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 Mg of compact Riemann surfaces. A recent work of Biswas, Favale, Pirola and Torelli shows that families of projective structures over Mg admit an equivalent characterization in terms of complex connections on the dual L of the determinant of the Hodge line bundle over Mg; 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 L2-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 obtained using the Arakelov metric on the universal curve, modified by Faltings’ delta invariant.