A Hodge theoretic projective structure on Riemann surfaces

Given any compact Riemann surface C, there is a canonical meromorphic 2-form ηˆ on C×C, with pole of order two on the diagonal Δ⊂C×C, constructed in \cite{cfg}. This meromorphic 2-form ηˆ produces a canonical projective structure on C. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface C. We prove that these two projective structures differ in general. This is done by comparing the (0,1)-component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. The (0,1)-component of the differential of the section corresponding to the projective structure given by the uniformization theorem was computed by Zograf and Takhtadzhyan in \cite{ZT} as the Weil-Petersson Kähler form ω_wp on the moduli space of curves. We prove that the (0,1)-component of the differential of the section of the moduli space of projective structures corresponding to ηˆ is the pullback of a nonzero constant scalar multiple of the Siegel form, on the moduli space of principally polarized abelian varieties, by the Torelli map.