Cohomology of Vector Bundles and Syzygies

Cohomology of Vector Bundles and Syzygies PDF

Author: Jerzy Weyman

Publisher: Cambridge University Press

Published: 2003-06-09

Total Pages: 404

ISBN-13: 9780521621977

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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.

Vector Bundles and Their Applications

Vector Bundles and Their Applications PDF

Author: Glenys Luke

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 259

ISBN-13: 1475769237

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The book is devoted to the basic notions of vector bundles and their applications. The focus of attention is towards explaining the most important notions and geometric constructions connected with the theory of vector bundles. Theorems are not always formulated in maximal generality but rather in such a way that the geometric nature of the objects comes to the fore. Whenever possible examples are given to illustrate the role of vector bundles. Audience: With numerous illustrations and applications to various problems in mathematics and the sciences, the book will be of interest to a range of graduate students from pure and applied mathematics.

Vector Bundles in Algebraic Geometry

Vector Bundles in Algebraic Geometry PDF

Author: N. J. Hitchin

Publisher: Cambridge University Press

Published: 1995-03-16

Total Pages: 359

ISBN-13: 0521498783

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This book is a collection of survey articles by the main speakers at the 1993 Durham symposium on vector bundles in algebraic geometry.

Vector Bundles on Complex Projective Spaces

Vector Bundles on Complex Projective Spaces PDF

Author: Christian Okonek

Publisher: Springer Science & Business Media

Published: 2013-11-11

Total Pages: 399

ISBN-13: 1475714602

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These lecture notes are intended as an introduction to the methods of classification of holomorphic vector bundles over projective algebraic manifolds X. To be as concrete as possible we have mostly restricted ourselves to the case X = Fn. According to Serre (GAGA) the classification of holomorphic vector bundles is equivalent to the classification of algebraic vector bundles. Here we have used almost exclusively the language of analytic geometry. The book is intended for students who have a basic knowledge of analytic and (or) algebraic geometry. Some funda mental results from these fields are summarized at the beginning. One of the authors gave a survey in the Seminaire Bourbaki 1978 on the current state of the classification of holomorphic vector bundles overFn. This lecture then served as the basis for a course of lectures in Gottingen in the Winter Semester 78/79. The present work is an extended and up-dated exposition of that course. Because of the introductory nature of this book we have had to leave out some difficult topics such as the restriction theorem of Barth. As compensation we have appended to each sec tion a paragraph in which historical remarks are made, further results indicated and unsolved problems presented. The book is divided into two chapters. Each chapter is subdivided into several sections which in turn are made up of a number of paragraphs. Each section is preceeded by a short description of iv its contents.

Vector Bundles

Vector Bundles PDF

Author: Andrej N. Tjurin

Publisher: Universitätsverlag Göttingen

Published: 2008

Total Pages: 330

ISBN-13: 3938616741

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This is the first volume of a three volume collection of Andrey Nikolaevich Tyurin's Selected Works. It includes his most interesting articles in the field of classical algebraic geometry, written during his whole career from the 1960s. Most of these papers treat different problems of the theory of vector bundles on curves and higher dimensional algebraic varieties, a theory which is central to algebraic geometry and most of its applications.

Koszul Cohomology and Algebraic Geometry

Koszul Cohomology and Algebraic Geometry PDF

Author: Marian Aprodu

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 138

ISBN-13: 0821849646

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The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well. This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.

Vector Bundles and Complex Geometry

Vector Bundles and Complex Geometry PDF

Author: Oscar García-Prada

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 218

ISBN-13: 0821847503

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This volume contains a collection of papers from the Conference on Vector Bundles held at Miraflores de la Sierra, Madrid, Spain on June 16-20, 2008, which honored S. Ramanan on his 70th birthday. The main areas covered in this volume are vector bundles, parabolic bundles, abelian varieties, Hilbert schemes, contact structures, index theory, Hodge theory, and geometric invariant theory. Professor Ramanan has made important contributions in all of these areas.

Syzygies

Syzygies PDF

Author: E. Graham Evans

Publisher: Cambridge University Press

Published: 1985-08-15

Total Pages: 137

ISBN-13: 0521314119

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This 1985 book covers from first principles the theory of Syzygies.

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