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arXiv · 2608.09757

Varieties with prescribed fundamental group schemes

Lingguang Li·Hao Wang
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Abstract

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 \'etale $k$-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 \'etale variants.11

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Lingguang Li, Hao Wang. 2026-08-10. Varieties with prescribed fundamental group schemes. https://arxiv.org/abs/2608.09757

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