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

Robin de Jong

Published 2013-09-04, updated 2014-01-06Version 2

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 We present alternative proofs of Kawazumi's (unpublished) results on the second variation of a_g on M_g.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.