_{λ}} B_{w,δ}, all holomorphically identified with {z ∈ ℂ : |z| ≤ δ}, with z = 0 corresponding to w ∈ D_{λ}. Likewise, the boundary ∂B_{w,δ} is identified with the aforementioned circle in Σ_{δ}^{λ}. We also identify X ∩ B_{w,δ} =: B_{w,δ}^{∗} with B_{δ}^{∗} := {z ∈ ℂ : 0 < |z| ≤ δ}, or with the semi-infinite cylinder A_{δ} := [log δ, ∞) × S^{1}.

The above map from Σ_{1}^{λ} to N induces a map from ∂A_{1} = {0} × S^{1} to N, mapping ∂A_{1} proportionally to arclength onto a closed geodesic γ in N. We denote this map by ḡ_{γ}(θ) (θ ∈ S^{1}). We extend ḡ_{γ} to all of A_{1} by putting

g_{γ}(s, θ) = ḡ_{γ}(θ).

Obviously, this defines a harmonic map from A_{1} onto the closed geodesic in N.

Performing this construction on all punctured disks B_{w,1}^{∗}, we obtain a map v_{1} from S_{1}^{λ} to N that is harmonic on each such punctured disk. Again, the construction is continuous in the intersection of two such sets S_{1}^{λ_{1}}, S_{1}^{λ_{2}}.

We shall perform a higherdimensional analogue of the construction of Wolf [48] and Lohkamp [32]. The essential point will be that v_{1} is absolutely energy minimizing in its homotopy class on annulus type compact subsets of each disk B_{w,1}^{∗}.

We recall the Poincaré metric on the punctured disk

−\frac{i}{2} \frac{∂^{2}}{∂z∂z̄} log log \frac{1}{|z|^{2}} dz ∧ dz̄ = \frac{i}{2} \frac{1}{|z|^{2} (log |z|^{2})^{2}} dz ∧ dz̄.

We put σ := ∏_{λ=1}^{l} σ_{λ}, choose M > sup_{X} |σ|, and may then construct a Poincaré type Kähler metric on X via

\frac{i}{2}γ_{αβ̄}dz^{α} ∧ dzβ̄ = −\frac{i}{2} \frac{∂^{2}}{∂z^{α} ∧ ∂zβ̄} log log \frac{M^{2}}{|σ|^{2}} dz^{α} ∧ dzβ̄ + cω_{αβ̄}dz^{α} ∧ dzβ̄

where ω_{αβ̄}dz^{α} ∧ dzβ̄ is some Kähler metric on X̄, and c is chosen so large that the above expression becomes positive definite. The resulting metric on X then is complete and of finite volume; see [6]. Also, when restricted to our above local disks near D, it behaves like the Poincaré metric.

We let

γ := det (γ_{αβ̄}) .

We put

v_{δ} := v_{1}|_{S_{δ}^{λ}},

and consider the harmonic map

u_{δ} : X \ S_{δ}^{λ} → N

with boundary values

u_{δ}|_{Σ_{δ}^{λ}} = v_{δ}|_{Σ_{δ}^{λ}}

and, of course, inducing the homomorphism ϱ : π_{1}(X) → I(Y). The existence of u_{δ} was shown by Schoen [45]. We shall show that the maps u_{δ} are equicontinuous on compact subsets of X as δ → 0 and induce a harmonic map u : X → N in the limit. In general, u will have infinite energy, but the blow-up of the energy density of u near D will be controlled sufficiently well in order to deduce that u is pluriharmonic.

We extend v_{1} smoothly from S_{1}^{λ} to all of X as a map to N in the required homotopy class, and denote this extension by

v : X → N.

We also put

A′_{δ} := [0, log δ] × S^{1}.

Thus

A_{1} = A_{δ} ∪ A′_{δ}.

Finally

S′_{δ}^{λ} := S_{1}^{λ} \ S_{δ}^{λ} , S′′_{δ} := X \ ⋃_{λ=1}^{l} S_{δ}^{λ} , S′_{δ} := ⋃_{λ=1}^{l} S′_{δ}^{λ}.

Let w ∈ D_{λ} be a regular point of D. Near w, we choose coordinates on X̄ such that z^{1} parametrizes B_{w,1}, and z^{2} parametrizes D_{λ}. If dim_{ℂ} X > 2, z^{2} will have more than one component, but this will not affect our subsequent reasoning. In the sequel, the index 2 will always stand for all those z^{2}-directions together. In our local coodinates, w will be represented as 0 = (0, 0). While the component γ_{1¯1} of the Kähler metric becomes singular at w, γ_{2¯2} extends smoothly to a neighborhood of w in D_{λ}. We write in our coordinates

γ_{2¯2}(z^{1}, z^{2}) = γ_{2¯2}(0, z^{2}) + O(|z^{1}|)

(see e.g. [21] for the extension property of γ_{2¯2}).

For the subsequent estimates, we assume that D has only one connected component, because the case of several components is handled by the same type of reasoning, but requires a more complicated notation.

The energy of u_{δ} is

E(u_{δ})

= ∫_{S′′_{δ}} γ^{1¯1}(z^{1}, z^{2})\frac{∂u_{δ}}{∂z^{1}} \frac{∂u_{δ}}{∂z̄^{1}} γ(z^{1}, z^{2})dz^{1} ∧ dz̄^{1} ∧ dz^{2} ∧ dz̄^{2}

+ ∫_{S′′_{δ}} ∑_{(α,β)≠(1,1)} γ^{αβ̄}(z^{1}, z^{2})\frac{∂u_{δ}}{∂z^{α}} \frac{∂u_{δ}}{∂z̄^{β}} γ(z^{1}, z^{2})dz^{1} ∧ dz̄^{1} ∧ dz^{2} ∧ dz̄^{2}.

Since at each point, the metric tensor can be diagonalized through a choice of holomorphic local coodinates whose first component is tangential to B_{w,1}, we conclude that

E(u_{δ}) ≥ ∫_{S′′_{δ}} γ^{1¯1}(z^{1}, z^{2})\frac{∂u_{δ}}{∂z^{1}} \frac{∂u_{δ}}{∂z̄^{1}} γ(z^{1}, z^{2})dz^{1} ∧ dz̄^{1} ∧ dz^{2} ∧ dz̄^{2}
= ∫_{S′′_{δ}} γ_{2¯2}(z^{1}, z^{2})\frac{∂u_{δ}}{∂z^{1}} \frac{∂u_{δ}}{∂z̄^{1}} dz^{1} ∧ dz̄^{1} ∧ dz^{2} ∧ dz̄^{2}
≥ ∫_{S′_{δ}} γ_{2¯2}(0, z^{2})\frac{∂u_{δ}}{∂z^{1}} \frac{∂u_{δ}}{∂z̄^{1}} dz^{1} ∧ dz̄^{1} ∧ dz^{2} ∧ dz̄^{2}

+ ∫_{S′_{δ}} O(|z^{1}|)\frac{∂u_{δ}}{∂z^{1}} \frac{∂u_{δ}}{∂z̄^{1}} dz^{1} ∧ dz̄^{1} ∧ dz^{2} ∧ dz̄^{2}.

Let l = l(c) be the length of the image geodesic c. Then

∫_{A′_{δ}} γ_{2¯2}(0, z^{2})\frac{∂u_{δ}}{∂z^{1}} \frac{∂u_{δ}}{∂z̄^{1}} dz^{1} ∧ dz̄^{1} ≥ |log δ| l^{2}γ_{2¯2}(0, z^{2}).

