Projective structures and Hodge theory

Abstract Every compact Riemann surface X admits a natural projective structure pᵤ 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 pₕ 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 pᵤ p u and pₕ p h are not the same structure.