Resumen de Weight decompositions on étale fundamental groups

J. P. Pridham

• Let $X_0$ be a smooth or proper variety defined over a finite field $k$, and let $X$ be the base extension of $X_0$ to the algebraic closure $\bar{k}$. The geometric \'etale fundamental group $\pi_1(X,\bar{x})$ of $X$ is a normal subgroup of the Weil group, so conjugation gives it a Weil action. For $l$ not dividing the characteristic of $k$, we consider the pro-${\Bbb Q}_l$-algebraic completion of $\pi_1(X,\bar{x})$ as a nonabelian Weil representation. Lafforgue's Theorem and Deligne's Weil II theorems imply that this affine group scheme is mixed, in the sense that its structure sheaf is a mixed Weil representation. When $X$ is smooth, weight restrictions apply, affecting the possibilities for the structure of this group. This gives new examples of groups which cannot arise as \'etale fundamental groups of smooth varieties.