Bøger af Janos Kollar

The aim of this monograph is to give a comprehensive introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on them. The book contains many exercises which makes it especially suitable as a graduate level textbook.

This book gives the first complete treatment of the moduli theory of varieties of dimension larger than one, aimed at researchers and graduate students in algebraic geometry and related areas. The first chapter provides a historical introduction to the subject, while later chapters provide all necessary background material.

Exploring the connections between arithmetic and geometric properties of algebraic varieties has been the object of much fruitful study for a long time, especially in the case of curves. The aim of the Summer School and Conference on "e;Higher Dimensional Varieties and Rational Points"e; held in Budapest, Hungary during September 2001 was to bring together students and experts from the arithmetic and geometric sides of algebraic geometry in order to get a better understanding of the current problems, interactions and advances in higher dimension. The lecture series and conference lectures assembled in this volume give a comprehensive introduction to students and researchers in algebraic geometry and in related fields to the main ideas of this rapidly developing area.

This book provides the first comprehensive introduction to the circle of ideas developed around Mori's program, the prerequisites being only a basic knowledge of the algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.

This book gives a comprehensive treatment of the singularities that appear in the minimal model program and in the moduli problem for varieties. The study of these singularities and the development of Mori's program have been deeply intertwined. Early work on minimal models relied on detailed study of terminal and canonical singularities but many later results on log terminal singularities were obtained as consequences of the minimal model program. Recent work on the abundance conjecture and on moduli of varieties of general type relies on subtle properties of log canonical singularities and conversely, the sharpest theorems about these singularities use newly developed special cases of the abundance problem. This book untangles these interwoven threads, presenting a self-contained and complete theory of these singularities, including many previously unpublished results.

Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, Janos Kollar provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollar goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.