Lectures on resolution of singularities.

*(English)*Zbl 1113.14013
Annals of Mathematics Studies 166. Princeton, NJ: Princeton University Press (ISBN 978-0-691-12923-5/pbk; 978-0-691-12922-8/hbk). vi, 208 p. (2007).

The existence of a resolution of singularities over a field of characteristic zero has been known since the famous work of H. Hironaka [Ann. Math. (2) 79, 109–203, 205–326 (1964; Zbl 0122.38603)]. Hironaka’s proof has been simplified during the last years. The actual proof given in the book covers only thirty pages and is based on the approach of J. Włodarczyk [J. Am. Math. Soc. 18, No. 4, 779–822 (2005; Zbl 1084.14018)]. The book is addressed to readers who are interested in the subject and want to understand one proof of resolution of singularities in characteristic \(0\). Chapter 1 is devoted to resolution of curve singularities. This can be used as a course for beginners in algebraic geometry because it is very elementary. Chapter 2 needs more technical background. It is about resolution of surface singularities including the Jungian method and the Albanese method using projections. The methods presented in Chapter 3 to prove the existence of resolution of singularities in general are again elementary. Several examples are given to motivate the approach and illustrate the proof. János Kollár succeeded in giving a very clear and understandable proof. It was very nice to read the book. It can be a good basis for a graduate course about this subject.

Reviewer: Gerhard Pfister (Kaiserslautern)