Torus bundles and 2-forms on the universal family of Riemann surfaces

TL;DR

This paper explores the relationships between natural cohomology classes, 2-forms, and invariants on the universal family of Riemann surfaces, revisiting Morita's results and connecting them with Arakelov geometry and Kawazumi's invariant.

Contribution

It provides new proofs of Kawazumi's results on the second variation of a_g and compares different natural 2-forms representing key cohomology classes.

Findings

Alternative proofs of Kawazumi's second variation results

Connections established between a_g, Faltings's delta-invariant, and Hain-Reed's beta-invariant

Comparison of 2-forms induced by different metrics on the universal family

Abstract

We revisit three results due to Morita expressing certain natural integral cohomology classes on the universal family of Riemann surfaces C_g, coming from the parallel symplectic form on the universal jacobian, in terms of the Miller-Morita-Mumford classes e and e_1. Our discussion will be on the level of the natural 2-forms representing the relevant cohomology classes, and involves a comparison with other natural 2-forms representing e, e_1 induced by the Arakelov metric on the relative tangent bundle of C_g over M_g. A secondary object called a_g occurs, which was discovered and studied by Kawazumi around 2008. We present alternative proofs of Kawazumi's (unpublished) results on the second variation of a_g on M_g. Also we review some results that were previously obtained on the invariant a_g, with a focus on its connection with Faltings's delta-invariant and Hain-Reed's beta-invariant.