Symmetric differentials, Kähler groups and ball quotients

Abstract

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients:

• Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kähler manifold X are controlled by the global symmetric differentials on X.

• An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).

Appendix: Proof of Theorem 1.6

1.1 A.1 Replacing GL(r) by any reductive group

Before proving Theorem 1.6 notice that in this theorem one can replace GL(r) by any reductive F-algebraic group G if one replaces the number r in the vanishing assumption as follows. Let T be a maximal F-split torus of G, denote by X ^∗(T) the free ℤ-module of characters of T,by $\mathfrak {h}^{*}:= X^{*}(\mathbf {T}) \otimes_{{\mathbb{Z}}}{\mathbb{R}}$ the real vector space generated by X ^∗(T), by Σ⊂X ^∗(T) the root system of T in G, W its Weyl group. The group W acts naturally on $\mathfrak {h}^{*}$ hence also on its symmetric algebra $S^{\bullet} \mathfrak {h}^{*}$. As W is a reflection group the ring of invariants $(S^{\bullet} \mathfrak {h}^{*})^{W}$ is a polynomial ring. The generators of this ring are not canonical but their degrees are. We denote by i _G the maximal degree of such generators. In the general version of Theorem 1.6 one requires that $H^{0}(X, S^{i} \varOmega^{1}_{X}) = 0$ for 0<i≤i _G .

In the case G=GL(r), r≥2, the group W is the symmetric group S _r acting on $\mathfrak {h}^{*} \simeq {\mathbb{R}}^{r}$ by permutation of the coordinates. The ring $(S^{\bullet} \mathfrak {h}^{*})^{W}$ is generated by the elementary symmetric polynomials on $\mathfrak {h}$ and i _GL(r)=r. Other examples are i _Sp(r)=2r−2, i _SO(2r)=2r. A list of the i _G ’s for G simple is provided in [20, p. 59].

The reader will easily adapt the proof we give for GL(r) to the general case.

1.2 A.2 The Archimedean case

In this section we give a proof of Theorem 1.6(i), which generalizes to the Kähler case the result of Arapura [1, Prop. 2.4] for smooth projective varieties. The proof of Arapura does not generalize directly: as far as I know the construction of the moduli space M _Dol(X,GL(r,ℂ)) of GL(r,ℂ)-Higgs bundles of semi-harmonic type on X exists in the literature only for X smooth projective.

We denote by M _Betti(Γ,GL(r))(ℂ)=Hom(Γ,GL(r,ℂ))//GL(r,ℂ) the moduli space of representations of Γ=π ₁(X) in GL(r,ℂ). We want to show that this space is a collection of points under our assumption $H^{0}(X, S^{i} \varOmega^{1}_{X})=0$, 0<i≤r. We use non-Abelian Hodge theory as developed by Hitchin (cf. [18, 19]) and Simpson (cf. [38–41]).

Let ρ:Γ⟶GL(r,ℂ) be a reductive representation and $(\mathcal{F}, D)$ the associated flat ℂ^r-bundle on X. Here $\mathcal{F}$ denotes the ℂ^r-bundle $\tilde{X} \times_{\varGamma,\rho} {\mathbb{C}}^{r}$ with flat connection $D \in A^{1}(\operatorname {End}\mathcal{F})$. As X is compact Kähler and ρ is reductive there exists an essentially unique ρ-equivariant harmonic map $f: \tilde{X} \longrightarrow \mathbf{GL}(r, {\mathbb{C}})/U(r)$ (where GL(r,ℂ)/U(r) denotes the symmetric space of Hermitian metrics on ℂ^r). The map f defines a metric K on $\mathcal{F}$ called the harmonic metric (cf. [39, p. 16]). The pair $(\mathcal{F}, K)$ is called a harmonic bundle. The flat connexion D decomposes uniquely as D=∇+α, where ∇ is compatible with K and $\alpha\in A^{1}(X, \operatorname {End}\mathcal{F})$ is K-selfadjoint. Decompose furthermore using types:

