Fundamental Group of Locally Symmetric Varieties

The geometry of moduli spaces of complex abelian varieties and of compactifications of those moduli spaces has been the object of much study in the last few years. Given such a compactified moduli space it is natural to ask about the fundamental group of a resolution of singularities. This problem has been studied in some special cases, for instance in [K], [HK] and [HS]. It follows from the results of [K] and the well-known fact that every arithmetic subgroup of Sp(2g, Q) is a congruence subgroup of some level that the fundamental group must be finite except when g = 1, i.e., except in the case of modular curves. The method in all these cases, and here, is to use the toroidal compactification of [SC], considering the moduli spaces as quotients of the Siegel upper half-space by arithmetic subgroups of the symplectic group. In this paper we treat the subject in greater generality. In many cases we are able to identify the fundamental group explicitly as a quotient of the arithmetic group in question. For this purpose we do not need to restrict ourselves to the symplectic group but instead may consider any locally symmetric variety. Later we return to the case of moduli of abelian varieties (specifically, to Siegel modular varieties) and calculate the fundamental group in some interesting special cases. In many cases, including those studied in the papers mentioned above, the fundamental group is trivial, but we give examples to show that this need not be true in general. The example in Proposition 3.1 is a modified version of one suggested to me by Professor M.S. Raghunathan. I am grateful to him for pointing it out to me. I am also grateful to Professors K. Hulek and W. Ebeling for useful remarks, and to Tôhoku University and especially to Professor Tadao Oda for their hospitality during a visit to Japan when I began this work.