On the other hand,

E (v|_{S′′_{δ}}) = E (v|_{S′_{δ}}) + E (v|_{X\S_{δ}})
≤ E (v|_{S′_{δ}}) + c_{1}

for some constant c_{1}

= ∫ |log δ| l^{2}γ_{2¯2}(0, z^{2})dz^{2} ∧ dz̄^{2} + c_{2}

for some constant c_{2}

using the same expansion as above and

E (v|_{A′_{δ}}) = |log δ| l^{2}.

Here, the constants c_{1}, c_{2} are independent of δ, and so will be all subsequent ones. Since u_{δ} as a harmonic map is energy minimizing among all maps on S′′_{δ} with the same boundary values, we get

E(u_{δ}) ≤ E (v|_{S′′_{δ}}) .

Therefore

∫_{S′′_{δ}} ∑_{(α,β)≠(1,1)} γ^{αβ̄}(z^{1}, z^{2})\frac{∂u_{δ}}{∂z^{α}} \frac{∂u_{δ}}{∂z^{β}} γ(z^{1}, z^{2})dz^{1} ∧ dz̄^{1} ∧ dz^{2} ∧ dz̄^{2} ≤ c_{3},

where c_{3} again is independent of δ.

If δ < η < 1, we may also write

E (v|_{S′′_{δ}}) = E (v|_{S′′_{η}}) + E (v|_{S′_{δ}\S′_{η}}) .

Since the energy of u_{δ} on S′_{δ} \ S′_{η} again is bounded from below by the corresponding one of v up to some additive constant c_{4}, from the energy minimizing property of u_{δ} we obtain

E (u_{δ}|_{S′′_{η}}) ≤ E (v_{δ}|_{S′′_{η}}) + c_{5},

with c_{5} independent of δ (but not of η).

Thus, the energy of u_{δ} on S′′_{η} for fixed η > δ is bounded independently of δ. Therefore, as δ → 0, some subsequence of u_{δ} converges on S′′_{η} to a harmonic map, using standard estimates for harmonic maps (see [17]). By a diagonal sequence construction, we then get a harmonic map

u : X → N.

Here, the reductivity assumption prevents the limit u from disappearing at infinity. Furthermore, if we replace the coordinate z^{1} by the coordinate (s, θ) on the cylinder A_{1}, we get a coordinate system in which all first derivatives of u are uniformly bounded. This results from the facts that the contribution of the energy of u_{δ} in the z^{2}-direction is bounded independently of u_{δ} and that the energy of u_{δ} on A′_{η} behaves like |log η| l^{2}. Consequently, in our original (z^{1}, z^{2}) coordinates,

|\frac{∂u}{∂z^{1}} (z^{1}, z^{2})| ≤ \frac{c_{6}}{|z^{1}|},
|\frac{∂u}{∂z^{2}} (z^{1}, z^{2})| ≤ c_{7}

for constants c_{6}, c_{7}.

Similar constructions apply near singularities of D, i.e., where two components of D meet. Here, we may use σ as our first coordinate and then obtain a cylindrical coordinate system by replacing σ by log σ. We conclude that in the σ-direction, the derivative of u behaves like \frac{1}{|σ|}, whereas in directions normal to σ, it is bounded. (By construction, this is the behaviour of v, and we compare u_{δ} and v as before up to some bounded terms.)

Lemma 1.1. u is pluriharmonic.

Proof. By Siu’s formula [46], using local coordinates on N and denoting the corresponding metric tensor by (g_{ij}), we obtain

∂ ¯∂(g_{ij} ¯∂u^{i} ∧ ∂u^{j}) ∧ ω^{m−2} = fω^{m}

(ω =Kähler form of some Kähler metric on X, m = dim_{ℂ} X), for some nonnegative^{1} function f, provided Y has nonpositive curvature operator, a condition satisfied in our applications. If Y is a Euclidean Tits building, the preceding formula holds in a weak sense, but the computations can be justified by the analysis of Gromov-Schoen [11]. If f ≡ 0, then u is pluriharmonic.

On a compact manifold, the left hand side of Siu’s formula integrates to 0, and one readily concludes f ≡ 0. In the present noncompact case, we shall verify this with the help of a cut-off argument. For simplicity of notation, we shall only consider the case m = 2, and the higher dimensional case is completely analogous.

1The signs in [46] are not quite correct, but this will be irrelevant for the structure of the argument.

If φ is a cut-off function that vanishes near D, we have

∫_{X} φ∂ ¯∂(g_{ij} ¯∂u^{i} ∧ ∂u^{j}) = ∫_{X} ∂ ¯∂φ ∧ g_{ij} ¯∂u^{i} ∧ ∂u^{j}.

For 0 < ϵ < 1, we choose the cut-off function

φ_{ϵ}(z) := \begin{cases}(1 − \frac{log|σ(z)|^{2}}{log ϵ^{2}} )^{2} & for ϵ ≤ |σ(z)| ≤ 1, \\ 0 & for |σ(z)| < ϵ, \\ 1 & for |σ(z)| > 1. \end{cases}

φ_{ϵ}(z) is of class C^{1,1} near |σ(z)| = ϵ. It is not of class C^{1,1} at |σ(z)| = 1, but the derivative of φ_{ϵ} near |σ(z)| = 1 tends to 0 as ϵ tends to 0, and therefore, this defect can be remedied by a straightforward interpolation that is left out.

We compute

\frac{∂}{∂z^{α}} φ_{ϵ} = −2 (1 − \frac{log |σ(z)|^{2}}{log ϵ^{2}} )^{2} \frac{1}{log ϵ^{2}} \frac{σ_{z^{α}}}{σ}

\frac{∂^{2}}{∂z^{α}∂z̄^{β}} φ_{ϵ} = \frac{1}{(log ϵ^{2})^{2}} \frac{σ_{z^{α}}σ̄_{zβ̄}}{|σ|^{2}} .

We then have

∫ ∂ ¯∂φ_{ϵ} ∧ g_{ij} ¯∂u^{i} ∧ ∂u^{j}

≤ c_{8} ∫_{|σ(z)|=ϵ}^{|σ(z)|=1} \frac{1}{(log ϵ^{2})^{2}} \frac{1}{|σ|^{2}}

(since the derivative of u is bounded in the directions normal to σ)

≤ c_{9} ∫_{ϵ}^{1} \frac{1}{(log ϵ^{2})^{2}} \frac{1}{r^{2}} rdr

(using polar coordinates σ = re^{iθ})

= \frac{−c_{9} log ϵ}{4(log ϵ)^{2}},

and this tends to 0 as ϵ → 0.

Therefore,

∫_{X} ∂ ¯∂(g_{ij} ¯∂u^{i} ∧ ∂u^{j}) = lim_{ϵ→0} ∫_{X} ∂ ¯∂φ_{ϵ} ∧ g_{ij} ¯∂u^{i} ∧ ∂u^{j} = 0.

Thus, in Siu’s formula, f ≡ 0, and u is pluriharmonic as desired. q.e.d.

Remark. Instead of Siu’s Bochner identity, we might as well employ one of the other Bochner type identities that have been shown to apply in the present situation; see e.g. [44] or [47].

Theorem 1 for the case of a Euclidean Tits building now follows from

Lemma 1.2. Let Y be a locally compact Euclidean Tits building, with isometry group operating transitively on the vertices. Then every isometry of Y is elliptic or hyperbolic.

Proof. Let γ ∈ I(Y), and let (y_{n})_{n∈ℕ} ⊂ Y be a sequence with

lim_{n→∞} d(y_{n}, γy_{n}) = inf_{y∈Y} d(y, γy).

