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Arithmetic Compactifications of PEL-Type Shimura Varieties$
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Kai-Wen Lan

Print publication date: 2013

Print ISBN-13: 9780691156545

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691156545.001.0001

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Structures of Semi-Abelian Schemes

Structures of Semi-Abelian Schemes

(p.143) Chapter Three Structures of Semi-Abelian Schemes
Arithmetic Compactifications of PEL-Type Shimura Varieties

Kai-Wen Lan

Princeton University Press

This chapter explains well-known notions important for the study of semi-abelian schemes. It first studies groups of multiplicative type and the torsors under them. A fundamental property of groups of multiplicative type is that they are rigid in the sense that they cannot be deformed. The chapter then turns to biextensions, cubical structures, semi-abelian schemes, Raynaud extensions, and certain dual objects for the last two notions extending the notion of dual abelian varieties. Such notions are, as this chapter shows, of fundamental importance in the study of the degeneration of abelian varieties. The main objective here is to understand the statement and the proof of the theory of degeneration data.

Keywords:   semi-abelian schemes, multiplicative type, torsors, biextensions, cubical structures, Raynaud extensions, dual objects, dual abelian varieties, degeneration data

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