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.
