Algebraic surfaces and holomorphic vector bundles
type: Item
instance of: scholarly article
title: Algebraic surfaces and holomorphic vector bundles
zbMATH Open document ID: 0902.14029
published in: Universitext
publication date: 1998-03-30T00:00:00Z
review text: This book gives a comprehensive account of the theory of complex projective algebraic surfaces and, simultaneously, of the present state of knowledge concerning the classification theory of holomorphic vector bundles over those surfaces. Basically, it is designed as a textbook for advanced graduate students in those areas of complex geometry that are interspersed with methods of algebraic geometry, for instance such as algebraic geometry itself, gauge theory in global analysis, four-manifold topology, quantum field theory in theoretical physics, and others. On the other hand, with regard to its thoroughness, comprehensiveness and up-to-dateness, this text does provide much more than an excellent introduction to the theory of complex algebraic surfaces and holomorphic vector bundles over them. In fact it also represents a respectable work of reference for researchers and teachers in these areas of mathematics and physics, all the more as there are still only very few reference books on vector bundles available within the existing literature. The book grew out of courses that the author taught, over the past ten years, at several American mathematical institutes. Originally, the goal of these lectures was to acquaint researchers in four-manifold topology with the classification of algebraic surfaces and with methods for describing moduli spaces of holomorphic vector bundles over algebraic surfaces, and that with a special view toward computing Donaldson invariants for four-manifolds. This approach to investigating the topology of smooth four-manifolds is described, in great variety, in the monograph ``Smooth four-manifolds and complex surfaces'' written by \textit{R. Friedman} and \textit{J. W. Morgan} in 1994 (Zbl 0817.14017). Although the focus of four-manifold topology has shifted since that time more and more to the purely analytic-topological aspects connected with the recently discovered Seiberg-Witten invariants, the author's method of teaching the theory of al…
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14Jxx
Mathematics Subject Classification ID: 14-01
Mathematics Subject Classification ID: 32C35
Mathematics Subject Classification ID: 32-01
zbMATH DE Number: 1134984
zbMATH Keywords: curves on a surface
zbMATH Keywords: gauge theory
zbMATH Keywords: four-manifold topology
zbMATH Keywords: quantum field theory
zbMATH Keywords: moduli spaces of holomorphic vector bundles over algebraic surfaces
zbMATH Keywords: Donaldson invariants
author: Robert Friedman
reviewed by: Werner Kleinert
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1134984