The main theme of this two volume monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. The first volume in the set offers a systematic presentation of ideas connected with classical notions of linear series and ample divisors on a projective variety from a modern point of view. Individual chapters are devoted to the basic theory of positivity for line bundles and Castelnuovo-Mumford regularity, asymptotic geometry of linear systems, geometric properties of projective subvarieties of small codimension, vanishing theorems for divisors and the theory of local positivity.

The second volume contains the second and the third parts of the monograph. The main theme of the second part is positivity for vector bundles of higher ranks. It contains a systematic exposition of the theory (formal properties of ample and nef bundles, higher rank generalizations of the Lefschetz hyperplane theorem and the Kodaira vanishing theorem and numerical properties of ample bundles). The third part describes ideas and methods connected with multiplier ideals (Kawamta-Viehweg vanishing theorems for Q-divisors, multiplier ideals and their applications, and asymptotic multiplier ideals). The book is written for mathematicians interested in the modern development of algebraic geometry. A knowledge of basic notions is assumed (a standard introductory courses is sufficient).