This two volume work on Positivity in Algebraic Geometry contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments.

Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II.

This two volume work on “Positivity in Algebraic Geometry” contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments.Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II. Both volumes are also available as hardcover editions as Vols. 48 and 49 in the series “Ergebnisse der Mathematik und ihrer Grenzgebiete”.

Review
From the reviews:

"This a ] book offers a comprehensive, up-to-date account on ampleness and positivity in complex algebraic geometry. a ] The book contains a wealth of material and aims at readers with a certain overview in complex algebraic geometry. a ] the text never gets bogged down in technicalities, and is a pleasure to read. The presentation nicely reveals historical developments and mathematical interplay between various results. a ] A fine book indeed." (Stefan SchrAer, Zentralblatt MATH, Vol. 1066, 2005)