Lectures On K Hler Manifolds
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Lectures on Kähler Manifolds
Author: Werner Ballmann
language: en
Publisher: European Mathematical Society
Release Date: 2006
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.
Lectures on Kähler Groups
Author: Pierre Py
language: en
Publisher: Princeton University Press
Release Date: 2025-03-25
"A natural question that sits at the nexus of algebraic geometry, differential geometry, and geometric group theory is: which groups can be realized as fundamental groups of compact Kähler manifolds, called "Kähler groups"? Roughly speaking, the fundamental group of a manifold measures the number of "holes." Many restrictions are known, and many examples are known; but mathematicians are far from having a precise conjecture about which groups are Kähler. The question serves as a fruitful connection between several major areas of geometry and complex analysis. Py's book is an up-to-date pedagogical survey of the central theorems and methods for the study of Kähler groups including, where illuminating, detailed proofs. It includes results of Gromov, Schoen, Napier, Ramachandran, Corlette, Simpson, Delzant, Arapura, and Nori. The charm of the subject is that different methods yield information of different flavors, and the challenge is to draw these threads together. This book leans toward geometric group theory, but it gives a coherent account of great value to anyone interested in Kähler groups - and in Kähler manifolds more broadly. The emphasis is on unity and cross-fertilization among approaches"--
Canonical Metrics in Kähler Geometry
There has been fundamental progress in complex differential geometry in the last two decades. For one, The uniformization theory of canonical Kähler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. This monograph gives an introduction to the theory of canonical Kähler metrics on complex manifolds. It also presents some advanced topics not easily found elsewhere.