$$ \nabla= \partial_K + \overline{\partial}, \qquad \alpha= \theta+ \theta^*, $$

where ∂ _K is of type (1,0), $\overline{\partial}$ is of type (0,1), $\theta\in A^{1,0}(\operatorname {End}\mathcal{F})$ and $\theta^{*} \in A^{0,1}(\operatorname {End}\mathcal{F})$ is the adjoint of θ with respect to K. Define D′=∂ _K +θ ^∗, $D'' = \overline{\partial}+ \theta$, thus D=D′+D″. As D is flat and the map f is harmonic, (D″)²=0, that is: $\overline{\partial}^{2} =\overline{\partial}(\theta) = [\theta, \theta]=0$. One says that the pair $(F = (\mathcal{F}, \overline{\partial}) , \theta)$ consisting of the holomorphic bundle F and the holomorphic one-form $\theta\in\varOmega^{1}(\operatorname {End}F)$ is a Higgs bundle.

The Hitchin map associates to the harmonic bundle $(\mathcal{F}, K)$ the element of $\bigoplus_{i=1}^{r} H^{0}(X, S^{i}\varOmega^{1}_{X})$ consisting of the coefficients of the characteristic polynomial $\det(T\cdot \operatorname {Id}- \theta)$. By our assumption the space $\bigoplus_{i=1}^{r} H^{0}(X, S^{i}\varOmega^{1}_{X})$ is trivial. It follows from the compactness property of the Hitchin map [39, Prop. 2.8] that the space of harmonic bundles on X is compact. As the map associating to an harmonic bundle $(\mathcal{F}, K)$ its monodromy ρ is continuous it follows that M _Betti(Γ,GL(r))(ℂ) is compact. However the algebraic variety M _Betti(Γ,GL(r)) is affine. Hence it has to be 0-dimensional. This concludes the proof of Theorem 1.6(i).

1.3 A.3 The non-Archimedean case

Let us prove Theorem 1.6(ii). We refer the reader to [22, 23, 47] for details on harmonic maps, buildings and spectral cover.

1.3.1 A.3.1 The reductive case

Let ρ:Γ⟶GL(r,F) be a reductive representation with unbounded image in GL(r,F). Let $\mathcal{BT}$ be the Bruhat–Tits building of GL(r,F). Following [14, Theorem 7.1, Lemma 8.1] and [14, Theorem 7.3]) there exists a Lipschitz pluriharmonic ρ-equivariant map $f : \tilde{X} \longrightarrow \mathcal{BT}$ of finite energy. Recall that a point x of $\tilde{X}$ is said regular for f if there exists an apartment F of $\mathcal{BT}$ containing the image by f of a neighbourhood of x [14, p. 225]. Otherwise the point x is called singular. Denote by R(X,ρ)⊂X the open set image in X of the set of regular points for f and by S(X,ρ) its complement. By [14, Theorem 6.4] the subspace S(X,ρ) of X has Hausdorff codimension at least 2 (in fact it is complex algebraic).

With the notations of Sect. A.1 let $\varSigma= \{ \beta_{1}, \ldots,\beta_{l}\} \subset \mathfrak {h}^{*}$ be the root system of T in GL(r,F). For each apartment $A \simeq \mathfrak {h}$ of $\mathcal {BT}$ we can consider the real 1-forms {dβ ₁,…,dβ _l } _A on A. If U⊂X is an open set satisfying f(U)⊂A the forms (f ^∗(dβ _i )_ℂ)^(1,0), 1≤i≤l, are holomorphic 1-forms on U as f is pluriharmonic. On the intersection of two apartments A and A′, the sets {dβ ₁,…,dβ _l } _A and {dβ ₁,…,dβ _l } _A′ coïncide up to permutation by an element of W (for its natural action on Σ). In particular there is a canonical map

$$h_\rho: \bigl(S^\bullet \mathfrak {h}_{\mathbb{C}}^* \bigr)^W \longrightarrow H^0\bigl(R(X, \rho), S^\bullet \varOmega^1_{R(X, \rho)}\bigr). $$

