Review of Hodge theory and algebraic cycles

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Exact Sequence ◽  
Hodge Theory ◽  
Algebraic Cycles ◽  
Chow Groups ◽  
Hodge Conjecture ◽  
Hodge Structures ◽  
Mixed Hodge Structures ◽  
Generalized Hodge Conjecture ◽  
Cycle Classes

This chapter provides the background for the studies to be undertaken in succeeding chapters. It reviews Chow groups, correspondences and motives on the purely algebraic side, cycle classes, and (mixed) Hodge structures on the algebraic–topological side. Emphasis is placed on the notion of coniveau and the generalized Hodge conjecture which states the equality of geometric and Hodge coniveau. The chapter first follows the construction of Chow groups, the application of the localization exact sequence, the functoriality and motives of Chow groups, and cycle classes. It then turns to Hodge structures; pursuing related topics such as polarization, Hodge classes, standard conjectures, mixed Hodge structures, and Hodge coniveau.

Hodge Theory (MN-49)

2017 ◽  
Author(s):  
Eduardo Cattani ◽  
Fouad El Zein ◽  
Phillip A. Griffiths ◽  
Lê Dung Tráng
Keyword(s):  
Summer School ◽  
Hodge Structure ◽  
Hodge Theory ◽  
Algebraic Cycles ◽  
Mixed Hodge Structure ◽  
Hodge Structures ◽  
De Rham ◽  
New Treatment ◽  
Variations Of Hodge Structure ◽  
Mixed Hodge Structures

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.

scholarly journals THE KUGA–SATAKE CONSTRUCTION UNDER DEGENERATION

2019 ◽  
Vol 19 (6) ◽  
pp. 2165-2182
Author(s):  
Stefan Schreieder ◽  
Andrey Soldatenkov
Keyword(s):  
Abelian Varieties ◽  
Hodge Theory ◽  
Hodge Structures ◽  
Mixed Hodge Structures

We extend the Kuga–Satake construction to the case of limit mixed Hodge structures of K3 type. We use this to study the geometry and Hodge theory of degenerations of Kuga–Satake abelian varieties associated with polarized variations of K3 type Hodge structures over the punctured disc.

scholarly journals Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes

2014 ◽  
Vol 66 (3) ◽  
pp. 505-524 ◽  
Author(s):  
Donu Arapura
Keyword(s):  
Projective Space ◽  
Prime Number ◽  
Hyperplane Arrangement ◽  
Hodge Theory ◽  
Cyclic Cover ◽  
Hodge Conjecture ◽  
Cyclic Covers ◽  
Generalized Hodge Conjecture ◽  
Ramification Locus

AbstractSuppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D and that U is the complement of the ramification locus in Y. The first theorem in this paper implies that the Beilinson-Hodge conjecture holds for U if certain multiplicities of D are coprime to the degree of the cover. For instance, this applies when D is reduced with normal crossings. The second theorem shows that when D has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of Y. The last section contains some partial extensions to more general nonabelian covers.

Chow groups of large coniveau complete intersections

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Chow Groups ◽  
Small Dimension ◽  
Complete Intersections ◽  
Powerful Method ◽  
Hodge Conjecture ◽  
The Right ◽  
Generalized Hodge Conjecture

This chapter first describes how to compute the Hodge coniveau of complete intersections. It then explains a strategy to attack the generalized Hodge conjecture for complete intersections of coniveau 2. The guiding idea is that although the powerful method of the decomposition of the diagonal suggests that computing Chow groups of small dimension is the right way to solve the generalized Hodge conjecture, it might be better to invert the logic and try to compute the geometric coniveau directly. And indeed, this chapter culminates with the proof of the fact that for very general complete intersections, the generalized Hodge conjecture implies the generalized Bloch conjecture.

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

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.

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