# Arithmetic Compactifications of PEL-Type Shimura Varieties

Kai-Wen Lan
Pages: 588
https://www.jstor.org/stable/j.ctt24hpwv

1. Front Matter
(pp. i-iv)
(pp. v-x)
3. Acknowledgments
(pp. xi-xii)
4. Introduction
(pp. xiii-xxvi)

The main subject of this book is the construction of good compactifications of PEL-type Shimura varieties in mixed characteristics. In this introduction, we will summarize the history of such a construction, clarify its aims, and describe what has been achieved (by others and by us). After this, we will also give an outline of the topics in each chapter of the book.

To avoid introducing a heavy load of notation and definitions (which is perhaps typical for this subject), we shall not attempt to give any precise mathematical statement here. (We shall try not to introduce general non-PEL-type Shimura varieties...

5. Chapter One Definition of Moduli Problems
(pp. 1-90)

In this chapter, we give the definition of the moduli problems providing integral models of PEL-type Shimura varieties that we will compactify.

Just to make sure that potential logical problems do not arise in our use of categories, we assume that a pertinent choice of auniversehas been made (see Section A.1.1 for more details). This is harmless for our study, and we shall not mention it again in our work.

The main objective in this chapter is to state Definition 1.4.1.4 with justifications. In order to explain the relation between our definition and those in the literature, we...

6. Chapter Two Representability of Moduli Problems
(pp. 91-142)

In this chapter, let us assume the same setting as in Section 1.4. Let us fix a choice of an open compact subgroup$\mathcal{H} \subset G({\widehat\mathbb{Z}^\square })$.

Our main objective is to prove Theorem 1.4.1.11, with Proposition 2.3.5.2 as a by-product. Technical results worth noting are Proposition 2.1.6.8 and Corollary 2.2.4.12. The proof of Theorem 1.4.1.11 is carried out by verifying Artin’s criterion in Section 2.3.4 (see, in particular, Theorems B.3.7, B.3.9, and B.3.11). For readers who might have wondered, let us make it clear that we will not need Condition 1.4.3.10 in this chapter.

Let us outline the strategy of...

7. Chapter Three Structures of Semi-Abelian Schemes
(pp. 143-174)

In this chapter, we review notions that are of fundamental importance in the study of degeneration of abelian varieties. The main objective is to understand the statement and the proof of the theory of degeneration data, to be presented in Chapter 4. Our main references for these will be [43], [61], and in particular, [103].

Technical results worth noting are Theorem 3.1.3.3, Propositions 3.1.5.1, 3.3.1.5, 3.3.1.7, and 3.3.3.6, Theorems 3.4.2.4 and 3.4.3.2, Lemma 3.4.3.1, and Proposition 3.4.4.1.

Definition 3.1.1.1 ([43, IX, 1.1]).A group(scheme)of multiplicative type over a scheme$S$is a commutative group scheme over$S$that...

8. Chapter Four Theory of Degeneration for Polarized Abelian Schemes
(pp. 175-284)

In this chapter we reproduce the theory of degeneration data for abelian varieties, following Mumford’s original paper [105] and the first three chapters of Faltings and Chai’s monograph [46]. Although there is essentially nothing new in this chapter, some modifications have been introduced to make the statements compatible with our understanding of the proofs. Moreover, since [105] and [46] have supplied full details only in the completely degenerate case, we will balance the literature by avoiding the special case and treating all cases equally.

The main objective in this chapter will be to state and prove Theorems 4.2.1.14, 4.4.16, and...

9. Chapter Five Degeneration Data for Additional Structures
(pp. 285-372)

In this chapter, we supply a theory of degeneration for abelian varieties with additional structures of PEL-type, based on the theory developed in Chapter 4. The running assumptions and notation in Chapter 4 (see 4.1) will be continued in this chapter without further remark. Moreover, we fix some choices of$B$,$O$, and$(L,\langle \cdot , \cdot \rangle ,h)$as in Section 1.4.

The main objective is to state and prove Theorem 5.3.1.19, with Theorem 5.3.3.1 and the notion of cusp labels in Section 5.4 as by-products. Technical results worth noting are Propositions 5.1.2.4, 5.2.2.23, 5.2.3.3, and 5.2.3.9, Theorem 5.2.3.14, and Proposition 5.4.3.8. The preparation...

10. Chapter Six Algebraic Constructions of Toroidal Compactifications
(pp. 373-446)

We will generalize the techniques in [46] and construct the toroidal compactifications of the moduli problems we considered in Chapter 1.

The main objective of this chapter is to state and prove Theorem 6.4.1, with by-products concerning Hecke actions given in Sections 6.4.2 and 6.4.3. Technical results worth noting are Propositions 6.2.2.4, 6.2.3.2, 6.2.3.18, and 6.2.5.18, Lemma 6.3.1.11, and Propositions 6.3.2.6, 6.3.2.10, 6.3.3.5, 6.3.3.13, and 6.3.3.17.

Let$H$be a group of multiplicative type of finite type over$S$, so that its character group$\underline X (H) = {\underline {{\rm{Hom}}} _S}(H,{G_{m,S}})$is an étale sheaf of finitely generated commutative groups.

DEFINITION 6.1.1.1.The cocharacter group$\underline X {(H)^{\rm{v}}}$...

11. Chapter Seven Algebraic Constructions of Minimal Compactifications
(pp. 447-486)

In this chapter we explain the construction of arithmetic toroidal compactifications and several other useful results as a by-product of the construction of arithmetic toroidal compactifications.

Although all results to be stated have their analytic analogues over the complex numbers, the analytic techniques in [18] and [17] do not carry over naively. We need the arithmetic toroidal compactifications and the positivity of Hodge invertible sheaves (to be reviewed in Section 7.2.1, based on the theory of theta constants) to establish the finite generation of certain natural sheaves of graded algebras. This should be considered as the main technical input of...

12. Appendix A Algebraic Spaces and Algebraic Stacks
(pp. 487-518)
13. Appendix B Deformations and Artin’s Criterion
(pp. 519-534)
14. Bibliography
(pp. 535-544)
15. Index
(pp. 545-561)