Positivity in Algebraic Geometry I

For the most part we follow generally accepted notation, as in [280]. We do however adopt a few specific conventions:

Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of projective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated behavior of linear series defined by line bundles that may not be ample. This led to particularly profound insights in the case of surfaces [623]. In yet another direction, the classical theorems of Lefschetz comparing the topology of a variety with that of a hyperplane section were understood from new points of view, and developed in surprising ways in [258] and [30].

This chapter contains the basic theory of positivity for line bundles and divisors on a projective algebraic variety.

This chapter presents some of the basic facts and examples concerning linear series on a projective variety X. The theme is to use the theory developed in the previous chapter to study the complete linear series |mD| associated to a divisor D on X that may not be ample or nef.

This chapter focuses on a number of results that in one way or another express geometric consequences of positivity. In the first section we prove the Lefschetz hyperplane theorem following the Morse-theoretic approach of Andreotti-Frankel [7]. Section 3.2 deals with subvarieties of small codimension in projective space: we prove Barth’s theorem and give an introduction to the conjectures of Hartshorne. The connectedness theorems of Bertini and Fulton-Hansen are established in Section 3.3, while applications of the Fulton-Hansen theorem occupy Section 3.4. Like the results from 3.2, these reflect the positivity of projective space itself. Finally, some extensions and variants are presented in Section 3.5.

This chapter is devoted to the basic vanishing theorems for integral divisors. The prototype is Kodaira’s result that if A is an ample divisor on a smooth complex projective variety X, then O _X (K _X + A) has vanishing higher cohomology. An important extension, due to Kawamata and Viehweg, asserts that the statement remains true assuming only that A is nef and big. These and related vanishing theorems have a vast number of applications to questions central to the focus of this book.

In this chapter we discuss local positivity. The theory originates with Demailly’s idea for quantifying how much of the positivity of an ample line bundle can be localized at a given point of a variety. The picture turns out to be considerably richer and more structured than one might expect at first glance, although the existing numerical results are (presumably!) not optimal.