If (y_{n}) is bounded, a subsequence converges, and the infimum is realized so that γ is elliptic or hyperbolic. If (y_{n}) is unbounded, we perform the following construction. We choose p_{0} ∈ Y and let c_{n}(t) be the geodesic from p_{0} to y_{n}, parametrized by arclength t ≥ 0. c′_{n}(t) := γc_{n}(t) then is the geodesic from γp_{0} to γy_{n}. Since (y_{n}) is unbounded, after selection of a subsequence, we obtain geodesic rays c(t) and c′(t) = γc(t). These two rays stay within bounded distance of each other. Therefore, they are asymptotically contained in a flat F. Thus, they are geodesic rays in the Euclidean space F, and they then have to be parallel. In particular, we may choose p_{0} so that

d(p_{0}, γp_{0}) = d(y_{n}, γy_{n})

for all n, and p_{0} realizes the infimum. Thus, γ has to be hyperbolic. q.e.d.

1.5. Arbitrary representations

In this section, we make the following assumption:

(B) Let c_{1}(t), c_{2}(t) be geodesic rays in Y parametrized by arclength

t ∈ ℝ^{+} with

lim_{t→∞} d(c_{1}(t), c_{2}(t)) = α.

Then there exist constants β > 0 and γ with d(c_{1}(t), c_{2}(t)) ≤ α + γe^{−βt} for all t > 0. (1.1)

Assumption (B) is satisfied if Y is a symmetric space of noncompact type. Therefore, the considerations of the present section will prove Theorem 1.1 in that case.

Actually, under the assumption (B), the case of parabolic elements can be handled by the analysis of our previous paper [24] because it does not cause infinite energy. Parabolic elements, however, also easily succumb to the treatment of quasihyperbolic elements that we are going to present. (Namely, our reasoning will not need that the constant λ_{g} introduced below is positive.)

We wish to proceed as in the preceding section, but we need to construct the comparison map v_{r} : Σ_{r} → N somewhat differently. In the notation of the preceding section, we let {0} × S^{1} ⊂ A_{1} correspond to g ∈ π_{1}(X), and choose a sequence (y_{n})_{n∈ℕ} ⊂ Y with

lim_{n→∞} d(y_{n}, ρ(g)y_{n}) = inf_{y∈Y} d(y, ρ(g)y) =: λ_{g}.

We also choose y_{0} ∈ Y. If ρ(g) is not elliptic or hyperbolic, (y_{n})_{n∈ℕ} diverges, and after selection of a subsequence, the geodesic arcs from y_{0} to y_{n} converge to a geodesic ray c_{0}(t), t being the arclength parameter. The geodesic arcs from ρ(g)y_{0} to ρ(g)y_{n} converge to the geodesic ray c_{1}(t) := ρ(g)c_{0}(t). We let γ_{t}(s) be the geodesic arc from c_{1}(t) to c_{2}(t) with parameter s ∈ [0, 2π] proportional to arclength (the proportionality factor depends on t). Using polar coordinates on our local disks transversal to D, on

{(ρ, ϑ) : 0 ≤ ρ ≤ r, 0 ≤ ϑ ≤ 2π} ,

we put

v_{r}(ρ, ϑ) := γ_{μ log(ρ+1)}(ϑ), (1.2)

where passing to the quotient N is implicitly understood, and μ > 0 is chosen in such a way that μβ > 1 with β as in (B). We have

(1.3)  ∥\frac{∂v_{r}}{∂ρ} ∥ ≤ c\frac{1}{ρ + 1} for some constant c,

and

∥\frac{∂v_{r}}{∂ϑ} ∥ ≤ \frac{1}{2π} (λ_{g} + γe^{−βμ log ρ}) = \frac{1}{2π} (λ_{g} + γ \frac{1}{ρ^{βμ}} ) , (1.4)

because of assumption (B). Since βμ > 1 by choice of μ,

E(v_{r}) ≤ \frac{1}{2}rλ_{g}^{2} + κ_{1} (1.5)

for some constant κ_{1}, and we may then proceed as in the previous section to conclude the proof. q.e.d.

1.6. Harmonic bundles

In order to formulate a corollary, we need to recall some constructions of Hitchin [13], Corlette [4], and Simpson [42]. Let V be a Gl(n, ℂ) bundle with a flat connection D. Introducing a metric g leads to the decomposition

D = D_{g} + ϑ,

where D_{g} preserves the metric. Let ρ : π_{1}(X) → Gl(n, ℂ) be the representation defined by the flat bundle V. A metric on V can be considered as a ρ-equivariant map

h : X̃ → Gl(n, ℂ)/U(n).

We have dh = ϑ, and h is harmonic iff

D_{g}^{∗}ϑ = 0.

In this case, one says that the metric is harmonic. We then decompose into types:

D_{g} = D′ + D′′,
ϑ = ϑ^{1,0} + ϑ^{0,1}.

We have (D′′)^{2} = 0, and

E := (V, D′′)

then is a holomorphic bundle. (The complex structure on E is different from the complex structure on V induced by the flat connection D, unless h is constant, i.e., ρ is a U(n) representation.) Thus (E, ϑ^{1,0}) is a Higgs bundle. A Higgs bundle on X consists of a holomorphic bundle E together with

ϑ^{1,0} : E → Ω^{1,0}(X) ⊗ E

satisfying the integrability condition

ϑ^{1,0} ∧ ϑ^{1,0} = 0.

(It was shown by Sampson [44] that the harmonicity of h implies this integrability condition.) Let (E, ϑ^{1,0}) be a Higgs bundle on X. The analogue of the ¯∂-operator on E now is

D^{2} = ¯∂ + ϑ^{1,0}.

A metric on E then defines

D^{1} = ∂ + ϑ^{0,1}

with

(ϑ^{0,1}e, f) = (e, ϑ^{0,1}f) for all e, f.

The metric again is preserved by

D_{g} = ∂ + ¯∂ ,

and the metric is harmonic iff D is flat. These two conditions are inverses of each other.

Corollary 1.1.

(i) Let (V, D) be a flat bundle on X defined by a reductive representation ρ. Let u of Theorem 1.1 define the corresponding harmonic metric g, and write

D = D_{g} + du = ∂ + ¯∂ + ϑ^{1,0} + ϑ^{0,1}

as above. Then the curvature of the metric connection D_{g} = ∂ + ¯∂ is bounded by a Poincaré type Kähler metric. (In particular, the curvature is in L^{1}.)

(ii) Let (E, ϑ^{1,0}) be a Higgs bundle on X with u of Theorem 1.1 defining a harmonic metric as before. Then the metric connection associated to the holomorphic structure ¯∂+ϑ^{1,0}, namely ∂+ ¯∂+ϑ^{1,0}+ ϑ^{0,1}, has curvature again bounded by a Poincaré type Kähler metric.

Proof. If we restrict everything to a curve transversal to D in the direction defined by σ, from the proof of Theorem 1.1, we have the estimate

∥du(z)∥^{2} ≤ \frac{const}{|z|^{2}} (using z = σ as a coordinate on the curve)

on that curve, while ∥du∥ is locally bounded in the other directions. The argument of Simpson (see [40, Section 2; main estimate; Thm. 1]) then implies that the curvatures R of the connections under consideration locally satisfy an estimate

∥R∥ ≤ \frac{const}{|z|^{2} (log |z|)^{2}} .

q.e.d.