As the map f is Lipschitz the symmetric differential $h_{\rho}(P) \in H^{0}(R(X, \rho),\allowbreak S^{\bullet}\varOmega^{1}_{R(X, \rho)})$ is bounded for any $P \in(S^{\bullet} \mathfrak {h}_{{\mathbb{C}}}^{*})^{W}$. As the singular set $S(\tilde{X}, \rho)$ has Hausdorff codimension at least 2 the symmetric differential h _ρ (P) uniquely extends to X. Finally we obtain a canonical map

$$h(\rho): \bigl(S^\bullet \mathfrak {h}_{\mathbb{C}}^*\bigr)^W \longrightarrow H^0\bigl(X, S^\bullet \varOmega^1_{X} \bigr). $$

The ring $(S^{\bullet} \mathfrak {h}_{{\mathbb{C}}}^{*})^{W}$ is a polynomial ring generated in degree less or equal to r. Our assumption $H^{0}(X, S^{i}\varOmega^{1}_{X})=0$, 0<i≤r, implies that the image of h _ρ vanishes. This forces all the local (f ^∗(dβ _i )_ℂ)^(1,0), 1≤i≤l, to vanish. Thus $f: \tilde{X} \longrightarrow \mathcal{BT}$ is constant. But then the group ρ(Γ) fixes the point $f(\tilde{X})$ in $\mathcal{BT}$ hence is bounded in GL(r,F).

This finishes the proof of Theorem 1.6(ii), for reductive representations.

1.3.2 A.3.2 The unipotent case

Suppose that the Zariski closure of ρ(Γ) is a unipotent subgroup U of GL(r)_/F . I claim that in this case U is finite. Indeed let U⊃U ₁⊃⋯⊃U _l be the central descending series for U. Each A _i :=U _i /U _i+1 is of additive type.

As $H^{0}(X, \varOmega^{1}_{X})=0$ one also has H ¹(Γ,ℂ)=H ¹(X,ℂ)=0 by Hodge theory. Hence any character of Γ has finite image. In particular the image of $\varGamma\stackrel{\rho}{ \longrightarrow } \mathbf{U}(F) \longrightarrow (\mathbf{U}/\mathbf{U}_{1})(F)$ is finite. By passing to a finite index subgroup of Γ one can assume that ρ(Γ)⊂U ₁(F). Considering the image to the successive quotients A _i our claim follows.

1.3.3 A.3.3 The general case in characteristic zero

Let ρ:Γ⟶GL(r,F) be any representation. Let G=R⋉U be the Levi decomposition of the Zariski-closure G of ρ(Γ) in GL(r,F), where U denotes the unipotent radical of G and R a Levi factor lifting the reductive quotient G/U (this Levi decomposition exists as F is perfect). Let l:G⟶R be the canonical quotient map. As ρ has Zariski-dense image in G the representation l∘ρ has Zariski-dense image in the reductive group R⊂GL(r). The reductive case implies that the image (l∘ρ)(Γ) is contained in a maximal compact subgroup K of R(F).

The exact sequence of groups $1 \longrightarrow \mathbf{U}(F) \longrightarrow \mathbf{G}(F) \stackrel {l}{\longrightarrow } \mathbf{R}(F) \longrightarrow 1$ induces by restriction an exact sequence of groups 1⟶U(F)⟶l ⁻¹(K)⟶K⟶1. As K is compact and F has characteristic zero this sequence splits: l ⁻¹(K)=U(F)×K. To show that the subgroup ρ(Γ)⊂l ⁻¹(K) has bounded image in GL(r,F) it is thus enough to show that the projection of ρ(Γ) on U(F) has bounded image. This follows from the unipotent case.