Differential Geometry of Complex Vector Bundles
Differential Geometry of Complex Vector Bundles

Author: Shoshichi Kobayashi

Publisher: Princeton University Press

Published: 2014-07-14

Total Pages: 317

ISBN-13: 1400858682

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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.

Birational Geometry, Kähler–Einstein Metrics and Degenerations
Birational Geometry, Kähler–Einstein Metrics and Degenerations

Author: Ivan Cheltsov

Publisher: Springer Nature

Published: 2023-05-23

Total Pages: 882

ISBN-13: 3031178599

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This book collects the proceedings of a series of conferences dedicated to birational geometry of Fano varieties held in Moscow, Shanghai and Pohang The conferences were focused on the following two related problems: • existence of Kähler–Einstein metrics on Fano varieties • degenerations of Fano varieties on which two famous conjectures were recently proved. The first is the famous Borisov–Alexeev–Borisov Conjecture on the boundedness of Fano varieties, proved by Caucher Birkar (for which he was awarded the Fields medal in 2018), and the second one is the (arguably even more famous) Tian–Yau–Donaldson Conjecture on the existence of Kähler–Einstein metrics on (smooth) Fano varieties and K-stability, which was proved by Xiuxiong Chen, Sir Simon Donaldson and Song Sun. The solutions for these longstanding conjectures have opened new directions in birational and Kähler geometries. These research directions generated new interesting mathematical problems, attracting the attention of mathematicians worldwide. These conferences brought together top researchers in both fields (birational geometry and complex geometry) to solve some of these problems and understand the relations between them. The result of this activity is collected in this book, which contains contributions by sixty nine mathematicians, who contributed forty three research and survey papers to this volume. Many of them were participants of the Moscow–Shanghai–Pohang conferences, while the others helped to expand the research breadth of the volume—the diversity of their contributions reflects the vitality of modern Algebraic Geometry.

The Geometry of Moduli Spaces of Sheaves
The Geometry of Moduli Spaces of Sheaves

Author: Daniel Huybrechts

Publisher: Cambridge University Press

Published: 2010-05-27

Total Pages: 345

ISBN-13: 1139485822

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This edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.

Contemporary Aspects Of Complex Analysis, Differential Geometry And Mathematical Physics - Procs Of The 7th Int'l Workshop On Complex Structures And Vector Fields
Contemporary Aspects Of Complex Analysis, Differential Geometry And Mathematical Physics - Procs Of The 7th Int'l Workshop On Complex Structures And Vector Fields

Author: Stancho Dimiev

Publisher: World Scientific

Published: 2005-07-04

Total Pages: 358

ISBN-13: 9814479977

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This volume presents the cutting-edge contributions to the Seventh International Workshop on Complex Structures and Vector Fields, which was organized as a continuation of the high successful preceding workshops on similar research.The volume includes works treating ambitious topics in differential geometry, mathematical physics and technology such as Bézier curves in space forms, potential and catastrophy of a soap film, computer-assisted studies of logistic maps, and robotics.

Holomorphic Morse Inequalities and Bergman Kernels
Holomorphic Morse Inequalities and Bergman Kernels

Author: Xiaonan Ma

Publisher: Springer Science & Business Media

Published: 2007-12-14

Total Pages: 432

ISBN-13: 3764381159

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This book examines holomorphic Morse inequalities and the asymptotic expansion of the Bergman kernel on manifolds by using the heat kernel. It opens perspectives on several active areas of research in complex, Kähler and symplectic geometry. A large number of applications are also included, such as an analytic proof of Kodaira's embedding theorem, a solution of the Grauert-Riemenschneider and Shiffman conjectures, compactification of complete Kähler manifolds of pinched negative curvature, Berezin-Toeplitz quantization, weak Lefschetz theorems, and asymptotics of the Ray-Singer analytic torsion.

Characteristic Classes
Characteristic Classes

Author: John Willard Milnor

Publisher: Princeton University Press

Published: 1974

Total Pages: 342

ISBN-13: 9780691081229

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The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

Holomorphic Vector Bundles over Compact Complex Surfaces
Holomorphic Vector Bundles over Compact Complex Surfaces

Author: Vasile Brinzanescu

Publisher: Springer

Published: 2006-11-14

Total Pages: 175

ISBN-13: 3540498451

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The purpose of this book is to present the available (sometimes only partial) solutions to the two fundamental problems: the existence problem and the classification problem for holomorphic structures in a given topological vector bundle over a compact complex surface. Special features of the nonalgebraic surfaces case, like irreducible vector bundles and stability with respect to a Gauduchon metric, are considered. The reader requires a grounding in geometry at graduate student level. The book will be of interest to graduate students and researchers in complex, algebraic and differential geometry.

An Introduction to Extremal Kahler Metrics
An Introduction to Extremal Kahler Metrics

Author: Gábor Székelyhidi

Publisher: American Mathematical Soc.

Published: 2014-06-19

Total Pages: 210

ISBN-13: 1470410478

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A basic problem in differential geometry is to find canonical metrics on manifolds. The best known example of this is the classical uniformization theorem for Riemann surfaces. Extremal metrics were introduced by Calabi as an attempt at finding a higher-dimensional generalization of this result, in the setting of Kähler geometry. This book gives an introduction to the study of extremal Kähler metrics and in particular to the conjectural picture relating the existence of extremal metrics on projective manifolds to the stability of the underlying manifold in the sense of algebraic geometry. The book addresses some of the basic ideas on both the analytic and the algebraic sides of this picture. An overview is given of much of the necessary background material, such as basic Kähler geometry, moment maps, and geometric invariant theory. Beyond the basic definitions and properties of extremal metrics, several highlights of the theory are discussed at a level accessible to graduate students: Yau's theorem on the existence of Kähler-Einstein metrics, the Bergman kernel expansion due to Tian, Donaldson's lower bound for the Calabi energy, and Arezzo-Pacard's existence theorem for constant scalar curvature Kähler metrics on blow-ups.

Cartan for Beginners
Cartan for Beginners

Author: Thomas Andrew Ivey

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 394

ISBN-13: 0821833758

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This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior differential systems and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems. It begins with the classical geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics.One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. The book features an introduction to $G$-structures and a treatment of the theory of connections. The Cartan machinery is also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.

Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics
Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics

Author: Y.-T. Siu

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 172

ISBN-13: 3034874863

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These notes are based on the lectures I delivered at the German Mathematical Society Seminar in Schloss Michkeln in DUsseldorf in June. 1986 on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics. The purpose of these notes is to present to the reader the state-of-the-art results in the simplest and the most comprehensible form using (at least from my own subjective viewpoint) the most natural approach. The presentation in these notes is reasonably self-contained and prerequisi tes are kept to a minimum. Most steps in the estimates are reduced as much as possible to the most basic procedures such as integration by parts and the maximum principle. When less basic procedures are used such as the Sobolev and Calderon-Zygmund inequalities and the interior Schauder estimates. references are given for the reader to look them up. A considerable amount of heuristic and intuitive discussions are included to explain why certain steps are used or certain notions introduced. The inclusion of such discussions makes the style of the presentation at some places more conversational than what is usually expected of rigorous mathemtical prese"ntations. For the problems of Hermi tian-Einstein metrics for stable bundles and Kahler-Einstein metrics one can use either the continuity method or the heat equation method. These two methods are so very intimately related that in many cases the relationship betwen them borders on equivalence. What counts most is the a. priori estimates. The kind of scaffolding one hangs the a.