Projective Geometry and Formal Geometry
By: by Lucian Bădescu
Series: Monografie Matematyczne, Instytut Matematyczny Polskiej Akademii Nauk (IMPAN) -- 65; Monografie Matematyczne, Instytut Matematyczny Polskiej Akademii Nauk (IMPAN) -- 65
Publisher: Birkhäuser Basel; Imprint: Birkhäuser
Published: 2004
Place of publication: Basel
Language: eng
Format: [electronic resource] /
Number of pages: 214
ISBN-13: 9783034896269; 9783034879361
Subjects: Geometry; Mathematics; Algebraic Geometry; Global analysis; Functions of complex variables
Description: The aim of this monograph is to introduce the reader to modern methods of projective geometry involving certain techniques of formal geometry. Some of these methods are illustrated in the first part through the proofs of a number of results of a rather classical flavor, involving in a crucial way the first infinitesimal neighbourhood of a given subvariety in an ambient variety. Motivated by the first part, in the second formal functions on the formal completion X/Y of X along a closed subvariety Y are studied, particularly the extension problem of formal functions to rational functions. The formal scheme X/Y, introduced to algebraic geometry by Zariski and Grothendieck in the 1950s, is an analogue of the concept of a tubular neighbourhood of a submanifold of a complex manifold. It is very well suited to study the given embedding Y\subset X. The deep relationship of formal geometry with the most important connectivity theorems in algebraic geometry, or with complex geometry, is also studied. Some of the formal methods are illustrated and applied to homogeneous spaces. The book contains a lot of results obtained over the last thirty years, many of which never appeared in a monograph or textbook. It addresses to algebraic geometers as well as to those interested in using methods of algebraic geometry.
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Table of contents: title: Introduction; title: I. Special Chapters of Projective Geometry (Projective Varieties, Zak Map, Quasi-homogeneous Singularities, Cohomological Dimension, Connectedness); title: II. Formal Functions in Algebraic Geometry (Basics, Lefschetz Theory, Meromorphic Functions, Connectedness, Homogeneous Spaces, Quasi-lines on Projective Manifolds); title: Bibliography; title: Notation; title: Index.