Skip to main content
main-content

Über dieses Buch

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.

Whereas Volume I is more elementary, the present Volume II is more at the research level and somewhat more specialized. Both volumes are also available as hardcover edition as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete".

Inhaltsverzeichnis

Frontmatter

Notation and Conventions

Notation and Conventions

Without Abstract
Robert Lazarsfeld

Positivity for Vector Bundles

Frontmatter

6. Ample and Nef Vector Bundles

Abstract
This chapter is devoted to the basic theory of ample and nef vector bundles. We start in Section 6.1 with the “classical” material from [274]. In Section 6.2 we develop a formalism for twisting bundles by Q-divisor classes, which is used to study nefness. The development parallels - and for the most part reduces to - the corresponding theory for line bundles. The next two sections constitute the heart of the chapter. In the extended Section 6.3 we present numerous examples of positive bundles arising “in nature,” as well as some methods of construction. Finally, we study in Section 6.4 the situation on curves, where there is a close connection between amplitude and stability: following Gieseker [224] one obtains along the way an elementary proof of the tensorial properties of semistability for bundles on curves.
Robert Lazarsfeld

7. Geometric Properties of Ample Vector Bundles

Abstract
This chapter is devoted to some geometric properties of positive vector bundles. We start in Section 7.1 with analogues and extensions of the theorems of Lefschetz and Barth. The next section concerns degeneracy loci: we prove that under suitable positivity hypotheses a map of bundles must drop rank (on a connected locus) whenever it is dimensionally predicted to do so, and give some applications. Finally, in the last section we briefly take up vanishing theorems for ample bundles.
Robert Lazarsfeld

8. Numerical Properties of Ample Bundles

Abstract
This chapter deals with the numerical properties of ample vector bundles. The theme is that amplitude implies the positivity of various intersection and Chern numbers.
Robert Lazarsfeld

Multiplier Ideals and Their Applications

Frontmatter

9. Multiplier Ideal Sheaves

Abstract
This chapter is devoted to the definition and basic properties of multiplier ideals.
Robert Lazarsfeld

10. Some Applications of Multiplier Ideals

Abstract
The machinery developed in the previous chapter already has many substantial applications, and we present a number of these here. We start in the first section with several results involving singularities of divisors. In Section 10.2 we prove Matsusaka’s theorem following the approach of Siu and Demailly. The next section is devoted to a theorem of Nakamaye describing the base loci of big and nef divisors. Results and conjectures concerning global generation of adjoint linear series occupy Section 10.4, where in particular we prove the theorem of Angehrn and Siu. Finally, in Section 10.5 we discuss applications of Skoda’s theorem to the effective Nullstellensatz.
Robert Lazarsfeld

11. Asymptotic Constructions

Abstract
As we have suggested on a number of occasions, an important use of multiplier ideals is to make it possible to apply vanishing theorems for Q-divisors without first passing to a normal crossing situation. While this can be extremely valuable, in many cases it constitutes mainly a conceptual and technical simplification of the direct approach: several of the theorems presented in Chapter 10, for example, were originally proven without the language of multiplier ideals.
Robert Lazarsfeld

Backmatter

Weitere Informationen