Arakelov inequalities and semistable families of curves uniformized by the unit ball

Abstract

The main object of study in this thesis is an Arakelov inequality

which bounds the degree of an invertible subsheaf of the direct image of

the pluricanonical relative sheaf of a semistable family of curves. A natural problem that arises is the characterization of those families for which the equality is satisfied in that Arakelov inequality, i.e. the case of Arakelov equality. Few examples of such families are known. In this thesis we provide some examples by proving that the direct image of the bicanonical relative sheaf of a semistable family of curves uniformized by the unit ball, all whose singular fibers are totally geodesic, contains an invertible subsheaf which satisfies Arakelov equality.