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 characterise those infinite groups which can arise as fundamental groups of compact Kähler manifolds. Most of these results are negative ones, saying which groups do not arise. They are proved using Hodge theory and its combinations with rational homotopy theory, with \(L^2\) -cohomology, with the theory of harmonic maps, and with gauge theory. There are a number of positive results as well, exhibiting interesting groups as fundamental groups of Kähler manifolds, in fact, of smooth complex projective varieties.

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.

Readership

Graduate students and research mathematicians interested in algebraic geometry and several complex variables and analytic spaces.

Reviews

"This book, presently the only one dealing with this subject, should be of interest to geometers ... and be accessible to graduate students interested in these topics as well."

-- Bulletin of the London Mathematical Society

Table of Contents

• Introduction
• Fibering Kähler manifolds and Kähler groups
• The de Rham fundamental group
• \(L^2\)-cohomology of Kähler groups
• Existence theorems for harmonic maps
• Applications of harmonic maps
• Non-Abelian Hodge theory
• Positive results for infinite groups
• Pro group theory (Appendix A)
• A glossary of Hodge theory (Appendix B)
• Bibliography
• Index