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Arithmetic compactifications of PEL-type Shimura varieties. (English) Zbl 1284.14004

London Mathematical Society Monographs Series 36. Princeton, NJ: Princeton University Press (ISBN 978-0-691-15654-5/hbk; 978-1-4008-4601-6/ebook). xxiii, 561 p. (2013).
The topic of this book is the construction of compactifications of integral models of Shimura varieties of PEL type with good reduction. Shimura varieties of PEL type are moduli spaces of abelian varieties with additional structure: polarization, endomorphism and level structure. The most classical example are modular curves, which parameterize elliptic curves. Considered over the complex numbers, these modular curves are Riemann surfaces which can be compactified by adding a finite number of points, the so-called cusps. In fact, the modular curves and their compactifications are defined over a number field. For general Shimura varieties of PEL type, again over the number field they are naturally defined over (the reflex field), several compactifications have been known for decades, notably the minimal, or Satake-Baily-Borel, compactification, and toroidal compactifications, and are an important cornerstone of the theory of Shimura varieties.
By extending the moduli problem of abelian varieties, there is a natural way to define integral models of these Shimura varieties (over the completion of the local ring of the reflex field at a finite prime, say). Then naturally the question arises, whether there are corresponding compactifications of these integral models. Such compactifications have been known for quite some time for Shimura varieties attached to “small” groups (e.g., the cases of modular curves [P. Deligne and M. Rapoport, in: Modular Functions of one Variable II, Proc. internat. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349, 143–316 (1973; Zbl 0281.14010)] and of Hilbert modular varieties [M. Rapoport, Compos. Math. 36, 255–335 (1978; Zbl 0386.14006)]).
G. Faltings and C.-L. Chai [Degeneration of abelian varieties. Berlin etc.: Springer-Verlag (1990; Zbl 0744.14031)] have constructed minimal and toroidal compactifications of the moduli space of principally polarized abelian varieties (which is attached to the group \(\mathrm{GSp}_{2g}\) of symplectic similitudes; in this case, there is no “endomorphism structure” imposed). A key ingredient of their construction is the work of Mumford about degenerations of abelian varieties. Using similar techniques, M. J. Larsen [in: The zeta functions of Picard modular surfaces. CRM Workshop, Montreal / Can. 1988, 31–45 (1992; Zbl 0760.14009)] has treated the case of Picard modular surfaces.
The book at hand constructs such arithmetic compactifications in the general PEL case with good reduction, i.e., for smooth integral models of arbitrary Shimura varieties of PEL type, by generalizing the methods of Faltings and Chai in order to take into account the additional structure imposed on the abelian varieties in the corresponding moduli problem. First, arithmetic toroidal compactifications are constructed (Theorem 6.4.1) – they depend on the choice of certain combinatorial data which makes them non-canonical; on the other hand, they are smooth and the boundary is a divisor with normal crossings. Second, the arithmetic minimal compactification is constructed (Theorem 7.2.4.1); it is normal but not in general smooth. For more precise statements we refer to the book itself; the statement alone of Theorem 6.4.1 extends over more than 2 pages.
While the methods are based on the work of Chai and Faltings (who make a remark to the effect that their method also works in the general PEL case), several issues arise which are not visible in the Siegel case.
The content of the individual chapters is as follows: The book starts with an introduction which recalls the characteristic \(0\) theory as well as previous results in the arithmetic setting, without going into the technical details. In the first chapter, some preliminaries are treated, and the relevant moduli problems are defined. In the second chapter, the represenatbility of proved; this is based on Artin’s criterion for the representability by algebraic stacks. In the third chapter, semi-abelian schemes are discussed. In the fourth and fifth chapters, the theory of degeneration of abelian varieties is investigated – first for principally polarized abelian schemes, and later for abelian schemes with additional structure. In Chapter 6, arithmetic toroidal compactifications are constructed, and in the final Chapter 7, arithmetic minimal compactifications are handled. Furthermore, the book has two appendices; one on algebraic spaces and algebraic stacks, and one on deformation theory and Artin’s criterion.
The book is well written. In the introduction, the author states that “It is our belief that it is the right of the reader, but not the author, to skip details”, and he sticks to this maxim. The book is based on the author’s PhD thesis (Harvard University, 2008). A list of errata is available on the author’s web page.
Finally, let us mention some of the more recent developments: B. Stroh [Bull. Soc. Math. Fr. 138, No. 2, 259–315 (2010; Zbl 1203.14048)]; Ann. Inst. Fourier 60, No. 3, 1035–1055 (2010; Zbl 1220.14028)] has constructed arithmetic compactifications in certain cases of bad reduction; the notion of degeneration data is rephrased in terms of “Mumford 1-motives”. K. M. Pera [“Toroidal compactifications of integral models of Shimura varieties of Hodge type”, arXiv:1211.1731] has constructed arithmetic toroidal and minimal compactifications for Shimura varieties of Hodge type, a larger class than PEL type Shimura varieties; this relies on the results of Faltings and Chai and result of K.-W. Lan [J. Reine Angew. Math. 664, 163–228 (2012; Zbl 1242.14022)] on the compatibility of algebraic and analytic toroidal compactifications.
In the paper [K.-W. Lan, “Compactifications of PEL-type Shimura varieties and Kuga families with ordinary loci”, preprint, http://math.umn.edu/~kwlan/articles/cpt-ram-ord.pdf], the author constructs arithmetic partial toroidal and minimal compactifications for ordinary loci in PEL Shimura varieties which need not have good reduction.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
14K10 Algebraic moduli of abelian varieties, classification
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