An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

Lecture Details

An Introduction to Riemann Surfaces and Algebraic Curves Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit httpwww.nptel.iitm.ac.insyllabus111106044Goals To extend the theory of topological coverings to that of holomorphic (complex analytic) coverings To show that any Riemann surface structure on the base space of a topological covering induces a Riemann surface structure on the covering space in such a way that the covering projection map is holomorpic. To achieve this using the technique of "pulling back charts from below" To see why the Riemann surface structure induced above is essentially unique In particular, we get a unique Riemann surface structure on the topological covering of a Riemann surface. The deck transformations therefore become holomorphic automorphisms of this Riemann surface structureKeywords Topological covering, holomorphic covering, admissible neighborhood, chart, pulling back charts by local homeomorphisms, locally biholomorphic, pulling back Riemann surface structures, holomorphicity or complex analyticity of continuous liftings, deck transformation