2. Strong rigidity of lattices in Hermitian symmetric spaces

Theorem 2.1. Let Y be an irreducible Hermitian symmetric space of noncompact type other than the hyperbolic plane. Let Γ be a lattice in Y, i.e., a discrete subgroup of I(Y) with a quotient Γ\Y of finite volume. Let X be a smooth quasiprojective variety (or, more generally, a Kähler manifold X that can be compactified as a Kähler manifold X̄ such that X̄ \ X is a divisor with at most simple normal crossings as singularities — note that in the quasiprojective case such a compactification always exists by Hironaka’s theorem) with contractible universal cover of dimension (larger or) equal to the dimension of Y and with π_{1}(X) isomorphic to Γ. Then the universal cover X̃ of X is ±biholomorphically^{2} equivalent to Y, and π_{1}(X) is conjugate to Γ as a group of automorphisms of Y.

Remark. We have formulated Theorem 2.1 here for lattices without fixed points only. It is known, however, that the general case can easily be reduced to that case by taking suitable finite covers. Therefore Theorem 2.1 continues to hold if X is a finite quotient of a smooth quasiprojective variety.

In the case where X is also locally Hermitian symmetric, the result is due to Mostow [36] (X compact), Prasad [38] (Y of rank 1) and Margulis [33]. If X is compact, the result was proved by Siu [46] with harmonic map techniques. In the general case, the result was shown by Jost-Yau [21] under an additional technical assumption on the projective compactification X̄ of X.

Before giving the actual proof of Theorem 2.1, we should describe the essential difficulty encountered in it. This difficulty stems from the fact that while X and Γ\Y are assumed to be homotopically equivalent, the compactifications of these spaces may a priori be quite different. Let us consider the simple example N := S^{1} × S^{1} × [0, 1]. This space may be considered as either

2this means biholomorphically or antibiholomorphically

N_{1} := S^{1} × {zϵℂ : 0 < |z| ≤ 1} or N_{2} := {zϵℂ : 0 < |z| ≤ 1} × S^{1},

and thus it may be compactified by adding either the first or the second S^{1} factor times {0}. If we equip {zϵℂ : 0 < |z| ≤ 1} with its complete hyperbolic metric, and N_{1} and N_{2} with the resulting product metrics, N_{1} and N_{2} become complete Riemannian manifolds with boundary S^{1} × S^{1}. The identity of N obviously induces a proper homotopy equivalence between N_{1} and N_{2}. In N_{1}, the first S^{1} factor corresponds to a hyperbolic element of the fundamental group, and the second S^{1} factor to a parabolic one, while for N_{2} these roles are exchanged. Consequently, the harmonic map between N_{1} and N_{2} which is homotopic to the identity produced by our method will have infinite energy because a parabolic element is mapped to a hyperbolic one.

Since, as mentioned, in the situation of Theorem 2.1, at the beginning we cannot identify the two compactifications of X and Γ\Y, we have to deal with such maps of infinite energy although in the end one of the conclusions of the proof has to be that the harmonic map under consideration after all does have finite energy, and the two spaces do possess isomorphic compactifications.

If Y has rank 1, then Γ\Y may be compactified by adding finitely many cusp points. Consequently, all elements of the fundamental group, i.e., of Γ, that can be deformed to infinity are parabolic. Therefore, in particular, the images of all parabolic elements of π_{1}(X) are parabolic again, and one may construct a harmonic map of finite energy. This is the case treated in [20]. Thus, in the sequel, we may assume that Y has rank at least 2, and by the theorem of Margulis (see [34]), Γ is arithmetic. This fact then gives additional information about possible compactifications of Γ\Y.

In [21], the situation was considered where one can produce a finite energy harmonic map in spite of the possibility that a parabolic element may be mapped to a hyperbolic or quasihyperbolic one. Namely even if that occurs and the energy density of any homotopy equivalence then has to blow up towards infinity, this might still be compensated by a sufficiently fast decay of the volume form of the domain near infinity so that the overall energy of some suitable map might still turn out to be finite. This requires an additional assumption on the compactification of X. Here, we do not wish to make such an assumption.

The reason for wanting to have a finite energy harmonic map was to make sure that the Bochner type identity of Siu is satisfied. In §1, however, we have produced a map of possibly infinite energy that still satisfies this Bochner identity. For this reason, the proof of Theorem 2.1 may proceed essentially as in [21], and we may therefore be somewhat brief in certain places.

After this discussion, let us now start with the

Proof of Theorem 2.1. Since π_{1}(X) is isomorphic to the lattice Γ, we obtain a reductive homomorphism

ρ : π_{1}(X) → I(Y) with ρ (π_{1} (X)) = Γ.

Theorem 1.1 yields a ρ-equivariant pluriharmonic map

u : X̃ → Y.

u can be considered as a pluriharmonic map

u : X → Γ \ Y.

Moreover, in the course of proof of Theorem 1.1 we have shown that u satisfies Siu’s Bochner identity. By our topological assumptions, u is a homotopy equivalence.

The work of Borel-Serre [3] implies that the Γ \ Y has nontrivial cohomology of degree 2 dim_{ℂ} Y − ℚ-rank(Γ), which is at least 2 dim_{ℂ} Y − rank(Y). Therefore, the maximal rank of the differential du of u is at least this number. This is large enough for Siu’s analysis [47] to apply. Thus we may conclude that u is ±holomorphic.

We equip X with a Poincaré type metric as in §1.4. By a result of Cornalba-Griffiths [6], such a metric has Ricci curvature bounded from below, and therefore Royden’s version [39] of Yau’s Schwarz Lemma [49] may be applied to u as u is ±holomorphic.

This Schwarz lemma says that

∥du(z)∥^{2} ≤ \frac{k_{1}}{k_{2}} ,

where k_{1} is a lower bound for the Ricci curvature of X, and k_{2} is an upper bound for the holomorphic sectional curvature of Y. With the standard normalizations, we have in fact k_{2} = −1. In particular, u has bounded energy density, and since the Poincaré type metric on X has finite volume (see [6], or also [21]), u has finite energy after all. Therefore, the rest of the proof may proceed essentially as in [21]. Nevertheless, we shall describe the key steps now.

The first step is the properness of u. The details of this step are provided in [21, §3d]. The essential idea is the following: Let us consider a loop around a component of X̄ \X that is contractible in X̄ but not in X (at least not in a neighborhood of X̄ \ X). We call such a loop short, because it is homotopic to loops of arbitrarily small length in X. The Schwarz lemma implies that its image under u is likewise short in this sense, i.e., it corresponds to a parabolic element of Γ. Consequently, if we take a sequence of such loops in a given homotopy class in X that contract to a point on X̄ \ X, the images of these loops have to go to infinity in Γ \ Y. From this observation, properness is deduced.

(In passing, we observe that it follows from Remmert’s proper mapping theorem that the image of X under the proper ±holomophic map u now can be seen to be an analytic subvariety of Γ \ Y.)

Also, one shows as in [21, §3d] that u extends as a continuous map from X̄ to the Baily-Borel compactification of Γ \ Y, essentially because the latter is the minimal compactification. Therefore, u has a well defined degree.

As u is a proper ±holomorphic map, the preimage of a point in Γ \ Y has to be zerodimensional, as otherwise it would be a compact subvariety, hence represent a nontrivial homology class in X. As u is a homotopy equivalence, such a class cannot be contracted to a point.

Consequently, u has nonzero degree. It remains to show that the absolute value of the degree is 1. For that purpose, one proceeds as in [21, §3e] and considers a proper homotopy equivalence g : Γ \ Y → X such that u ∘ g is homotopic to the identity of Γ \ Y.

As above, we may deform u∘g into a harmonic homotopy equivalence h which then again turns out to be proper and ±holomorphic.

The Schwarz lemma implies

|degh| ≤ 1,

either directly by observing that the constants in the Schwarz lemma of Yau-Royden are sharp, or by considering the iterates h^{n} which would violate the inequality of the Schwarz lemma for sufficiently large n if |degh| were larger than 1.

