## Riemann Surfaces and Generalized Theta FunctionsThe investigation of the relationships between compact Riemann surfaces (al gebraic curves) and their associated complex tori (Jacobi varieties) has long been basic to the study both of Riemann surfaces and of complex tori. A Riemann surface is naturally imbedded as an analytic submanifold in its associated torus; and various spaces of linear equivalence elasses of divisors on the surface (or equivalently spaces of analytic equivalence elasses of complex line bundies over the surface), elassified according to the dimensions of the associated linear series (or the dimensions of the spaces of analytic cross-sections), are naturally realized as analytic subvarieties of the associated torus. One of the most fruitful of the elassical approaches to this investigation has been by way of theta functions. The space of linear equivalence elasses of positive divisors of order g -1 on a compact connected Riemann surface M of genus g is realized by an irreducible (g -1)-dimensional analytic subvariety, an irreducible hypersurface, of the associated g-dimensional complex torus J(M); this hyper 1 surface W- r;;;, J(M) is the image of the natural mapping Mg- -+J(M), and is g 1 1 birationally equivalent to the (g -1)-fold symmetric product Mg- jSg-l of the Riemann surface M. |

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### Contents

Complex Manifolds and Vector Bundles | 1 |

Riemann Surfaces | 17 |

Generalized Theta Functions | 39 |

Copyright | |

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Abelian differentials Abelian integral analytic line bundle analytic relatively automorphic analytic vector bundle analytically trivial automorphy of rank base point bundle homomorphism characteristic class characteristic matrix compact Riemann surface complex analytic function complex analytic line complex analytic mapping complex analytic Prym complex analytic relatively complex analytic subvariety complex analytic vector complex manifold complex tori conclude the proof consequence constant coordinate neighborhoods coordinate transformations Corollary defined described differential form differentials associated exact sequence factor of auto factor of automorphy fixed points flat factor function f function of rank fundamental polygon hence isomorphic lattice subgroup linear linearly independent marked Riemann surface meromorphic function meromorphic Prym differentials morphy nonsingular open neighborhood open subset path period matrix points z1 points zie precisely Prym differentials associated rank g relatively automorphic function represent distinct points residue Riemann-Roch theorem satisfies scalar factor singularities space of complex theta function torus universal covering space vector space zero