In the literature there are several definitions of "rigidity" (or "super-rigidity") of representations, adapted to the circumastances. I wonder what are the relations between them; I excuse in advance if the answers are known, after some research I could not find them. I have singled out a few definitions for concreteness.
Notation first: let $\Gamma = \pi_1(X, x_0)$ where $X$ is Kähler, and $G$ a (semi-)simple connected algebraic group.
- (classic rigidity): Every $\phi$ close to $\rho$ (in the analytic topology of $\text{Hom}(\Gamma, G)$) is conjugated to $\rho$ by an element of $G$.
- (super-rigidity, e.g. Margulis): The harmonic $\rho$-equivariant map $f \colon \tilde{X} \to G/K$ is totally geodesic.
- (Hermitian super-rigidity, e.g. Siu): Here we suppose $G/K$ is Hermitian symmetric. Then $f$ as above is holomorphic.
- (from Kim-Pansu, Duke Math. J., to appear): (Here we may need to suppose $X$ to be a surface of high enough genus) $\rho$ is rigid if every $\phi$ close enough to $\rho$ is not smooth in $\text{Hom}(\Gamma, G)$ (equivalently, under the assumption on $X$: not Zariski-dense).
The last one is a bit "ad hoc" to their situation, since it allows them to prove that $\text{Hom}(\Gamma, G)$ splits in connected components, either entirely rigid or where Zariski dense points form a dense subset.
I think that 3. has been introduced as an intermediate step to 2. (when $\tilde{X}$ is a Hermitian symmetric space), but I do not know how general this is. To me, the only clear implication is 1. $\implies$ 4. (if $\phi_n \to \rho$ are smooth, then $\dim T_{\phi_n}\text{Hom}(\Gamma, G) = vdim = (1-\chi(X)) \dim(G)$ but if a neighborhood of $\phi_n$ is conjugated to $\rho$ then $\dim T_{\phi_n}\text{Hom}(\Gamma, G) \leq \dim G$).
I also think that despite the "super-" name there is a good chance that 1. $\implies$ 3. (at least if $X$ is a surface); so, main question:
Is there any implication between 1. and 3.? Or counter-example?
More with general-knowledge purposes:
Is there any other general implication / counterexample of implications between the points above? For example, how generally 3. $\implies$ 2.? Is 4. much weaker then any other one?
Of course, I would also be happy with answers for $\Gamma$ a cocompact lattice in $H$ and $\tilde{X} = H/K'$, or even just $X$ a Riemann surface.