Since degh = degu · degg, we also have

|degu| ≤ 1.

Altogether, |degu| = 1.

Finally, as in [46] and [21, §3e] one verifies that u is ±biholomorphic. Namely, let V be the set of points where the Jacobian of u vanishes. If V were not empty, it would be a complex subvariety of complex codimension 1 whereas u(V) would have codimension at least 2 as u is of degree ±1.

Thus, the preimage of a generic point in u(V) would be a nontrivial analytic subvariety, and compact as u is proper.

Again, this is not compatible with u being a homotopy equivalence. We conclude that V is empty. Thus, u is ±biholomorphic. q.e.d.

Remark. The proceeding proof made heavy use of the fact that Γ is known to be arithmetic by the work of Margulis. Ideally, the harmonic map approach should not use this result, but rather deduce it as a corollary. We thus obtain a generalization of the superrigidity theorem of Margulis, but not a new proof of the original version. In this sense, the harmonic map approach to superrigidity in the noncompact case is still not completely satisfactory.

3. Some applications to representations of π_{1} of algebraic varieties

3.1. Holomorphic 1-forms and spectral coverings

We now study the holomorphic 1-forms arising from the harmonic map u : X̃ → Δ. Let X̃_{r} be the set of regular points of u. A point x_{0} ∈ X̃ is called a regular point of u if there exist a ball B(x_{0}, δ_{0}) of radius δ_{0} > 0 and a rk_{K_{p}} G-flat F ⊂ Δ(G) with B(x_{0}, σ_{0}) ⊂ F (see [11, p.68]). And S(u) := X̃ \ X̃_{r} is called the singular set of u. Let A (≃ R^{r})be an apartment of Δ and W = Z^{r} × W̄ be the affine Weyl group of δ(G). Here, W̄ is the usual Weyl group of G(K_{p}) which operates on A as a finite linear subgroup generated by reflections, and Z^{r} acts on A as the usual translations.

The main point in this section is to construct the spectral covering of u. The spectral covering for the Higgs bundle case has been intensively studied by R. Donagi [8] and C. Simpson [42]. For the case where u is a harmonic map into a building we have a similar construction. The basic idea is to construct the coefficients of the characteristic polynomial directly from the differential of u.

Let R = {β_{1}, …, β_{l}} be the root system of W̄, where β_{i} are normalized vectors in R^{r} and β^{⊥} are the reflection hyperplanes. We can consider β_{i} as coordinate functions on A by orthogonal projection from A to β_{i}. By taking the differential, we obtain a collection of differential 1-forms {dβ_{1}, …, dβ_{l}}_{A} on A. It has the property that on the common part of any two apartments A and A′ these collections of 1-forms {dβ_{1}, …, dβ_{l}}_{A} and {dβ_{1}, …, dβ_{l}}_{A′} coincide as sets, and the indices of the 1-forms in these two collections differ by a permutation from W̄. Here, a differential form on Δ means that its restriction to each apartment is a usual differential form.

Let s_{1}, …, s_{l} be the basic symmetric polynomials of l variables. We set σ_{1} = s_{1}(β_{1}, …, dβ_{l}) , …,σ_{l} = s_{l}(β_{1}, …, dβ_{l}). Since they are invariant under the W̄ -action, they piece together into differential forms σ_{1}, …, σ_{l} on Δ in the symmetric tensor product. It is clear that {dβ_{1}, …, dβ_{l}} are roots of the polynomial t^{l} + σ_{1}t^{l−1} + ⋯ + σ_{l}.

We then take the complexified pull back u^{∗c}(σ_{1}), …, u^{∗c}(σ_{l}) via the differential du. Their (1, 0)-parts are holomorphic forms on X̃ in the symmetric tensor product which are called again σ_{1}, …, σ_{l}. This can be seen as follows. Clearly, σ_{1}, …, σ_{l} are holomorphic on X̃^{r} = X̃ \ S. Since u is Lipschitz, du is bounded near S and codim S ≥ 2, they can be extended over S. Because u is equivariant, σ_{1}, …, σ_{l} is π_{1}(X)- invariant, they descend to some holomorphic forms on X that we call again σ_{1}, …, σ_{l}. Because of the controlled growth at infinity, σ_{1}, …, σ_{r} have at most log poles along D_{∞}.

Now the characteristic polynomial t^{l} + σ_{1}t^{l−1} + ⋯ + σ_{l} defines a subvariety Q in the total space T_{X}^{∗} of the holomorphic cotangent bundle of X (see [42] for details). It is called the spectral variety of u and has the property that the restriction of the projection to Q induces a ramified covering (maybe nonreduced) p : Q → X and the preimage p^{−1}(x) coincides with the roots of the polymonial t^{l} + σ_{1}(x)t^{l−1} + ⋯ + σ_{l}(x). From this property we see that the embedding of Q ⊂ T_{X}^{∗} defines tautologically a holomorphic 1-form ω_{1} on Q such that ω_{1} is a root of the polymonial t^{l} + p^{∗}σ_{l}t^{l−1} + ⋯ + p^{∗}σ_{l}.

Furthermore we take the Galois closure of the function field extension K(Q)/K(X), and obtain a Galois covering σ : X^{s} → X and l 1-forms ω_{1}, …, ω_{l} ∈ Γ(X^{s}, σ^{∗}Ω_{X}^{1}) having at most log-poles along D_{∞} such that they are the roots of the polynomial t^{l} + σ^{∗}σ_{l}t^{l−1} + ⋯ + σ^{∗}σ_{l}, and the ramification divisor R ⊂ X^{s} of σ is contained in the union of the loci ⋃_{ω_{i}≠ω_{j}}(ω_{i} − ω_{j})_{0}. Vie the construction we see that the pull back σ^{∗}u is the equivariant pluriharmonic map for σ^{∗}p, and d′(σ^{∗}u)β_{1}, …, d′(σ^{∗}u)β_{l} glue together and yield ω_{1}, …, ω_{l}. The image of the singular set u(S(u)) is contained in the closed faces of all rk_{K_{p}} Gsimplices of Δ(G). By taking sequences along normal directions going to faces and using the estimate (1), we obtain the following.

Lemma 3.1.

1) There exists a finite ramified Galois covering σ : X^{s} → X so that the differentials d′(uσ)β_{1}, …, d′(uσ)β_{l} of the coordinate functions on all apartments of Δ(G(K_{p})) chosen as above piece together and yield l holomorphic 1-forms ω_{1}, …, ω_{l} ∈ Γ(X^{s}, σ^{∗}Ω_{X}^{1}) having at most log poles along D_{∞} and the ramification divisor R ⊂ X^{s} ⊂ ⋃_{ω_{i}≠ω_{j}}(ω_{i} − ω_{j})_{0}.

2) The singular set σ^{∗}S(u) of the harmonic map σ^{∗}u is contained in the union of the zero loci of some holomorphic 1-forms which are linear combinations of ω_{1}, …, ω_{l}.

