Description
Book SynopsisThe aim of this book is to study various geometric properties and algebraic invariants of smooth projective varieties with infinite fundamental groups. This approach allows for much interplay between methods of algebraic geometry, complex analysis, the theory of harmonic maps, and topology. Making systematic use of Shafarevich maps, a concept previ
Trade Review"I believe Kollar has written a powerful book, and not coincidentally, a fairly demanding one. However, the explanations are clear, and I think that more than half of it would be accessible to anyone who has mastered the basics of complex algebraic geometry... There are a number of sections ... which could be read independently of the rest of the book and would be of interest to almost any algebraic geometer. But to read only those sections would be a shame, because one would be missing a great deal of wonderful mathematics."--Bulletin of the American Mathematical Society
Table of ContentsPrefaceAcknowledgmentsIntroduction3Ch. 1Lefschetz-Type Theorems for [pi][subscript 1]19Ch. 2Families of Algebraic Cycles27Ch. 3Shafarevich Maps and Variants36Ch. 4The Fundamental Group and the Classification of Algebraic Varieties49Ch. 5The Method of Poincare59Ch. 6The Method of Atiyah71Ch. 7Subjectivity of the Poincare Map81Ch. 8Ball Quotients92Ch. 9The Kodaira Vanishing Theorem105Ch. 10Generalizations of the Kodaira Vanishing Theorem115Ch. 11Vanishing of L[superscript 2]-Cohomologies127Ch. 12Rational Singularities and Hodge Theory133Ch. 13The Method of Gromov141Ch. 14Nonvanishing Theorems151Ch. 15Plurigenera in Etale Covers161Ch. 16Existence of Automorphic Forms167Ch. 17Applications to Abelian Varieties175Ch. 18Open Problems and Further Remarks183References191Index201
