Introduction to classification theory of algebraic varieties and compact complex spaces

AbstractFor a smooth manifold X equipped with a volume form, let 𝓛0 (X) be the Lie algebra of volume preserving smooth vector fields on X. Lichnerowicz proved that the abelianization of 𝓛0 (X) is a finite-dimensional vector space, and that its dimension depends only on the topology of X. In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties with trivial canonical bundle, which include certain algebraic surfaces and linear algebraic groups. The proofs are based on a remarkable result of Grothendieck on the cohomology of affine varieties, and some techniques that were introduced with the purpose of extending the Andersén–Lempert theory, which was originally developed for the complex spaces ℂn, to the larger class of Stein manifolds that satisfy the density property.

Download Full-text

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

10.23943/princeton/9780691160504.001.0001 ◽

2017 ◽

Author(s):

Claire Voisin

Keyword(s):

Recent Work ◽

Chow Groups ◽

Ring Structure ◽

Complete Intersections ◽

Algebraic Varieties ◽

Chow Ring ◽

Hodge Structures ◽

Chow Rings ◽

Geometric Methods ◽

The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Download Full-text

Branched Coverings and Minimal Free Resolution for Infinite-Dimensional Complex Spaces

Georgian Mathematical Journal ◽

10.1515/gmj.2003.37 ◽

2003 ◽

Vol 10 (1) ◽

pp. 37-43

Author(s):

E. Ballico

Keyword(s):

Projective Space ◽

Projective Spaces ◽

Free Resolution ◽

Complete Intersections ◽

Vanishing Theorem ◽

Sheaf Cohomology ◽

Minimal Free Resolution ◽

Branched Coverings ◽

Infinite Dimensional ◽

Complex Spaces

Abstract We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P(V ).

Download Full-text

ALGEBRAIC VARIETIES OVER FIELDS WITH DIFFERENTIATION

Mathematics of the USSR-Sbornik ◽

10.1070/sm1969v009n03abeh002054 ◽

1969 ◽

Vol 9 (3) ◽

pp. 389-413

Author(s):

Ju R Vaĭnberg

Keyword(s):

Algebraic Varieties

Download Full-text