Vector Bundles on Complex Projective Spaces

Author: Christian Okonek,Heinz Spindler,Michael Schneider
Publisher: Springer Science & Business Media
ISBN: 1475714602
Category: Mathematics
Page: 389
View: 7270

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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.

Complex Projective Geometry

Selected Papers
Author: G. Ellingsrud
Publisher: Cambridge University Press
ISBN: 9780521433525
Category: Mathematics
Page: 340
View: 380

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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? These questions are addressed in this cohesive volume, where results, work in progress, conjectures, and modern accounts of classical ideas are presented.

Complex Algebraic Varieties

Proceedings of a Conference held in Bayreuth, Germany, April 2-6, 1990
Author: Klaus Hulek,Thomas Peternell,Michael Schneider,Frank-Olaf Schreyer
Publisher: Springer
ISBN: 3540467866
Category: Mathematics
Page: 179
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The Bayreuth meeting on "Complex Algebraic Varieties" focussed on the classification of algebraic varieties and topics such as vector bundles, Hodge theory and hermitian differential geometry. Most of the articles in this volume are closely related to talks given at the conference: all are original, fully refereed research articles. CONTENTS: A. Beauville: Annulation du H(1) pour les fibres en droites plats.- M. Beltrametti, A.J. Sommese, J.A. Wisniewski: Results on varieties with many lines and their applications to adjunction theory.- G. Bohnhorst, H. Spindler: The stability of certain vector bundles on P(n) .- F. Catanese, F. Tovena: Vector bundles, linear systems and extensions of (1).- O. Debarre: Vers uns stratification de l'espace des modules des varietes abeliennes principalement polarisees.- J.P. Demailly: Singular hermitian metrics on positive line bundles.- T. Fujita: On adjoint bundles of ample vector bundles.- Y. Kawamata: Moderate degenerations of algebraic surfaces.- U. Persson: Genus two fibrations revisited.- Th. Peternell, M. Szurek, J.A. Wisniewski: Numerically effective vector bundles with small Chern classes.- C.A.M. Peters: On the rank of non-rigid period maps in the weight one and two case.- A.N. Tyurin: The geometry of the special components of moduli space of vector bundles over algebraic surfaces of general type.

Discriminants, Resultants, and Multidimensional Determinants

Author: Israel M. Gelfand,Mikhail Kapranov,Andrei Zelevinsky
Publisher: Springer Science & Business Media
ISBN: 0817647716
Category: Mathematics
Page: 523
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"This book revives and vastly expands the classical theory of resultants and discriminants. Most of the main new results of the book have been published earlier in more than a dozen joint papers of the authors. The book nicely complements these original papers with many examples illustrating both old and new results of the theory."—Mathematical Reviews

The Geometry of Moduli Spaces of Sheaves

A Publication of the Max-Planck-Institut für Mathematik, Bonn
Author: Daniel Huybrechts,Manfred Lehn
Publisher: Vieweg+Teubner Verlag
ISBN: 9783663116257
Category: Technology & Engineering
Page: 270
View: 7254

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This book is intended to serve as an introduction to the theory of semistable sheaves and at the same time to provide a survey of recent research results on the geometry of moduli spaces. The first part introduces the basic concepts in the theory: Hilbert polynomial, slope, stability, Harder-Narasimhan filtration, Grothendieck's Quot-scheme. It presents detailed proofs of the Grauert-Mülich Theorem, the Bogomolov Inequality, the semistability of tensor products, and the boundedness of the family of semistable sheaves. It also gives a self-contained account of the construction of moduli spaces of semistable sheaves on a projective variety à la Gieseker, Maruyama, and Simpson. The second part presents some of the recent results of the geometry of moduli spaces of sheaves on an algebraic surface, following work of Mukai, O'Grady, Gieseker, Li and many others. In particular, moduli spaces of sheaves on K3 surfaces and determinant line bundles on the moduli spaces are treated in some detail. Other topics include the Serre correspondence, restriction of stable bundles to curves, symplectic structures, irreducibility and Kodaira-dimension of moduli spaces.


Author: E. Graham Evans,Phillip Griffith
Publisher: Cambridge University Press
ISBN: 0521314119
Category: Mathematics
Page: 124
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This 1985 book covers from first principles the theory of Syzygies.

Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor

Author: Peter B. Gilkey
Publisher: World Scientific
ISBN: 9812799699
Category: Mathematics
Page: 316
View: 3895

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A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold. The full curvature tensor is in general quite difficult to deal with. This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition. The book presents algebraic preliminaries and various Schur type problems; deals with the skew-symmetric curvature operator in the real and complex settings and provides the classification of algebraic curvature tensors whose skew-symmetric curvature has constant rank 2 and constant eigenvalues; discusses the Jacobi operator and a higher order generalization and gives a unified treatment of the Osserman conjecture and related questions; and establishes the results from algebraic topology that are necessary for controlling the eigenvalue structures. An extensive bibliography is provided. Results are described in the Riemannian, Lorentzian, and higher signature settings, and many families of examples are displayed. Contents: Algebraic Curvature Tensors; The Skew-Symmetric Curvature Operator; The Jacobi Operator; Controlling the Eigenvalue Structure. Readership: Researchers and graduate students in geometry and topology.

The Adjunction Theory of Complex Projective Varieties

Author: Mauro C. Beltrametti,Andrew J. Sommese
Publisher: Walter de Gruyter
ISBN: 3110871742
Category: Mathematics
Page: 418
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The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

Algebraic Geometry I

Complex Projective Varieties
Author: David Mumford
Publisher: Springer Science & Business Media
ISBN: 9783540586579
Category: Mathematics
Page: 186
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From the reviews: "Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's "Volume I" is, together with its predecessor the red book of varieties and schemes, now as before one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!" Zentralblatt

Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics

Author: CLAUDIO BARTOCCI,Ugo Bruzzo,Daniel Hernández Ruipérez
Publisher: Springer Science & Business Media
ISBN: 0817646639
Category: Science
Page: 418
View: 6314

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Integral transforms, such as the Laplace and Fourier transforms, have been major tools in mathematics for at least two centuries. In the last three decades the development of a number of novel ideas in algebraic geometry, category theory, gauge theory, and string theory has been closely related to generalizations of integral transforms of a more geometric character. "Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics" examines the algebro-geometric approach (Fourier–Mukai functors) as well as the differential-geometric constructions (Nahm). Also included is a considerable amount of material from existing literature which has not been systematically organized into a monograph. Key features: Basic constructions and definitions are presented in preliminary background chapters - Presentation explores applications and suggests several open questions - Extensive bibliography and index. This self-contained monograph provides an introduction to current research in geometry and mathematical physics and is intended for graduate students and researchers just entering this field.

Compact Manifolds with Special Holonomy

Author: Dominic D. Joyce
Publisher: Oxford University Press on Demand
ISBN: 9780198506010
Category: Mathematics
Page: 436
View: 708

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This is a combination of a graduate textbook on Riemannian holonomy groups, and a research monograph on compact manifolds with the exceptional holonomy groups G2 and Spin (7). It is the first book on compact manifolds with exceptional holonomy, and contains much new research material and many new examples.

Vector Bundles and Complex Geometry

Conference on Vector Bundles in Honor of S. Ramanan on the Occasion of His 70th Birthday, June 16-20, 2008, Miraflores de la Sierra, Madrid, Spain
Author: S. Ramanan
Publisher: American Mathematical Soc.
ISBN: 0821847503
Category: Mathematics
Page: 206
View: 6243

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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.