Mixed Hodge Structures and compactifications of Siegel’s space

Let Hg be the Siegel upper half-space of order g, and let Γ = Sp(g,Z) be the Siegel modular group. The quotient Hg/Γ is a normal analytic variety, which we shall call Siegel’s space. It admits certain natural compactifications; in particular, the minimal compactification of Satake [12] and the various toroidal compactifications of Mumford [10] (see also [1], [2], [13], [14]). The aim of this report is to describe a compactification which is natural from the point of view of mixed Hodge theory: The boundary components will be distinguished quotients of classifying spaces for mixed Hodge structures. This compactification is also good from the standpoint of curve degenerations: the limit point at the boundary detects the extension-theoretic part (cf. [3]) of the limiting mixed Hodge structure, and this in turn carries geometric information about the central fiber of the degeneration. Although the compactification presented here for Siegel’s space turns out to agree with Mumford’s smooth compactification for arithmetic quotients of Hermitian symmetric spaces, we have, nevertheless, included a direct construction in the hope that this will clarify some of the subtler geometric aspects of the toroidal techniques, while at the same time lay a foundation for the study of Hodge-theoretic compactifications for classifying spaces of Hodge structures of higher weight. To this end, the emphasis of this report is on descriptions and motivations, rather than on the technical details. These will be supplied in a subsequent paper where we shall also discuss some important aspects of the compactification which are not treated here: analytic structure and extension of period mappings, among others. The first two sections are of an introductory nature: after recalling some basic facts about degenerations of Hodge structure, Schmid’s nilpotent orbit