Rigidity of maximal holomorphic representations of Kähler groups

In this paper I study the representations of fundamental groups of compact Kähler manifolds $X$, and in particular their Toledo invariant $\tau(\rho)$. Adapting some techniques of Koziarz and Maubon, see [1], I give a bound for the Toledo invariant under the (strong) additional hypothesis that the representation admit a holomorphic equivariant map. In particular, this proves that in some sense if the Kähler manifold is not a ball quotient, then such representations must "lose" some of their Toledo invariant, due to the metric having non-constant sectional curvature. On the other hand, if the Toledo invariant is maximal, then the Kähler manifold is a ball quotient, and the representation is rigid.

Since this paper was written, two major papers came out that have almost settled the case where, without any holomorphicity assumption, one already knows that the Kähler manifold $X$ is a ball quotient: See Pozzetti [2] where, so far, some hypothesis on the Zariski closure of the representation is needed (but $X$ may be non-compact) and Koziarz-Maubon [3] where that hypothesis does not appear, but $X$ has to be compact.

Abstract: We investigate representations of Kähler groups $\Gamma = \pi_1(X)$ to a semisimple non-compact Hermitian Lie group $G$ that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor-Wood inequality similar to those found by Burger-Iozzi and Koziarz-Maubon. Thanks to the study of the case of equality in Royden's version of the Ahlfors-Schwarz Lemma, we can completely describe the case of maximal holomorphic representations. If $\dim_{\mathbb{C}}X \geq 2$, these appear if and only if $X$ is a ball quotient, and essentially reduce to the diagonal embedding $\Gamma < \text{SU}(n,1) \to \text{SU}(nq,q) \hookrightarrow \text{SU}(p,q)$. If $X$ is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, that thus appear as preferred elements of the respective maximal connected components.

Link:The preprint can be found on the arXiv.

[1] V. Koziarz, J. Maubon: Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type.
[2] M. B. Pozzetti: Maximal representations of complex hyperbolic lattices in SU(m,n).
[3] V. Koziarz, J. Maubon: Maximal representations of uniform complex hyperbolic lattices.