Hodge-Arakelov inequalities for family of surfaces fibered by curves

TL;DR

This paper explores Arakelov inequalities for families of surfaces fibered by curves, using Hodge theory and fibrations to understand the relationships between Hodge invariants, degrees of Hodge bundles, and their correction terms.

Contribution

It introduces new methods to analyze Arakelov inequalities in surface families using Hodge identities and fibrations, aiming to establish Arakelov equalities.

Findings

Derived identities relating Hodge numbers and degrees of Hodge bundles.

Compared Fujita decomposition of Hodge bundles in different fibrations.

Proposed relations between degrees of Hodge bundles in two related families.

Abstract

The Hodge numerical invariants of a variation of Hodge structure over a smooth quas--projective variety are a measure of complexity for the global twisting of the limit mixed Hodge structure when it degenerates. These invariants appear in inequalities which they may have correction terms, called Arakelov inequalities. One may investigate the correction term to make them into equalities, also called Arakelov equalities. We investigate numerical Arakelov type (in)equilities for a family of surfaces fibered by curves. Our method uses Arakelov identities in a weight 1 and also in a weight 2 variations of Hodge structure (cf. \cite{GGK}), in a commutative triangle of fibrations. We have proposed to relate the degrees of the Hodge bundles in the two families. We also compare the Fujita decomposition of Hodge bundles in these fibrations. We examine various identities and relations between Hodge numbers and degrees of the Hodge bundles in different levels.