Read e-book online Positivity in Algebraic Geometry II: Positivity for Vector PDF

By R.K. Lazarsfeld

ISBN-10: 3540225315

ISBN-13: 9783540225317

Quantity paintings containing a modern account on "Positivity in Algebraic Geometry".

Both volumes additionally to be had as hardcover variations as Vols. forty eight and forty nine within the sequence "Ergebnisse der Mathematik und ihrer Grenzgebiete".

A good buy of the cloth has no longer formerly seemed in booklet form.

Volume II is extra on the study level and a bit extra really good than quantity I.

Volume II features a survey of positivity for vector bundles, and strikes directly to a scientific improvement of the speculation of multiplier beliefs and their applications. 

Contains many concrete examples, purposes, and tips that could additional developments

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Extra resources for Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and Multiplier Ideals

Example text

Bogomolov’s construction). Let Y1 , . . , Ym be smooth projective varieties of dimension d ≥ 1, each having big cotangent bundle, 9 and let X ⊆ Y1 × . . × Ym be a general complete intersection of very ample divisors. Bogomolov proves that if d(m + 1) dim X ≤ , 2d + 1 then X has ample cotangent bundle. Bogomolov and Debarre deduce from this that there exists a projective variety X having ample cotangent bundle with the additional property that π1 (X) is any group arising as the fundamental group of a smooth projective variety: in particular, X can be simply connected.

3 Examples and Constructions In order to add substance to the general theory, we present in this section several examples and constructions of ample and nef vector bundles. Our hope is to convey a sense of some of the many settings in which positivity of bundles arises “in nature”, and to illustrate a few of the methods that have been used to detect and exploit it. In the first two subsections, we discuss the geometric consequences of positivity conditions on tangent, cotangent and normal bundles.

1] proves in the situation of the Proposition that the formal cohomology groups H i M , E are finite dimensional for all i < d. 5] that if in addition M is projective, then for every coherent sheaf F on M , the cohomology group H i M − X, F is finite dimensional whenever i ≥ n−d. There have been a number of attempts to find global geometric consequences of the amplitude of the normal bundle N = NX/M . 30). 5] concerning what one might expect in higher codimension: Hartshorne’s Conjecture A. If X ⊆ M is a smooth subvariety with ample normal bundle, then a sufficiently high multiple of [X] should move (as a cycle) in a large algebraic family.

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Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and Multiplier Ideals by R.K. Lazarsfeld


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