  Every compact Riemann surface $X$ admits a natural projective structure $p_u$
as a consequence of the uniformization theorem. In this work we describe the
construction of another natural projective structure on $X$, namely the Hodge
projective structure $p_h$, related to the second fundamental form of the
period map. We then describe how projective structures correspond to
$(1,1)$-differential forms on the moduli space of projective curves and, from
this correspondence, we deduce that $p_u$ and $p_h$ are not the same structure.
