By Dror Varolin

This ebook establishes the fundamental functionality idea and intricate geometry of Riemann surfaces, either open and compact. a number of the equipment utilized in the publication are diversifications and simplifications of tools from the theories of a number of complicated variables and complicated analytic geometry and might function first-class education for mathematicians desirous to paintings in complicated analytic geometry. After 3 introductory chapters, the e-book embarks on its primary, and positively such a lot novel, objective of learning Hermitian holomorphic line bundles and their sections. between different issues, finite-dimensionality of areas of sections of holomorphic line bundles of compact Riemann surfaces and the triviality of holomorphic line bundles over Riemann surfaces are proved, with quite a few purposes. possibly the most results of the e-book is Hörmander's Theorem at the square-integrable answer of the Cauchy-Riemann equations. The crowning software is the facts of the Kodaira and Narasimhan Embedding Theorems for compact and open Riemann surfaces. The meant reader has had first classes in actual and complicated research, in addition to complicated calculus and uncomplicated differential topology (though the latter topic isn't really crucial). As such, the ebook may still entice a large part of the mathematical and clinical group. This e-book is the 1st to offer a textbook exposition of Riemann floor idea from the point of view of confident Hermitian line bundles and Hörmander $\bar \partial$ estimates. it's extra analytical and PDE orientated than past texts within the box, and is a wonderful advent to the tools used at the moment in complicated geometry, as exemplified in J. P. Demailly's on-line yet another way unpublished publication "Complex analytic and differential geometry." I used it for a one area path on Riemann surfaces and located it to be truly written and self-contained. It not just fills an important hole within the huge textbook literature on Riemann surfaces yet is usually fairly indispensible if you happen to want to educate the topic from a differential geometric and PDE standpoint.

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