Let $C$ be a compact Riemann surface, and let $U$ be a Zariski open subset in $C.$ Let $L$ be a local system (with coefficients $\mathbb C$ or $\mathbb Q_{\ell}$) on $U.$ For each point $z_i\in C-U,$ let $M_i$ be the monodromy matrix of $L$ around $z_i.$ If we identify $L$ as a representation of $\pi_1(U),$ and let $\gamma_i$ be a small loop in $U$ around $z_i,$ then $M_i$ is the image of $\gamma_i$ under the representation $\rho_L:\pi_1(U,a)\to GL(L_a)$ (where $a$ is a base point and $L_a$ is the fiber of $L$ at $a$).
Here's my question.
It seems to me that there should be some relation between (the traces of) these monodromies (as well as the monodromies around loops in $\pi_1(C)$) and (Betti numbers of) the cohomology groups $H^i_c(U,L)$ of $L$ with compact support, but I don't know the precise formulation or reference.
As an example, if $C$ has genus $g$ and $L=\mathbb C$ is constant of rank 1, then $H^1_c(U,\mathbb C)$ has dimension $2g+n-1,$ where $n$ is the number of points at infinity. This number is the rank of $\pi_1(U),$ and each “canonical generator" of $\pi_1(U)$ has trace 1.
Best Answer
Suppose that $C = \mathbb P^1$ and $U = \mathbb P^1\setminus \{0,\infty\}.$ Then $\pi_1(U)$ is cyclic, freely generated by a loop around $0$.
The local system $L$ is thus given by the vector space $L_a$, equipped with an invertible operator, call it $T$, corresponding to the action of the generator of $\pi_1(U)$. This operator $T$ is the monodromy matrix.
Now $H^1_c(U,L) = $ the space of $T$-invariants of $L_a$, while $H^2_c(U,L) = $ the space of $T$-coinvariants of $L_a$. If you think of the possible Jordan decompositions of $T$, and the fact that the trace is insensitive to the unipotent part, but just depends on the semi-simple part, you'll see that the it's going to be hard to find any interesting relation of the type that you want. (E.g. if $L_a$ is $n$-dimensional, and $T$ acts by the identity, or by a maximally non-trivial unipotent element, in both cases the trace of $T$ is equal to $n$, but in the first case the Betti numbers are also both equal to $n$, while in the second, they are both equal to $1$.)
You might also wonder about the Euler characteristic $H^2_c(U,L) - H^1_c(U,L)$, but this is always equal to (rank $L$) $\cdot \chi(U)$, and so is insensitive to the monodromy matrices.