Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series

by R.K. Lazarsfeld

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.

Genres: Mathematics

Hardcover

First published January 1, 2004

Reviews

[Andrew · Rating 3 out of 5]

The results and subject have a very classical character -- they are rather "hands on", so the proofs are readable but require effort. Here and there Lazarsfeld shows nice connections to different subjects, which are often more analytic in character, such as differential geometry and Morse theory. The highlight of the book might be the lucid treatment of Kodaira's vanishing and its variants, where the result is strong for obvious reasons. Some results seem more esoteric, and at times the taxonomy or concepts (like Seshadri constants) feel a bit scattered.