The fundamental group of an extension in aTannakian category and the unipotent radical of the Mumford–Tate group of an opencurve

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 \ 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.