Universiteit Leiden

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Dissertation

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

The main object of study in this thesis is an Arakelov inequalitywhich bounds the degree of an invertible subsheaf of the direct image ofthe pluricanonical relative sheaf of a semistable family of curves.

Author
Damjanovic, N.
Date
14 June 2018
Links
Thesis in Leiden Repository

The main object of study in this thesis is an Arakelov inequalitywhich bounds the degree of an invertible subsheaf of the direct image ofthe pluricanonical relative sheaf of a semistable family of curves. A naturalproblem that arises is the characterization of those families for which the equalityis 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 examplesby proving that the direct image of the bicanonical relative sheaf ofa semistable family of curves uniformized by the unit ball, all whose singularfibers are totally geodesic, contains an invertible subsheaf which satisfiesArakelov equality.

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