In our previous paper (Commun. Algebra, 45(8)
(2017) 3422–3448), we studied the category of semifinite bundles on a proper variety defined over a field of characteristic 0. As a result, we obtained the fact that for a smooth projective curve defined over an algebraically closed field of characteristic 0 with genus \(g>1\), Nori fundamental group acts faithfully on the unipotent fundamental group of its universal covering. However, it was not mentioned about any explicit module structure. In this paper, we prove that the Chevalley–Weil formula gives a description of it.
Fundamental group schemes vector bundles Tannaka duality
Mathematics Subject Classification
14L15 14H30 14H60
This is a preview of subscription content, log in to check access.
The author would like to thank Professor Takao Yamazaki for many discussions clarifying the relation between his previous work and the Chevalley–Weil formula. The author also thanks Professor Takuya Yamauchi for suggesting many examples of projective smooth higher dimensional varieties with infinitely abelian fundamental group. The author is supported by JSPS, Grant-in-Aid for Scientific Research for JSPS fellows (16J02171).
Chevalley C, Weil A and Hecke E, Über das verhalten der integrale 1, gattung bei automorphismen des funktionenkörpers, Abh. Math. Sem. Univ. Hamburg, 10(1) (1934) 358–361CrossRefMATHGoogle Scholar
Deligne P and Milne J, Tannakian Categories, Lectures Notes in Mathematics 900 (1982) (Berlin-New York: Springer-Verlag)MATHGoogle Scholar
Grothendieck A, Représentations linéaires et compactification profinie des groupes discrets, Manuscr. Math., 2 (1970) 375–396CrossRefMATHGoogle Scholar
Grothendieck A, Revêtements étales et groupe fondamental, SGA1, Lecture Notes in Mathematics 224 (1971) (Berlin-New York: Springer-Verlag)Google Scholar