Characteristic classes for the classifying spaces of hodge structures

K-Theory ◽  
1987 ◽  
Vol 1 (3) ◽  
pp. 237-270 ◽  
Author(s):  
Ruth Charney ◽  
Ronnie Lee
Keyword(s):  
Characteristic Classes ◽  
Classifying Spaces ◽  
Hodge Structures

On the homotopy theory of sheaves of simplicial groupoids

1996 ◽  
Vol 120 (2) ◽  
pp. 263-290 ◽  
Author(s):  
André Joyal ◽  
Myles Tierney
Keyword(s):  
Homotopy Theory ◽  
Characteristic Classes ◽  
Spectral Sequences ◽  
Classifying Spaces ◽  
Sheaf Cohomology ◽  
Natural Context ◽  
Simplicial Sheaves ◽  
Cohomology Operations

The aim of this paper is to contribute to the foundations of homotopy theory for simplicial sheaves, as we believe this is the natural context for the development of non-abelian, as well as extraordinary, sheaf cohomology.In [11] we began constructing a theory of classifying spaces for sheaves of simplicial groupoids, and that study is continued here. Such a theory is essential for the development of basic tools such as Postnikov systems, Atiyah-Hirzebruch spectral sequences, characteristic classes, and cohomology operations in extraordinary cohomology of sheaves. Thus, in some sense, we are continuing the program initiated by Illusie[7], Brown[2], and Brown and Gersten[3], though our basic homotopy theory of simplicial sheaves is different from theirs. In fact, the homotopy theory we use is the global one of [10]. As a result, there is some similarity between our theory and the theory of Jardine[8], which is also partially based on [10]

Introduction to Variations of Hodge Structure

2017 ◽  
Author(s):  
Eduardo Cattani
Keyword(s):  
Asymptotic Behavior ◽  
Hodge Structure ◽  
Flat Connection ◽  
Classifying Spaces ◽  
Hodge Structures ◽  
Projective Varieties ◽  
Local Systems ◽  
Period Mapping ◽  
Variations Of Hodge Structure ◽  
Smooth Projective Varieties

This chapter emphasizes the theory of abstract variations of Hodge structure (VHS) and, in particular, their asymptotic behavior. It first studies the basic correspondence between local systems, representations of the fundamental group, and bundles with a flat connection. The chapter then turns to analytic families of smooth projective varieties, the Kodaira–Spencer map, Griffiths' period map, and a discussion of its main properties: holomorphicity and horizontality. These properties motivate the notion of an abstract VHS. Next, the chapter defines the classifying spaces for polarized Hodge structures and studies some of their basic properties. Finally, the chapter deals with the asymptotics of a period mapping with particular attention to Schmid's orbit theorems.

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