The fundamental group of an extension in a Tannakian category and the unipotent radical of the Mumford–Tate group of an open curve

Payman Eskandari and V. Kumar Murty

Vol. 325 (2023), No. 2, 255–279

DOI: 10.2140/pjm.2023.325.255

Abstract

In the first part, we give a self-contained account of Tannakian fundamental groups of extensions, generalizing a result of Hardouin (2008; 2011). In the second part, we use Hardouin’s characterization of Tannakian groups of extensions to give a characterization of the unipotent radical of the Mumford–Tate group of an open complex curve. Consequently, we prove a formula that relates the dimension of the unipotent radical of the Mumford–Tate group of an open complex curve $X\setminus S$ with $X$ smooth and projective and $S$ a finite set of points to the rank of the subgroup of the Jacobian of $X$ supported on $S$.

Keywords

Tannakian categories, Mumford–Tate groups, mixed Hodge structures

Mathematical Subject Classification

Primary: 11G30, 14C30

Secondary: 18D30

Milestones

Received: 20 August 2021

Revised: 23 March 2023

Accepted: 22 April 2023

Published: 3 November 2023