Arakelov-type inequalities for Hodge bundles

TL;DR

This paper generalizes classical Arakelov inequalities to the degrees of Hodge bundles in families of curves, under unipotent monodromy assumptions, extending the understanding of their geometric properties.

Contribution

It provides a new proof of generalized Arakelov inequalities for Hodge bundles in the context of families of curves with unipotent monodromy.

Findings

Proved inequalities for degrees of Hodge bundles in specific geometric settings

Extended classical Arakelov inequalities to broader Hodge bundle contexts

Clarified conditions under which these inequalities hold

Abstract

We give a proof of generalizations of the classical Arakelov inequality valid for the degree $d$ of the relative canoincal bundle of a family of curves of genus $g$ over a complete curve of genus $p$ under the assumption that the monodromy around the singular fibers is unipotent. This relative canonical bundle is the (canonical extension of) the Hodge bundle and the inequality is generalized to the degrees of the Hodge bundles of a complex variation of Hodge structures.