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Introduction to Variations of Hodge Structure
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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.
Title: Introduction to Variations of Hodge Structure
Description:
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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