I am reading Katz’s Rigid Local Systems ⟦cite:Kat96⟧ . Here are my notes. One starts complex analytically with $U=\mathbb P^1-\{m\ \textrm{pts}\}$. Riemann introduced local systems on $U$, i.e. a locally constant sheaf of complex vector spaces, to study the solutions to $n$-th order linear differential equations with singularities at $\mathbb P^1-U$ ⟦cite:Rie57⟧ . If there are no “accessory parameters”, we say this local system is rigid, i.e. determiend by its local monodromies.

Generalities Concerning Rigid Local Systems over $\mathbb C$

Let $X$ be a smooth projective connected curve over $\mathbb C$ of genus $g$. Let $S\subseteq X(\mathbb C)$ a nonempty finite subset and set $U:=X-S$ the open complement. For each base point $u$ on the complex manifold $U^{\mathrm{an}}$, there is an equivalence of symmetric monoidal categories induced by the fibre functor $\mathcal F\mapsto \mathcal F_u$ where $\mathcal F$ is a local system on $U$.

All vector spaces and representations here are assumed finite dimensional. We say the local system is irreducible if the corresponding representation is. For every singular point $s\in S$, the punctured neighborhood

has fundamental group $I(s)=\pi_1(D^*(s),u)=\mathbb Z$, where $u$ is any base point.

However, physically rigid is a notion that only works for genus $0$. If $g(X)\ge 1$, then no local system on $U^{\mathrm{an}}$ is physically rigid. Let $\mathcal L$ be a rank $1$ local system on $X^{\mathrm{an}}$ (they correspond to characters of $\pi_1(X^{\mathrm{an}})^{\mathrm{ab}}\cong \mathbb Z^{2g}$) no tensor power of which is trivial. Let $j:U^{\mathrm{an}}\to X^{\mathrm{an}}$ be the inclusion. The pushforward $j_*:\pi_1(U^{\mathrm{an}},u)\rightarrow \pi_1(X^{\mathrm{an}},u)$ is surjective, so no tensor power of $j^*\mathcal L$ is trivial, and it has trivial local monodromy. Thus $\mathcal F$ and $\mathcal F\otimes j^*\mathcal L$ have isomorphic local monodromy for any $\mathcal F\in\mathrm{LocSys}_{\mathbb C}(U^{\mathrm{an}})$. Suppose $\mathcal F\cong \mathcal F\otimes j^*\mathcal L$ then $\mathrm{det}(\mathcal F)\cong \mathrm{det}(\mathcal F\otimes j^*\mathcal L)\cong \mathrm{det}(\mathcal F)\otimes (j^*\mathcal L)^{\otimes\mathrm{rank}(\mathcal F)}$. By tensoring both sides with $\mathrm{det}(\mathcal F)^\lor$, we have $(j^*\mathcal L)^{\otimes\mathrm{rank}(\mathcal F)}=\underline{\mathbb C}_U$, which is a contradiction. Hence $\mathcal F=0$. Hence no nonzero local system is physically rigid when $g\ge 1$.

Numerical Criterion for $\mathbb P^1$

With same setting above and let $X=\mathbb P^1$.

References