Leseprobe
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Lectures on Riemann Surfaces
Robert C. Gunning
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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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Irreducible component, Tangent space, Torelli theorem, Submanifold, Differential of the first kind, Linear map, Complex manifold, Meromorphic function, Vector space, Complex multiplication, Existential quantification, Algebraic operation, Cotangent space, Complex torus, Open set, Riemann surface, Homology (mathematics), Projective line, Special case, Cauchy's integral formula, Hypersurface, Parity (mathematics), Line bundle, Principal part, Subset, Analytic function, Sheaf (mathematics), Riemann sphere, Topological space, Cohomology, Complex analysis, Compact Riemann surface, Tangent bundle, Holomorphic function, Linear space (geometry), Bernhard Riemann, Identity theorem, Group homomorphism, Identity (mathematics), Homeomorphism, Combinatorics, Quotient space (topology), Inner automorphism, Cohomology ring, Projective variety, Algebraic geometry, Diagram (category theory), Variable (mathematics), Commutative property, Projective space, Homomorphism, Intersection (set theory), Divisor, Algebraic curve, Scientific notation, Algebraic variety, Clifford's theorem, Endomorphism, Differential form, Equation, Mathematical induction, Chern class, Neighbourhood (mathematics), Pole (complex analysis), Theorem, Group (mathematics), Analytic continuation, Complex projective space, Induced homomorphism, Analytic manifold