Theta bundle, Quillen connection and the Hodge theoretic projective structure

TL;DR

This paper constructs and describes the connection on the Hodge line bundle over the moduli space of Riemann surfaces associated with the Hodge theoretic projective structure, linking it to various geometric and analytic metrics.

Contribution

It explicitly constructs the connection on the Hodge line bundle corresponding to the Hodge theoretic projective structure and describes it via three different geometric and analytic frameworks.

Findings

Connection described via the Chern connection of the L^2-metric.

Connection expressed as a root of the Quillen metric from the Theta line bundle.

Connection obtained through the Arakelov metric and Faltings' delta invariant.

Abstract

There are two canonical projective structures on any compact Riemann surface of genus at least two: one coming from the uniformization theorem, and the other from Hodge theory. They produce two (different) families of projective structures over the moduli space $M_g$ of compact Riemann surfaces. A recent work of Biswas, Favale, Pirola, and Torelli shows that families of projective structures over $M_g$ admit an equivalent characterization in terms of complex connections on the dual $\mathcal{L}$ of the determinant of the Hodge line bundle over $M_g$; the same work gave the connection on $\mathcal L$ corresponding to the projective structures coming from uniformization. Here we construct the connection on $\mathcal L$ corresponding to the family of Hodge theoretic projective structures. This connection is described in three different ways: firstly as the connection induced on $\mathcal L$ by the Chern connection of the $L^2$-metric on the Hodge bundle, secondly as an appropriate root of the Quillen metric induced by the (square of the) Theta line bundle on the universal family of abelian varieties, endowed with the natural Hermitian metric given by the polarization, and finally as Quillen connection gotten using the Arakelov metric on the universal curve, modified by Faltings' delta invariant.