The lecture notes under review, based on the author’s course at the 2000 Trieste school on vanishing theorems and effective results in algebraic geometry, are an extended version of the author’s CIME lectures [in: Transcendental methods in algebraic geometry. Lect. 3rd sess. CIME , Cetraro, Italy, 1994. Lect. Notes Math. 1646, 1–97 (1996; Zbl 0883.14005)]. They give a comprehensive survey on the application of analytic methods to algebraic geometry, especially to vanishing theorems. Aimed at non-specialists (in the author’s words, they are “written with the idea of serving as an analytic toolbox for algebraic geometers”), they provide a lot of historical and introductory material on the subject, as well as very advanced topics and recent developments.
For a detailed account see the review of [loc. cit.]. The main changes are due to the incorporation of Y. T. Siu’s new result on the deformation invariance of plurigenera of varieties of general type [Invent. Math. 134, No.3, 661-673 (1998; Zbl 0955.32017)].
For the entire collection see [Zbl 0986.00053].