Arakelov inequalities in higher dimensions

TL;DR

This paper introduces a Hodge theoretic invariant for higher-dimensional families of projective manifolds, generalizing classical Arakelov inequalities and proving boundedness for families with ample canonical bundles.

Contribution

It develops a new invariant that measures the failure of Arakelov inequalities in higher dimensions and proves its boundedness for certain families, answering a longstanding question.

Findings

Invariant is uniformly bounded for families with ample canonical bundle

Higher-dimensional families satisfy the generalized Arakelov inequality

Answers a question posed by Viehweg regarding these inequalities

Abstract

We develop a Hodge theoretic invariant for families of projective manifolds that measures the potential failure of an Arakelov-type inequality in higher dimensions, one that naturally generalizes the classical Arakelov inequality over regular quasi-projective curves. We show that for families of manifolds with ample canonical bundle this invariant is uniformly bounded. As a consequence we establish that such families over a base of arbitrary dimension verify the aforementioned Arakelov inequality, answering a question of Viehweg.