Analytic methods in algebraic geometry.

*(English)*Zbl 1271.14001
Surveys of Modern Mathematics 1. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-234-3/pbk). 231 p. (2012).

The book under review is based upon a series of lectures given by Jean-Pierre Demailly at the Park City Mathematics Institute in 2008, it was partly published in [IAS/Park City Mathematics Series 17, 295–370 (2010; Zbl 1222.32043)]. The main aim is to give a presentation of analytic techniques especially as it relates to positivity of vector bundles.

The book begins by briefly reviewing the concepts of sheaf cohomology, plurisubharmonic functions, currents and other topics. In Chapter 4, it moves on to the Bochner technique and applications to the Akizuki-Nakano-Kodaira vanishing theorem. Chapter 5 covers \(L^2\) estimates and multiplier ideal sheaves. Chapter 6 covers pseudo effective and nef line bundles and Kawamata-Viehweg vanishing and applications. The next chapter covers applications of these ideas, results towards Fujita’s conjecture, such as Reider’s Theorem and work of Siu. Chapter 8 covers Holomorphic Morse inequalities as introduced by Demailly. Chapter 9 covers effective versions of Matsusaka’s big theorem. Chapter 11 covers the question of surjectivity of global sections for maps of vector bundles and an application to the Briancon-Skoda Theorem. Chapter 12 covers the Ohsawa-Takegoshi \(L^2\) Extension Theorem and Skoda’s division theorem. Chapter 13 focuses on approximation of positive currents and plurisubharmoic functions and relations to the Hodge Conjecture. The remainder of the book (chapters 15 through 20), cover various topics in higher dimensional algebraic geometry such as Subadditivity of multiplier ideals, invariance of plurigenera, the Kähler cone and Pseudo-efective cone, abundance, and many other topics.

The book begins by briefly reviewing the concepts of sheaf cohomology, plurisubharmonic functions, currents and other topics. In Chapter 4, it moves on to the Bochner technique and applications to the Akizuki-Nakano-Kodaira vanishing theorem. Chapter 5 covers \(L^2\) estimates and multiplier ideal sheaves. Chapter 6 covers pseudo effective and nef line bundles and Kawamata-Viehweg vanishing and applications. The next chapter covers applications of these ideas, results towards Fujita’s conjecture, such as Reider’s Theorem and work of Siu. Chapter 8 covers Holomorphic Morse inequalities as introduced by Demailly. Chapter 9 covers effective versions of Matsusaka’s big theorem. Chapter 11 covers the question of surjectivity of global sections for maps of vector bundles and an application to the Briancon-Skoda Theorem. Chapter 12 covers the Ohsawa-Takegoshi \(L^2\) Extension Theorem and Skoda’s division theorem. Chapter 13 focuses on approximation of positive currents and plurisubharmoic functions and relations to the Hodge Conjecture. The remainder of the book (chapters 15 through 20), cover various topics in higher dimensional algebraic geometry such as Subadditivity of multiplier ideals, invariance of plurigenera, the Kähler cone and Pseudo-efective cone, abundance, and many other topics.

Reviewer: Karl Schwede (University Park)