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Given any compact Riemann surface $C$, there is a canonical meromorphic2--form $\widehat\eta$ on $C\times C$, with pole of order two on the diagonal$\Delta\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic2--form $\widehat\eta$ produces a canonical projective structure on $C$. On theother hand the uniformization theorem provides another canonical projectivestructure on any compact Riemann surface $C$. We prove that these twoprojective structures differ in general. This is done by comparing the$(0,1)$--component of the differential of the corresponding sections of themoduli space of projective structures over the moduli space of curves. The$(0,1)$--component of the differential of the section corresponding to theprojective structure given by the uniformization theorem was computed by Zografand Takhtadzhyan in \cite{ZT} as the Weil--Petersson K\"ahler form$\omega_{wp}$ on the moduli space of curves. We prove that the$(0,1)$--component of the differential of the section of the moduli space ofprojective structures corresponding to $\widehat{\eta}$ is the pullback of anonzero constant scalar multiple of the Siegel form, on the moduli space ofprincipally polarized abelian varieties, by the Torelli map.

A Hodge theoretic projective structure on Riemann surfaces