Lectures on Kähler Groups

Pierre Py

• Publisher: Princeton University Press
• Year: 2025
• Print ISBN: 9780691247151
• eBook ISBN: 9780691247168
• Topic: Mathematics
• Subtopic: Algebraic Geometry
• Index: Mathematics

An introduction to the state of the art in the study of Kähler groups

This book gives an authoritative and up-to-date introduction to the study of fundamental groups of compact Kähler manifolds, known as Kähler groups. Approaching the subject from the perspective of a geometric group theorist, Pierre Py equips readers with the necessary background in both geometric group theory and Kähler geometry, covering topics such as the actions of Kähler groups on spaces of nonpositive curvature, the large-scale geometry of infinite covering spaces of compact Kähler manifolds, and the topology of level sets of pluriharmonic functions.

Presenting the most important results from the past three decades, the book provides graduate students and researchers with detailed original proofs of several central theorems, including Gromov and Schoen’s description of Kähler group actions on trees; the study of solvable quotients of Kähler groups following the works of Arapura, Beauville, Campana, Delzant, and Nori; and Napier and Ramachandran’s work characterizing covering spaces of compact Kähler manifolds having many ends. It also describes without proof many of the recent breakthroughs in the field.

Lectures on Kähler Groups also gives, in eight appendixes, detailed introductions to such topics as the study of ends of groups and spaces, groups acting on trees and Hilbert spaces, potential theory, and L² cohomology on Riemannian manifolds.

Table of contents

• Cover
• Contents
• Preface
• 1. Introduction
• 2. Riemann Surfaces and Orbifolds
• 3. The Fibration Problem: From Infinite to Finite Covers
• 4. The Theorem of Castelnuovo–de Franchis and Its Variants
• 5. Many Fibration Criteria
• 6. Kähler Groups and Trees
• 7. Covering Spaces of Compact Kähler Manifolds and Ends
• 8. Representations into PSL2(C)
• 9. Harmonic Maps to Locally Symmetric Spaces
• 10. Lattices and Groups of Hodge Type
• 11. The Bieri–Neumann–Strebel Invariant
• 12. The Green–Lazarsfeld Set
• 13. Actions on Real Trees
• Appendices
• Bibliography
• Index