  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.
