Projective structures and Hodge theory

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AbstractEvery compact Riemann surface X admits a natural projective structure $$p_u$$ 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 $$p_h$$ 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 $$p_u$$ p u and $$p_h$$ p h are not the same structure.