Fundamental Groups of Compact Kahler Manifolds
Fundamental Groups of Compact Kahler Manifolds

Author: Jaume Amorós

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 154

ISBN-13: 0821804987

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This book is an exposition of what is currently known about the fundamental groups of compact Kähler manifolds. This class of groups contains all finite groups and is strictly smaller than the class of all finitely presentable groups. For the first time ever, this book collects together all the results obtained in the last few years which aim to characterize those infinite groups which can arise as fundamental groups of compact Kähler manifolds. Most of these results are negative ones, saying which groups don not arise. The methods and techniques used form an attractive mix of topology, differential and algebraic geometry, and complex analysis. The book would be useful to researchers and graduate students interested in any of these areas, and it could be used as a textbook for an advanced graduate course. One of its outstanding features is a large number of concrete examples. The book contains a number of new results and examples which have not appeared elsewhere, as well as discussions of some important open questions in the field.

Symplectic Manifolds with no Kaehler structure
Symplectic Manifolds with no Kaehler structure

Author: Alesky Tralle

Publisher: Springer

Published: 2006-11-14

Total Pages: 216

ISBN-13: 3540691456

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This is a research monograph covering the majority of known results on the problem of constructing compact symplectic manifolds with no Kaehler structure with an emphasis on the use of rational homotopy theory. In recent years, some new and stimulating conjectures and problems have been formulated due to an influx of homotopical ideas. Examples include the Lupton-Oprea conjecture, the Benson-Gordon conjecture, both of which are in the spirit of some older and still unsolved problems (e.g. Thurston's conjecture and Sullivan's problem). Our explicit aim is to clarify the interrelations between certain aspects of symplectic geometry and homotopy theory in the framework of the problems mentioned above. We expect that the reader is aware of the basics of differential geometry and algebraic topology at graduate level.

Infinite Dimensional Kähler Manifolds
Infinite Dimensional Kähler Manifolds

Author: Alan Huckleberry

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 385

ISBN-13: 3034882270

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Infinite dimensional manifolds, Lie groups and algebras arise naturally in many areas of mathematics and physics. Having been used mainly as a tool for the study of finite dimensional objects, the emphasis has changed and they are now frequently studied for their own independent interest. On the one hand this is a collection of closely related articles on infinite dimensional Kähler manifolds and associated group actions which grew out of a DMV-Seminar on the same subject. On the other hand it covers significantly more ground than was possible during the seminar in Oberwolfach and is in a certain sense intended as a systematic approach which ranges from the foundations of the subject to recent developments. It should be accessible to doctoral students and as well researchers coming from a wide range of areas. The initial chapters are devoted to a rather selfcontained introduction to group actions on complex and symplectic manifolds and to Borel-Weil theory in finite dimensions. These are followed by a treatment of the basics of infinite dimensional Lie groups, their actions and their representations. Finally, a number of more specialized and advanced topics are discussed, e.g., Borel-Weil theory for loop groups, aspects of the Virasoro algebra, (gauge) group actions and determinant bundles, and second quantization and the geometry of the infinite dimensional Grassmann manifold.

Riemannian Holonomy Groups and Calibrated Geometry
Riemannian Holonomy Groups and Calibrated Geometry

Author: Dominic D. Joyce

Publisher: Oxford University Press

Published: 2007

Total Pages: 314

ISBN-13: 019921560X

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Riemannian Holonomy Groups and Calibrated Geometry covers an exciting and active area of research at the crossroads of several different fields in mathematics and physics. Drawing on the author's previous work the text has been written to explain the advanced mathematics involved simply and clearly to graduate students in both disciplines.

Differential Geometry: Geometry in Mathematical Physics and Related Topics
Differential Geometry: Geometry in Mathematical Physics and Related Topics

Author: Robert Everist Greene

Publisher: American Mathematical Soc.

Published: 1993

Total Pages: 681

ISBN-13: 0821814958

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The second of three parts comprising Volume 54, the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July 1990 (ISBN for the set is 0-8218-1493-1). Among the subjects of Part 2 are gauge theory, symplectic geometry, complex ge

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.

Lectures on Kähler Manifolds
Lectures on Kähler Manifolds

Author: Werner Ballmann

Publisher: European Mathematical Society

Published: 2006

Total Pages: 190

ISBN-13: 9783037190258

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These notes are based on lectures the author gave at the University of Bonn and the Erwin Schrodinger Institute in Vienna. The aim is to give a thorough introduction to the theory of Kahler manifolds with special emphasis on the differential geometric side of Kahler geometry. The exposition starts with a short discussion of complex manifolds and holomorphic vector bundles and a detailed account of the basic differential geometric properties of Kahler manifolds. The more advanced topics are the cohomology of Kahler manifolds, Calabi conjecture, Gromov's Kahler hyperbolic spaces, and the Kodaira embedding theorem. Some familiarity with global analysis and partial differential equations is assumed, in particular in the part on the Calabi conjecture. There are appendices on Chern-Weil theory, symmetric spaces, and $L^2$-cohomology.

Geometry and Topology
Geometry and Topology

Author: Mccrory

Publisher: CRC Press

Published: 2020-12-17

Total Pages: 366

ISBN-13: 1000110842

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This book discusses topics ranging from traditional areas of topology, such as knot theory and the topology of manifolds, to areas such as differential and algebraic geometry. It also discusses other topics such as three-manifolds, group actions, and algebraic varieties.

Infinite Groups: Geometric, Combinatorial and Dynamical Aspects
Infinite Groups: Geometric, Combinatorial and Dynamical Aspects

Author: Laurent Bartholdi

Publisher: Springer Science & Business Media

Published: 2006-03-28

Total Pages: 419

ISBN-13: 3764374470

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This book offers a panorama of recent advances in the theory of infinite groups. It contains survey papers contributed by leading specialists in group theory and other areas of mathematics. Topics include amenable groups, Kaehler groups, automorphism groups of rooted trees, rigidity, C*-algebras, random walks on groups, pro-p groups, Burnside groups, parafree groups, and Fuchsian groups. The accent is put on strong connections between group theory and other areas of mathematics.

Representations of Fundamental Groups of Algebraic Varieties
Representations of Fundamental Groups of Algebraic Varieties

Author: Kang Zuo

Publisher: Springer

Published: 2006-11-14

Total Pages: 142

ISBN-13: 3540484248

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Using harmonic maps, non-linear PDE and techniques from algebraic geometry this book enables the reader to study the relation between fundamental groups and algebraic geometry invariants of algebraic varieties. The reader should have a basic knowledge of algebraic geometry and non-linear analysis. This book can form the basis for graduate level seminars in the area of topology of algebraic varieties. It also contains present new techniques for researchers working in this area.