3.2. The quasi Albanese map, related fibrations and a Lefschetz-type theorem

For a quasi projective smooth variety X = X̄ \ D_{∞}, Iitaka [14] has defined the quasi Albanese map

ψ : X \underrightarrow{∫_{x_{0}}^{x} ω} Alb(X)

by taking integrals of holomorphic 1-forms of X which have at most log poles along D_{∞}. Alb(X) is a semi abelian variety which is a group extension of ℂ^{∗d} by Alb(X̃). The morphism ψ extends to a rational map

ψ̄ : X̄ → \overline{Alb(X)},

where \overline{Alb(X)} is an algebraic P^{d}-fibre bundle over Alb(X̄). Using the map ψ̄, one obtains (as in [51], [24], and [25])

Lemma 3.2 ([24, Lemma 6.1]). Given r 1-forms

ω_{j} ∈ H^{0} (X̄, Ω1 ; X̄ (log D_{∞}(j))) ,

1 ≤ j ≤ r, suppose that the pulled back forms satisfy di_{B_{j}}ω_{j} = 0 along an effective divisor i_{B_{j}} : B_{j} ↪ X̄ and B_{j} ∩ D_{∞}(j) = ∅. Then there exists the following diagram

[figure: Z̄; X̄; W̄; τ; g]

where τ is a surjective, generically finite Galois map, g is a fibration over W̄ with dim W̄ ≤ r, and G^{−1}(ω) ∩ (∑_{1≤j≤r} B_{j}) = ∅ for generic ω.

Given again r holomorphic 1-forms ω_{j} on X̄, which have at most log poles at infinity, we let X̃ → X be the universal covering of X, and consider the integrals

X̃ ⟶^{(∫_{x_{0}}^{x} ω_{1},... ,∫_{x_{0}}^{x} ω_{r})=:h} ℂ^{r}.

Lemma 3.3 ([25], Section 3, Theorem 3.3). If ∀ 1 ≤ k ≤ r there is no collection of k linearly independent 1-forms in the span of ω_{1}, …, ω_{r} that factor through a surjective morphism g : X → W of dim W ≤ k, then h defined as above is surjective and all fibres of h are connected.

Lemma 3.4. Let f : X → Y be a surjective morphism with connected fibres, and ρ : π_{1}(X) → G be a Zariski dense representation into a simple algebraic group G. Suppose that f^{−1}(y) is a smooth fibre. Then either

a) the restriction ρ|_{f^{−1}(y)} is again Zariski dense, or

b) ρ|_{f^{−1}(y)} = {1}.

c) If ϱ|_{f^{−1}(y)} = {1}, then after passing to a blowingup and a finite etale covering e : X′ → X, the pull back e^{∗}ϱ factors through the Stein-factorization of fe.

Proof. The proof of a) and b) is very simple, just using the fact that π_{1}(f^{−1}(y)) is a normal subgroup in π_{1}(X). We only need to show c).

The main point in the proof is to use the fact that any finitely generated group of matrices is residually finite. Recall that a group is said to be residually finite if the intersection of all subgroups of finite index consists of the identity element alone.

We adopt some arguments used in [29] Section 4, and [43] Lemmata 6 and 6.4. Let Y_{0} ⊂ Y be the open subset over which f is regular. By blowing up we may assume that Y \ Y_{0} = ⋃ B_{j} is a divisor with normal crossings only. On the regular part, the exact homotopy sequence

π_{1}(F) → π_{1}(f^{−1}(Y_{0})) → π_{1}(Y_{0}) → 1

implies that ϱ factors through a representation τ : π_{1}(Y_{0}) → GL_{r}.

Claim 3.1. Let f^{∗}(B_{j}) = ∑_{i=1}^{m_{j}} b_{ij}B_{ij} and γ_{j} be a short loop wich goes around B_{j}. Then τ(γ_{j}) has finite order, and the order n_{j} satisfies n_{j}|b_{ij}, 1 ≤ i ≤ m_{j}.

Proof. Let Δ_{j} be a small disc transversal to B_{j} at the generic point, and let Δ_{ij} ⊂ f^{−1}(Δ_{j}) be a small disc in a small neighborhood of the generic point of B_{ij} ∩ f^{−1}(Δ_{j}). Choose coordinates x on Δ_{ij} and z on Δ_{j} such that f : Δ_{ij} → Δ_{j} is given by z = x^{b_{ij}}. Take a short loop γ_{ij} which lies in Δ_{ij}^{∗} and goes around B_{ij}. Thus we have f_{∗}(γ_{ij}) = γ_{j}^{b_{ij}}, and τ(γ_{j})^{b_{ij}} = ϱ(γ_{ij}) = 1. Hence Claim 3.1 is complete.

For a point z ∈ ⋃ B_{j}, let U_{z} denote a small ball in Y centered in z, and let Υ_{z} denote the fundamental group of U_{z} \ ⋃ B_{j}. It is known that Υ_{z} is a free abelian group generated freely by the short loops which go around the components of ⋃ B_{j} passing through z.

Claim 3.2.

1) There exists a subgroup Γ ⊂ τπ_{1}(Y_{0}) of finite index such that Γ ∩ τ(Υ_{z}) = 1 ∀z ∈ ⋃ B_{j}.

2) Let e : Y′_{0} → Y_{0} be the finite etale covering corresponding to the finite index subgroup τ^{−1}Γ ⊂ π_{1}(Y_{0}), Then there is a smooth completion Y′ ⊃ Y′_{0} such that the extended map e : Y′ → Y is a branched covering with the branching order n_{j} along B_{j}, and e^{∗}τ can be extended across Y′.

Proof. 1) Since ⋃ B_{j} has only finitely many components, only finitely many images {Υ_{z} → π_{1}(Y_{0}), z ∈ ⋃ B_{j}} can occur. Note that τΥ_{z} is finite by Claim 3.1. Taking a decreasing sequence of finite index subgroups {Γ_{i}} of τπ_{1}(Y_{0}) with ⋂_{i} Γ_{i} = 1, we find a Γ_{l} such that Γ_{l} ∩ τΥ_{z} = 1 ∀z ∈ ⋃ B_{j}.

2) Let e : Y′_{0} → Y_{0} be the finite etale covering corresponding to τ^{−1}Γ_{l}. For each point z ∈ ⋃ B_{j}, let B_{1}, …, B_{d} be the components of ⋃ B_{j} which contain z. We may take a small neighborhood of z in the form Δ^{d} ×Δ^{k−d} and with coordinates z_{1}, …, z_{d}, z_{d+1}, …, z_{k}, such that z_{j} = 0 defines B_{j}, 1 ≤ j ≤ d. The local covering

e : e^{−1}(Δ^{∗d} × Δ^{k−d}) → Δ^{∗d} × Δ^{k−d}

corresponds to the subgroup τ^{−1}(Γ_{l}) ∩ Υ_{z}, which is generated by the loops γ_{1}^{n_{1}}, …, γ_{d}^{n_{d}}. Thus, the corresponding covering is given by

e : Δ^{∗d} × Δ^{k−d} → Δ^{∗d} × Δ^{k−d}

with

z_{1} = x_{1}^{n_{1}}, …, z_{d} = x_{d}^{n_{d}}, z_{d+1} = x_{d+1}, …, z_{k} = x_{k}.

The completion Y′ is smooth, and the extended map e : Y′ → Y has the branching order n_{j} along B_{j} via its construction.

Now π_{1}(Y′) is the quotient of π_{1}(Y′_{0}) divided by the short loops around the components of the preimage e^{−1}(⋃ B_{j}). Let γ′_{j} be a short loop around a component of e^{−1}(B_{j}), Then e_{∗}(γ′_{j}) = γ_{j}^{n_{j}}. This shows that e^{∗}τ factors through the quotient π_{1}(Y′). We are done with Claim 3.2.

Finally, we take the normalization of the fibre product

e : X′ =: (X ×_{Y} Y′)^{nor} → X.

Since f^{∗}(B_{j}) = ∑_{i=1}^{m_{j}} b_{ij}B_{ij} and the covering e : Y′ → Y has the branching order n_{j} along B_{j} with n_{j}|b_{ij}, 1 ≤ i ≤ m_{j} by Claim 3.1, it follows that e : X′ → X is etale. Let f′ be the Stein-factorization of fe. Then it is clear that f′^{∗}e^{∗}τ = e^{∗}ϱ. c) is proved. q.e.d.

