Riemann Surfaces and Complex analytic geometry

Semalti, Navprabhat (2025) Riemann Surfaces and Complex analytic geometry. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Navprabhat Semalti (20MS036)) 20MS036_Thesis_file.pdf - Submitted Version Download (408kB)

Abstract

The goal of this thesis is to study some key aspects of the theory of 1-complex manifolds called ”Riemann surfaces” that are also 2-smooth manifolds. Riemann surfaces are nicer to study in the sense that behaviour of holomorphic functions on the complex plane can be translated onto them upto some degree, this makes their theory slightly less different and less challenging than higher dimensional complex manifolds. In general, people try to study Riemann surfaces as compact Riemann surfaces and non-compact Riemann surfaces(Open Riemann surfaces) since the phenomenon obtained as such vary depending upon this choice. We have studied the behaviour of holomorphic mappings of Riemann surfaces, which when restricted to compact connected settings give a quantification by Riemann-Hurwitz formula[4]. Along the way, for broader techniques on Riemann surfaces we have worked with holomorphic line bundles that allows us to use L2 theory on the space of their sections[4]. These techniques further allows one to probe into the phenomenon of analytic continuation which is also studied through del-bar problem on domains in Cn[4].

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