Let $X$ be a complex algebraic variety, and let $F=(F\_n)$ be a lisse $\mathbb Z\_{\ell}$-sheaf on $X.$ Does there exist an analytic open covering of $X(\mathbb C),$ such that $F$ is "(locally) constant" on each open subset? 



lisse sheaf on complex varieties
Dear Shenghao, If you really do mean a lisse sheaf on the etale site of $X$, then it doesn't make sense a priori to evaluate it on analytic open subsets of $X(\mathbb C)$, since these are not in the etale site of the algebraic variety $X$. However, $F$ corresponds to a representation of the (profinite) etale $\pi\_1$ of $X$, which in turn is the profinite completion of the
topological $\pi\_1$ of $X(\mathbb C)$. So there is a corresponding locally constant (in the analytic topology) $\mathbb Z\_{\ell}$
sheaf on $X(\mathbb C)$, which, being locally constant, will be constant on sufficiently small analytic open subsets.


