Theta Bundle, Quillen Connection and the Hodge Theoretic Projective Structure

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 L of
the determinant of the Hodge line bundle
over M_g; the same work gave the connection on L corresponding to the projective
structures coming from uniformization. Here we construct the
connection on L corresponding to the family of Hodge theoretic projective structures. This
connection is described in three different
ways: Firstly as the connection induced on 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.