Compactifying moduli spaces for abelian varieties
In the past five decades, beginning with the precise definition of the concept of moduli problems in algebraic geometry introduced by David Mumford in the early 1960s, both the construction of moduli spaces for certain classes of algebro-geometric objects and the study of their geometric properties have played a central role in algebraic geometry, arithmetic geometry, complex-analytic geometry, and increasingly also in mathematical physics. In attempting to study any established moduli space \(M\), one of the basic first steps is to find a suitable compactificatfon \(\overline M\) of \(M\). Preferably, such a compactification \(\overline M\) should have reasonably good geometric properties, its boundary \(\overline M\setminus M\) should be a manageable divisor in \(\overline M\), and the space \(\overline M\) should also have a reasonable moduli interpretation, with boundary points corresponding to degenerate objects. In this general context, a prominent example is provided by the well-known Deligne-Mumford compactification \(\overline M_{g,n}\) of the moduli space \(M_{g,n}\) of smooth projective curves of genus \(g\) with \(n\) marked points. The analoguous problem of compactifying the moduli space \(A_g\) of principally polarized abelian varieties of dimension \(g\) has a just as long and rich history, but the situation in this important special case has turned out to be much more difficult than for moduli spaces of curves. In fact, there are natural compactifications of \(A_g\), namely the so-called toroidal compactifications constructed by D. Mumford and his coworkers [cf.: \textit{A. Ash}, \textit{D. Mumford}, \textit{M. Rapoport} and \textit{Y. S. Tai}, Smooth compactification of locally symmetric varieties. Lie Groups: History, Frontiers and Applications. Vol. IV. Brookline, Mass.: Math Sci Press. IV (1975; Zbl 0334.14007)] over the ground field \(\mathbb{C}\) the variant proposed by \textit{Y. Namikawa} [Toroidal compactification of Siegel spaces. Lecture Notes in Mathematics. 812. Berlin-Heidelberg-New York: Springer-Verlag. VIII (1980; Zbl 0466.14011)], and the general compactifications over \(\mathbb{Z}\) by \textit{C.-L. Chai} and \textit{G. Faltings} [Degeneration of abelian varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 22. Berlin etc.: Springer-Verlag. xii (1990; Zbl 0744.14031)]. However, like the classical Satake compactification \(A^*_g\) of \(A_g\), these ompactifications have the crucial drawback of lacking a simple modular interpretation, that is, they do not represent any reasonable moduli problem in the category of algebraic schemes. In the monograph under review, the author develops an approach to compactifying \(A_g\) that does allow a reasonable modular interpretation. The starting point for his work is the more recent construction of a canonical compactification of \(A_g\) due to \textit{V. Alexeev} and \textit{I. Nakamura} [Tohoku Math. J., II. Ser. 51, No. 3, 399--420 (1999; Zbl 0989.14003)]. This construction is based on the observation that there is a moduli problem representable by a proper algebraic stack over \(\mathbb{Z}\), which contains \(A_g\) as a dense open subset of one of its irreducible components [cf.: \textit{V. Alexeev}, Ann. Math. (2) 155, No. 3, 611-708 (2002; Zbl 1052.14017)]. Now, using this recently established fact and applying the framework of logarithmic structures in the sense of Fontaine-Illusie-Kato [cf.: \textit{F. Kato}, Logarithmic structures of Fontaine-Illusie, in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191--224 (1989; Zbl 0776.14004)], the author describes, in the present monograph, a precise moduli problem defining the main component of Alexeev's stack, that is, the irreducible component containing the moduli space \(A_g\) of principally polarized abelian varieties over \(\mathbb{Z}\). Overall, the author's approach is utmost general and far-reaching. However, because of the many technical details involved with his constructions in full generality, the author has chosen to present the theory by means of three important, illustrating examples. More precisely, he explains three moduli problems of increasing technical difficulty, thereby illuminating the power of his general approach in a representative manner. As to the contents, the text is organized in six chapters, each of which is divided into several sections. After a very lucid and motivating introduction, Chapter 1 provides a brief primer on algebraic stacks. As stack-theoretic techniques are heavily used in the sequel, this chapter is meant as both a reference and a compilation of prerequisites. Chapter 2 summarizes the remaining necessary background material for the rest of the text, including the basics of abelian schemes and torsors, bi-extensions, logarithmic geometry, and a summary of V. Alexeev's above-mentioned results. Chapter 3 turns to the first example illustrating the logarithmic interpretation of the main component in Alexeev's moduli spaces. This chapter, dealing with moduli spaces for the class of ``broken toric varieties'', contains many of the basic ideas used later for the study of abelian varieties, without being technically too complicated. This chapter is independent of the other chapters, but very helpful for understanding the author's general approach to compactifying moduli spaces. Also, as the moduli problems discussed here have been studied in other contexts, mainly with a view toward Chow quotients and surgery on Grassmannians, this chapter is of independent interest likewise. Chapter 4 gives the modular interpretation of the normalization of the main component of Alexeev's moduli stack \(A^{\text{Alex}}_{g,1}\). In this case, the author's general construction yields an explicit Artin stack \(K_g\) containing \(A_g\) as an open substack. This stack \(K_g\) is proper over \(\text{Spec}(\mathbb{Z})\), admits a good analytic theory at the boundary \(K_g\setminus A_g\) generalizing the theory of the Tate curve for elliptic curves, has only toroidal singularieties, and its associated log stack is log smooth. Chapter 5 generalizes the constructions given in the previous chapter to the more general case of moduli spaces of abelian varieties with polarizations of higher degree. The main new ingredient needed to tackle the full theory is a detailed study of the degenerations of the theta group. Finally, Chapter 6 is devoted to the construction of compact moduli stacks for abelian varieties with polarizations and level structure. Using the results of the previous sections, the author presents two different approaches. While the first approach is based on logarithmic étale cohomology, the second approach uses again the deformation theory of the theta group developed in Chapter 5. The latter theory appears to be a higher-dimensional version of the moduli theory of generalized elliptic curves introduced by \textit{P. Deligne} and \textit{M. Rapoport} [cf.: Les schémas de modules de courbes elliptiques. Modular Functions of one Variable II, Proc. internat. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349, 143--316 (1973; Zbl 0281.14010)]. No doubt, the work presented in this research monograph is a fundamental contribution to the compactification theory of moduli spaces in general, and of moduli spaces for abelian varieties in particular. It must be seen as a major step forward in this direction with further applications and generalizations to follow. Although being of highly advanced nature, the present monograph is written in a very lucid, comprehensive, largely self-contained and enlightening style, including numerous additional remarks and hints.
research exposition (algebraic geometry)
scientific article; zbMATH DE number 5306366