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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?

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  • $\begingroup$ The correct definition is that $L$ corresponds to an isolated point of $M(\pi_1(X),\operatorname{GL}_n)$, which is equivalent to your second conditions because GIT quotients are submersive. Where did you see the first definition? $\endgroup$ – R. van Dobben de Bruyn Apr 18 '18 at 1:43
  • $\begingroup$ @R.vanDobbendeBruyn The first definition is the one from Katz's book. People ofter state the second definition, or the equivalent one you mentioned, and then cite Katz's book, indicating to me that results about the first definition should carry over to the second? $\endgroup$ – user2520938 Apr 18 '18 at 6:43
  • $\begingroup$ @R.vanDobbendeBruyn It's still true that with the second definition there are no rigid rank $1$ local systems right? Because the quotient map is an isomorphism $\endgroup$ – user2520938 Apr 18 '18 at 8:38
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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.

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