Asymptotic invariants of line bundles
Abstract
Let X be a smooth complex projective variety of dimension d. It is classical that ample line bundles on X satisfy many beautiful geometric, cohomological, and numerical properties that render their behavior particularly tractable. By contrast, examples due to Cutkosky and others have led to the common impression that the linear series associated to nonample divisors are in general mired in pathology. However starting with fundamental work of Fujita, Nakayama, and Tsuji, it has recently become apparent that arbitrary effective (or "big") divisors in fact display a surprising number of properties analogous to those of ample line bundles. The key is to study the properties in question from an asymptotic perspective. At the same time, many interesting questions and open problems remain. The purpose of the present expository note is to give an invitation to this circle of ideas. In the hope that this informal overview might serve as a jumping off point for the technical literature in the area, we sketch many examples but provide no proofs. We focus on one particular invariant  the "volume" of a line bundle  that measures the rate of growth of the number of sections of powers of the bundle in question.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 arXiv:
 arXiv:math/0505054
 Bibcode:
 2005math......5054E
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables;
 Mathematics  Differential Geometry
 EPrint:
 Informal expository overview, 20 pages