Equivalent definitions of rigid local systems

Question

There are (at least) two definitions for rigidity of a local system on $X = \mathbb{P}^1\setminus\{p_1,\dots,p_n\}$:

1. A local system $L$ on $X$ is rigid if any other local system with conjugate monodromy around each of the $p_i$ is isomorphic to $L$
2. A local system $L$ of rank $n$ on $X$ is rigid its preimage under the map $Hom(\pi_1(X),GL(n))\to M(X,n)$ is open.

In the literature people seem to treat these definitions as being equivalent, but I don't see how they could be. For example rank $1$ local systems are clearly always rigid in the first sense. But the map $Hom(\pi_1(X),GL(1))\to M(X,1)$ is an isomorphism, so the preimage of a (closed) point will never be open.

So what is the relation between these two notions of ridigity?

Answer 1

To avoid notational overlap, let $X = \mathbb P^1 \setminus \{p_1,\dots,p_m\}$.

The two definitions are equivalent for irreducible local systems if we restrict 2 to the subset of $M(X,n)$ with fixed local monodromy at the points $p_1,\dots,p_m$. (This uses the fact that we are working in $GL_n$, otherwise the first condition may be stronger.)

As far as I know, anyone dropping the condition on the local monodromy in the second definition is either working with proper varieties (e.g. Carlos Simpson in his paper Higgs bundles and local systems) or is making a mistake, most likely caused by copying from a paper working in the proper case and not making the appropriate adjustments.

The second definition is absurd in the setting of $\mathbb P^1 - \{p_1,\dots,p_m\}$ as it is easy to see there are no such representations except when $m=1$, in which case all representations are trivial.