Projective structures and Hodge theory

Every compact Riemann surface X admits a natural projective structure pᵤ 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 pₕ, 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 pᵤ and pₕ are not the same structure.