Varieties with prescribed fundamental group schemes

We study the realization problem for Tannakian fundamental group schemes: given an affine $k$-group scheme $G$, when does there exist a smooth projective connected pointed $k$-variety $(X,x)$ whose fundamental group scheme is isomorphic to $G$? We establish exact sequences for fundamental group schemes associated with principal bundles, and combine them with a Godeaux–Serre construction and Lefschetz-type theorems. As applications, we show that every finite étale $k$-group scheme is realized as the $S$-, Nori, and extended Nori fundamental group scheme of a smooth projective variety, while every finite constant group scheme is realized as the $F$- and étale variants.11