Chapter

Algebraic Constructions of Toroidal Compactifications

Kai-Wen Lan

in Arithmetic Compactifications of PEL-Type Shimura Varieties

Published by Princeton University Press

Published in print March 2013 | ISBN: 9780691156545
Published online October 2017 | e-ISBN: 9781400846016
Algebraic Constructions of Toroidal Compactifications

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This chapter explains the algebraic construction of toroidal compactifications. For this purpose the chapter utilizes the theory of toroidal embeddings for torsors under groups of multiplicative type. Based on this theory, the chapter begins the general construction of local charts on which degeneration data for PEL structures are tautologically associated. The next important step is the description of good formal models, and good algebraic models approximating them. The correct formulation of necessary properties and the actual construction of these good algebraic models are the key to the gluing process in the étale topology. In particular, this includes the comparison of local structures using certain Kodaira–Spencer morphisms. As a result of gluing, this chapter obtains the arithmetic toroidal compactifications in the category of algebraic stacks. The chapter is concluded by a study of Hecke actions on towers of arithmetic toroidal compactifications.

Keywords: toroidal compactifications; toroidal embeddings; good algebraic models; étale topology; Kodaira–Spencer morphisms; algebraic stacks; Hecke actions; arithmetic toroidal compactifications

Chapter.  46005 words. 

Subjects: Geometry

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