A sequel to Lectures on Riemann Surfaces (Mathematical Notes, 1966),
this volume continues the discussion of the dimensions of spaces of
holomorphic cross-sections of complex line bundles over compact Riemann
surfaces. Whereas the earlier treatment was limited to results
obtainable chiefly by one-dimensional methods, the more detailed
analysis presented here requires the use of various properties of Jacobi
varieties and of symmetric products of Riemann surfaces, and so serves
as a further introduction to these topics as well.
The first chapter consists of a rather explicit description of a
canonical basis for the Abelian differentials on a marked Riemann
surface, and of the description of the canonical meromorphic
differentials and the prime function of a marked Riemann surface.
Chapter 2 treats Jacobi varieties of compact Riemann surfaces and
various subvarieties that arise in determining the dimensions of spaces
of holomorphic cross-sections of complex line bundles. In Chapter 3, the
author discusses the relations between Jacobi varieties and symmetric
products of Riemann surfaces relevant to the determination of dimensions
of spaces of holomorphic cross-sections of complex line bundles. The
final chapter derives Torelli's theorem following A. Weil, but in an
analytical context.
Originally published in 1973.
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