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The study of vector bundles over algebraic varieties has been stimulated over the last few years by successive waves of migrant concepts, largely from mathematical physics, whilst retaining its roots in old questions concerning subvarieties of projective space. The 1993 Durham Symposium on Vector Bundles in Algebraic Geometry brought together some of the leading researchers in the field to explore further these interactions. This book is a collection of survey articles by the main speakers at the symposium and presents to the mathematical world an overview of the key areas of research involving vector bundles. Topics covered include those linking gauge theory and geometric invariant theory such as augmented bundles and coherent systems; Donaldson invariants of algebraic surfaces; Floer homology and quantum cohomology; conformal field theory and the moduli spaces of bundles on curves; the Horrocks-Mumford bundle and codimension 2 subvarieties in P4 and P5; exceptional bundles and stable sheaves on projective space.
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Holomorphic vector bundles have become objects of interest not only to algebraic and differential geometers and complex analysts but also to low dimensional topologists and mathematical physicists working on gauge theory. This book, which grew out of the author's lectures and seminars in Berkeley and Japan, is written for researchers and graduate students in these various fields of mathematics.Originally published in 1987.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Vector bundles. --- Vanishing theorems. --- Theorems, Vanishing --- Complex manifolds --- Fiber bundles (Mathematics) --- Homology theory --- Vector bundles --- Fiber spaces (Mathematics)
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The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.
Syzygies (Mathematics). --- Syzygies (Mathematics) --- Vector bundles. --- Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Fiber spaces (Mathematics) --- Syzygy theory (Mathematics) --- Categories (Mathematics) --- Rings (Algebra) --- Vector bundles --- Homology theory
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The theory of isomonodromic deformations enables the production of systems of non-linear differential equations or of their partial complex derivatives, beginning with one equation or a system of linear equations of a complex variable. The notion of a Frobenius structure on a complex analytic manifold is a beautiful application.
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Vector bundles - Vol 1
Vector bundles. --- Characteristic classes. --- Fibrés vectoriels --- Classes caractéristiques --- 514.76 --- Characteristic classes --- Vector Bundles --- Fiber spaces (Mathematics) --- Classes, Characteristic --- Differential topology --- Geometry of differentiable manifolds and of their submanifolds --- 514.76 Geometry of differentiable manifolds and of their submanifolds
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The aim of this book, which was originally published in 1985, is to cover from first principles the theory of Syzygies, building up from a discussion of the basic commutative algebra to such results as the authors' proof of the Syzygy Theorem. In the last three chapters applications of the theory to commutative algebra and algebraic geometry are given.
Syzygies (Mathematics) --- Rings (Algebra) --- Vector bundles. --- Fiber spaces (Mathematics) --- Algebraic rings --- Ring theory --- Algebraic fields --- Syzygy theory (Mathematics) --- Categories (Mathematics) --- Vector bundles --- 512.55 --- 512.55 Rings and modules --- Rings and modules
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This volume is devoted to the use of helices as a method for studying exceptional vector bundles, an important and natural concept in algebraic geometry. The work arises out of a series of seminars organised in Moscow by A. N. Rudakov. The first article sets up the general machinery, and later ones explore its use in various contexts. As to be expected, the approach is concrete; the theory is considered for quadrics, ruled surfaces, K3 surfaces and P3(C).
Vector bundles. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Fiber spaces (Mathematics) --- Helices (Algebraic topology) --- Helix (Algebraic topology) --- Fiber bundles (Mathematics)
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Algebraic geometers have renewed their interest in the interplay between algebraic vector bundles and projective embeddings. New methods have been developed for questions such as: what is the geometric content of syzygies and of bundles derived from them? how can they be used for giving good compactifications of natural families? which differential techniques are needed for the study of families of projective varieties? Such problems have often been reformulated over the last decade; often the need for a deeper analysis of the works of classical algebraic geometers was recognised. These questions were addressed at successive conferences held in Trieste and Bergen. New results, work in progress, conjectures and modern accounts of classical ideas were presented. This collection represents a development of the work conducted at the conferences; the Editors have taken the opportunity to mould the papers into a cohesive volume.
Geometry, Algebraic --- Vector bundles --- Embeddings (Mathematics) --- Algebraic varieties --- Varieties, Algebraic --- Linear algebraic groups --- Imbeddings (Mathematics) --- Immersions (Mathematics) --- Fiber spaces (Mathematics)
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Vector bundles and their associated moduli spaces are of fundamental importance in algebraic geometry. In recent decades this subject has been greatly enhanced by its relationships with other areas of mathematics, including differential geometry, topology and even theoretical physics, specifically gauge theory, quantum field theory and string theory. Peter E. Newstead has been a leading figure in this field almost from its inception and has made many seminal contributions to our understanding of moduli spaces of stable bundles. This volume has been assembled in tribute to Professor Newstead and his contribution to algebraic geometry. Some of the subject's leading experts cover foundational material, while the survey and research papers focus on topics at the forefront of the field. This volume is suitable for both graduate students and more experienced researchers.
Vector bundles. --- Moduli theory. --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Fiber spaces (Mathematics)
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This book presents the most up-to-date and sophisticated account of the theory of Euclidean lattices and sequences of Euclidean lattices, in the framework of Arakelov geometry, where Euclidean lattices are considered as vector bundles over arithmetic curves. It contains a complete description of the theta invariants which give rise to a closer parallel with the geometric case. The author then unfolds his theory of infinite Hermitian vector bundles over arithmetic curves and their theta invariants, which provides a conceptual framework to deal with the sequences of lattices occurring in many diophantine constructions. The book contains many interesting original insights and ties to other theories. It is written with extreme care, with a clear and pleasant style, and never sacrifices accessibility to sophistication. .
Algebraic geometry. --- Number theory. --- Algebraic Geometry. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Algebraic geometry --- Geometry --- Vector bundles. --- Fiber spaces (Mathematics)
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