Projective Structures and Theta Functions

This thesis studies two new projective structures on a compact Riemann surfaces, obtained from Jacobian and Prym varieties. The first is the Theta Projective Structure, denoted βθ, constructed canonically for any curve C by pulling back a specific line of sections from the canonical bundle 2Θ on the Jacobian variety J(C) via the difference map ϕ(p,q) = OC(p−q). The second is the Prym Projective Structure, βP, built by applying the same principle to the Prym variety of an ´etale double cover π : C → C, from which we obtain a projective structure on the base curve C. The central results of the thesis are that the theta projective structure βθ is distinct from all previously known canonical projective structures, namely the Fuchsian and the Hodge projective structures, and that the Prym projective structure βP generally depends on the choice of the ´etale double covering π and consequently does not descend to a canonical structure on the moduli space Mg of curves. A key technique for studying projective structures is the computation of their ∂-derivative, interpreted as a (1,1)-form on the corresponding moduli space. Much of the work of this thesis is dedicated to these computations.