The following lemma is well known.

Lemma 3.5 [25]. Suppose that K is a complete field with respect to a discrete valuation and F is a flat of positive dimension in the Tits building Δ(G(K)). Then the isotropy group I_{F} ⊂ G(K) of F is not Zariski dense.

3.3. Factorization theorems for nonrigid and unbounded representations

Theorem 3.1. Let G be a simple algebraic group over ℂ. If ρ ∈ χ_{∞}^{−1}(t) is a Zariski dense representation, and ρ is nonrigid in χ_{∞}^{−1}, then ρ factors through a morphism f : X → Y with dim Y ≤ rk_{ℂ} G.

Remark. The Zariski density of ρ is not a serious restriction. In general we consider a semisimple representation ρ, and take its Zariski closure in G. It is a direct product of almost simple algebraic groups.

Theorem 3.2. Let G be a simple algebraic group over a complete field K with a discrete valuation. Suppose that Δ(G(K)) is locally compact. If ρ is a Zariski dense representation, unbounded with respect to this valuation, then ρ factors through a morphism f : X → Y with dim Y ≤ rk_{K}(G).

Remark. Without the local compactness of Δ(G(K)), but assuming that ρ is stabilizing at infinity, Theorem 3.2 is Theorem 2 of [25]. For G = SL_{r+1}(ℂ), there are two proofs for Theorem 3.1. The first proof uses Higgs bundles as dicussed below, and the second one reduces it to Theorem 2 in [25] for SL_{r+1} over a function field. This latter proof works for an arbitrary simple group.

Proof of Theorem 3.2. Let ω_{1}, …, ω_{r}, r ≤ rk_{K} G, be a base of the holomorphic 1-forms from the harmonic map in section 3.1. Using Lemma 3.4 and Lemma 3.3, we obtain the following property (see [25], Claim 4.2):

There are two possibilities: either

1) There is a decomposition of the span ⟨ω_{1}, …, ω_{r}⟩ = U_{1} ⊕ U_{2} with dim U_{2} > 0, and a subvariety V ⊂ X^{s} of dim V ≥ dim X^{s} −dim U_{1} such that the holomorphic map

Ṽ \underrightarrow{∫ U_{2}:=h} ℂ^{dim U_{2}}

is surjective and with connected fibres and τρ|_{V} is Zariski dense;

or

2) ω_{1}, …, ω_{r} all factor through a morphism g : X^{s} → W with dim W ≤ r and g has connected fibres.

Case 1) in fact cannot happen (see 3.1). This can be seen as follows. The map Re h

Ṽ ⟶^{h} ℂ^{dim U_{2}} ⟶^{Re} ℝ^{dim U_{2}}

is surjective and has connected fibres. Since the image of the singular part h(S(uτ)) of uτ : Ṽ → Δ(G) is an analytic subvariety of ℂ^{dim U_{2}} of codimension at least 1 and any fibre of Re : ℂ^{dim U_{2}} → ℝ^{dim U_{2}} is a real vector space ℝ^{dim U_{2}} in ℂ^{dim U_{2}} after a translation on ℂ^{dim U_{2}}, the fibres of Re can not be contained in h(S(uτ)). Hence,

(Re H)^{−1}(x) ∩ S(uτ)

is a measure zero subset of (Re h)^{−1}(x). Since U_{2} comes from the complexified differential of uτ, d(Re h) and d(uτ) coincide of the regular part Ṽ_{r}. This implies that a fibre of Re h is a connected component of a fibre of uτ. Therefore, we have a factor map:

[figure: Ṽ; ℝ^{dim U_{2}}; Δ(G); Re h; ϕ; uτ]

The differential dϕ is equal to 1 with respect to some isometric coordinate systems on ℝ^{dim u_{2}} and Δ(G). Hence, ϕ is an isometric submersion and its image is a flat submanifold T × ℝ^{l} where T is a torus. Since it is the image of the pluriharmonic map uτ, if one direction is bounded, then this direction vanishes completly. Therefore, ϕ(ℝ^{dim U_{2}}) = ℝ^{dim U_{2}}, a flat in Δ(G). Because τ^{∗}ρ|_{V} fixes this flat, it cannot be Zariski dense. A contradiction.

We want to show that τ^{∗}ρ factors through g in Case 2). Pulling g back to the universal coverings, uτ factors through g̃,

[figure: X̃^{s}; W̃; Δ(G); g̃; v; uτ]

and the action of τ^{∗}ρ on uτ(X̃^{s}) factors through π_{1}(W). Let Δ_{uτ} be the minimal convex subcomplex containing uτ(X̃^{s}). Since any γ ∈ π_{1}(g^{−1}(ω)) goes to 1 ∈ π_{1}(W), τ^{∗}ρ(π_{1}(g^{−1}(ω))) fixes uτ(X̃^{s}) and hence Δ_{uτ}. Since uτ(X̃^{s}) is unbounded, Δ_{uτ} contains at least a geodesic line L, and τ^{∗}ρ(π_{1}(g^{−1}(ω))) fixes L. By Lemma 3.5, τρ(π_{1}(g^{−1}(ω))) cannot be Zariski dense. Since τ^{∗}ρ is Zariski dense, Lemma 3.4 says that τρ factors through g. We again take the Shafarevich map with respect to τ^{∗}ρ. It descends to an f : X → Y with dim Y ≤ dim W ≤ rk_{K}(G), and ρ factors through f.

Proof of Theorem 3.1. The algebraic group G ⊂ SL_{n} is in fact defined over a number field K after some conjugations. Hence, the morphism Rep(G) → M_{B}(G) from the space of representations of π_{1} into G to its moduli space is defined over Spec(O_{K}) by Seshadri’s geometric reductivity theorem over an arbitrary base. We take a completion K_{p} ⊃ K for a prime o, and let O_{K_{p}} be the ring of the algebraic integers of K_{p}. The O_{K_{p}}-valued points in M_{B}(G) correspond to the representations into G(O_{K_{p}}) after conjugations. If ϱ is not an isolated point in M(G), we may find an irreducible algebraic curve S in M(G) passing through ϱ and containing infinitely many nonintegral point {ϱ_{s}}_{s∈S_{0}}, valued in some finite extensions E_{s} ⊃ K_{p}. Since the subset of Zariski dense representations into a semisimple algebraic group is Zariski open, we may assume that all {ϱ_{s}}_{s∈S_{0}} are also Zariski dense.

Let u_{s} : X̃ → Δ(G(E_{s})) be the equivariant harmonic map for ϱ_{s}. Since rk G(E_{s}) ≤ rk G(K̄_{p}) where K̄_{p} is the Galois closure of K_{p}, the cardinal numbers of the Weyl groups W̄_{s} of G(E_{s}) are bounded, hence the cardinal numbers of the root systems of W̄_{s} are also bounded. This implies that the characteristic polynomials p_{s}(t) for u_{s} (see Lemma 3.1) have bounded degrees. So, we may assume that all p_{s}(t) for s ∈ S_{0} have the same degree d.

