The first L2-Betti number of certain homogeneous spaces, including classifying spaces for variations of Hodge structures

2004 ◽  
Vol 249 (4) ◽  
pp. 817-828
Author(s):  
J. Jost ◽  
Y. L. Xin
Keyword(s):  
Betti Number ◽  
Homogeneous Spaces ◽  
Classifying Spaces ◽  
Hodge Structures

scholarly journals Classifying spaces of degenerating mixed Hodge structures, IV: The fundamental diagram

2018 ◽  
Vol 58 (2) ◽  
pp. 289-426
Author(s):  
Kazuya Kato ◽  
Chikara Nakayama ◽  
Sampei Usui
Keyword(s):  
Classifying Spaces ◽  
Fundamental Diagram ◽  
Hodge Structures ◽  
Mixed Hodge Structures

scholarly journals Classifying spaces of degenerating mixed Hodge structures, II: Spaces of $\operatorname{SL}(2)$ -orbits

2011 ◽  
Vol 51 (1) ◽  
pp. 149-261 ◽  
Author(s):  
Kazuya Kato ◽  
Chikara Nakayama ◽  
Sampei Usui
Keyword(s):  
Classifying Spaces ◽  
Hodge Structures ◽  
Mixed Hodge Structures

scholarly journals Rational cohomologies of classifying spaces for homogeneous spaces of small rank

2016 ◽  
Vol 5 (4) ◽  
pp. 225-237
Author(s):  
Hirokazu Nishinobu ◽  
Toshihiro Yamaguchi
Keyword(s):  
Homogeneous Spaces ◽  
Classifying Spaces ◽  
Small Rank

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.

Maximal Homotopy Lie Subgroups of Maximal Rank

1988 ◽  
Vol 40 (04) ◽  
pp. 769-787 ◽  
Author(s):  
John A. Frohliger
Keyword(s):  
Lie Group ◽  
Homogeneous Spaces ◽  
Maximal Rank ◽  
Homotopy Equivalent ◽  
Classifying Spaces ◽  
Loop Spaces ◽  
Connected Subgroup ◽  
Image Position ◽  
Lie Subgroup ◽  
Connected Lie Group

Let G be a compact connected Lie group with H a connected subgroup of maximal rank. Suppose there exists a compact connected Lie subgroup K with H ⊂ K ⊂ G. Then there exists a smooth fiber bundle G/H → G/K with K/H as the fiber. (See for example [13].) This can be incorporated into a diagram involving the classifying spaces as follows: (1) Here π, ϕ, ϕ 1 and ϕ 2 denote fibrations. We also know that the homogeneous spaces and the Lie groups, which are homotopy equivalent to the loop spaces of their respective classifying spaces, are homotopy equivalent to connected finite complexes. Now suppose H is a maximal subgroup. Can there still exist spaces, which we will call BK, K/H, and G/K, and fibrations so that diagram (1) is still valid? This paper will show that in many cases either G/K or K/H will be homo topically trivial.

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
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