Let Γ := ⨁_{i=1}^{d} Γ(X, Sym^{i}Ω_{X}^{1}). Then the subset of all characteristic polynomials of degree d is an algebraic subvariety W ⊂ Γ (see [42] Section ”Hitchin proper map”). By taking the Zariski closure

V := {p_{s}(t)}_{s_{∈}S_{0}} ⊂ W,

we obtain an algebraic family of coverings {σ_{v} : X_{v}^{s} → X}_{v∈V} constructed in Lemma 3.1. Using Lemma 3.3 in the relative case, the constructions in the proof of Theorem 3.2 can be performed relative to V, in particular, the fibrations {f_{v}^{s} : X_{v}^{s} → Y_{v}^{s}}_{v∈V} constructed in the proof of Theorem 3.2 are relative to V. Clearly, we may find a Zariski open subset V_{0} ⊂ V so that the generic fibres of f_{v} for v ∈ V_{0} are homotopy equivalent. Furthermore, since {p_{s}(t)}_{s∈S_{0}} is Zariski dense, there are infinitely many s ∈ S_{0} such that p_{s}(t) ∈ V_{0}; we call this subset S′_{0}.

Fixing one fibre f_{v_{0}}^{s−1}, v_{0} ∈ V_{0}, then σ^{∗}ϱ_{s}|_{f_{v_{0}}^{s−1}} = 1 for s ∈ S′_{0} by a) and b) in Lemma 3.4. Since {f_{v}^{s} : X_{v}^{s} → Y_{v}^{s}}_{v∈V} is Gal({X_{v}^{s}/X}_{v∈V})- equivariant, it descends to a family of fibrations {f_{v} : X → Y_{v}}_{v∈V} with ϱ_{s}|_{f_{v_{0}}^{−1}} = 1 for s ∈ S′_{0}. Because the subset {ϱ_{s}|_{f_{v_{0}}^{−1}} = 1|s ∈ S} is closed in the Zariski topology on S and S is irreducible, this implies ϱ_{s}|_{f_{v_{0}}^{−1}} = 1, ∀s ∈ S. In particular, ϱ_{s}|_{f_{v_{0}}^{−1}} = 1. Finally, by c) in Lemma 3.4, ϱ factors through f_{v_{0}} after passing to a finite etale covering and a blowingup.

References

[1] S.I. Al’ber, On n-dimensional problems in the calculus of variations in the large, Soviet. Math. Dokl. 5 (1964) 700–704.

[2] , Spaces of mappings into a manifold with negative curvature, Soviet. Math. Dokl. 9 (1967) 6–9.

[3] A. Borel & J.P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436–491.

[4] K. Corlette, Flat G-bundles with canonical metrics, J. Differential Geom. 28 (1988) 361–382.

[5] , Archimedean superrigidity and hyperbolic geometry, Ann. of Math. 135 (1992) 165–182.

[6] M. Cornalba & P. Griffiths, Analytic cycles and vector bundles on non compact algebraic varieties, Invent. Math. 28 (1975) 1–106.

[7] J. Carlson & D. Toledo, Harmonic mappings of Kähler manifolds to locally symmetric spaces, Inst. Hautes Études Sci. Publ. Math. 69 (1989) 173–201.

[8] S. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987) 127–131.

[9] K. Diederich & T. Ohsawa, Harmonic mappings and disk bundles over compact Kähler manifolds, Publ. Res. Inst. Math. Sci. 21 (1985) 819–833.

[10] J. Eells & J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 85 (1964) 109–160.

[11] M. Gromov & R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992) 165–246.

[12] P. Hartman, On homotopic harmonic maps, Canad. J. Math. 19 (1967) 673–687.

[13] N. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126.

[14] S. Iitaka, Logarithmic forms on algebraic varieties, J. Fac. Sci., Univ. Tokyo, Sect. 1A Math., 23 (1976) 525–544.

[15] J. Jost, Equilibrium maps between metric spaces, Calculus Variations 2 (1994) 173–204.

[16] , Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comm. Math. Helv. 70 (1995) 659–673.

[17] , Riemannian geometry and geometric analysis, Springer, Berlin, 1995.

[18] , Generalized harmonic maps between metric spaces, (ed. J. Jost), Geometric Analysis and the Calculus of Variations for Stefan Hildebrandt, Internat. Press, 1996, 143–174.

[19] J. Jost & S. T. Yau, Harmonic mappings and Kähler manifolds, Math. Ann. 262 (1983) 145–166.

[20] , The strong rigidity of locally symmetric complex manifolds of rank one and finite volume, Math. Ann. 275 (1986) 291–304.

[21] , On the rigidity of certain discrete groups and algebraic varieties, Math. Ann. 278 (1987) 481–496.

[22] , Harmonic maps and group representations, Differential Geometry and Minimal Submanifolds, (B. Lawson and K. Tenenblat eds.), Longman Scientific, 1991, 241–260.

[23] , Harmonic maps and superrigidity, Proc. Sympos. Pure Math. 54 (1993) Part I, 245–280.

[24] J. Jost & K. Zuo, Harmonic maps and Sl(r, ℂ)-representations of fundamental groups of quasi projective manifolds, J. Algebraic Geom. 5 (1996) 77–106.

[25] , Harmonic maps into Tits buildings and factorization of non rigid and non arithmetic representations of π_{1} of algebraic varieties, Preprint, 1993.

[26] , Harmonic maps of infinite energy and rigidity results for quasiprojective varieties, Math. Res. Lett. 1 (1994) 631–638.

[27] L. Katzarkov, Factorization theorems for the representations of the fundamental groups of quasiprojective varieties and some applications, Preprint.

[28] N. Korevaar & R. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993) 561–659.

[29] J. Kollar, Shavarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993) 165–215.

[30] F. Labourie, Existence d’applications harmoniques tordues à valeurs dans les variétés à courbure négative, Proc. Amer. Math. Soc. 111 (1991) 877–882.

[31] J. Y. Li, Hitchin’s self-duality equations on complete Riemannian manifolds, Math. Ann., to appear.

[32] J. Lohkamp, An existence theorem for harmonic maps, Manuscripta Math. 67 (1990) 21–23.

[33] G. A. Margulis, Discrete groups of motion of manifolds of nonpositive curvature, Trans. Amer. Math. Soc. 190 (1977) 33–45.

[34] , Discrete subgroups of semisimple Lie groups, Springer, Berlin, 1991.

[35] J. Morgan, Γ-trees and their applications, Bull. Amer. Math. Soc. 26 (1992) 87–112.

[36] G. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Stud. 78, Princeton, 1973.

[37] N. Mok, Y.-T. Siu & S.-K. Yeung, Geometric superrigidity, Invent. Math. 113 (1993) 57–84.

[38] G. Prasad, Strong rigidity of Q-rank 1 lattices, Invent. Math. 21 (1973) 255–286.

[39] H. Royden, The Ahlfors-Schwarz lemma in several complex variables, Comment. Math. Helv. 55 (1980) 547–558.

[40] C. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990) 713–769.

[41] , A Lefschetz theorem for π_{0} of the integral leaves of a holomorphic oneform, Compositio Math. 87 (1993) 99–113.

[42] , Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992) 5–95.

[43] , Moduli of representation of the fundamental group of a smooth projective variety I and II, Inst. Hautes Études Sci. Publ. Math.

[44] J. Sampson, Applications of harmonic maps to Kähler geometry, Contemp. Math. 49 (1986) 125–134.

[45] R. Schoen, Thesis, Stanford, 1978.

[46] Y. T. Siu, The complex analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980) 73–111.

[47] , Strong rigidity of compact quotients of exceptional symmetric domains, Duke Math. J. 48 (1981) 857–871.

[48] M Wolf, Infinite energy maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991) 487–539.

[49] S. T. Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978) 197–203.

[50] K. Zuo, Some structure theorems for semi-simple representations of π_{1} of algebraic manifolds, Math. Ann. 295 (1993) 365–382.

[51] , Factorization of nonrigid Zariski dense representation of π_{1} of projective algebraic manifolds, Invent. Math. 118 (1994) 37–46.

Max-Planck-Institute for Mathematics in the Sciences University Kaiserslautern, Germany