A compactification of Igusa varieties

We investigate the notion of Igusa level structure for a one-dimensional Barsotti–Tate group over a scheme X of positive characteristic and compare it to Drinfeld’s notion of level structure. In particular, we show how the geometry of the Igusa covers of X is useful for studying the geometry of its Drinfeld covers (e.g. connected and smooth components, singularities). Our results apply in particular to the study of the Shimura varieties considered in Harris and Taylor (On the geometry and cohomology of some simple Shimura varieties. Princeton University Press, Princeton, 2001). In this context, they are higher dimensional analogues of the classical work of Igusa for modular curves and of the work of Carayol for Shimura curves. In the case when the Barsotti–Tate group has constant p-rank, this approach was carried-out by Harris and Taylor (On the geometry and cohomology of some simple Shimura varieties. Princeton University Press, Princeton, 2001).     

A new compactification of the Siegel space and degeneration of Abelian varieties. II

This is a continuation of the article with the same title appeared in this journal, to the introduction of which we referr the reader for the summary of contents of this article.Dedicated to Professor K. Kodaira on his 60th birthday.     

Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel compactification

We study the degeneration in the Baily-Borel compactification of variations of Hodge structure on Shimura varieties. Our main result, Theorem 2.6, expresses the degeneration of variations given by algebraic representations in terms of Hochschild, and abstract group cohomology. It is the Hodge theoretic analogue of Pink's theorem on degeneration of étale and ?-adic sheaves [Math. Ann. 292 (1992) 197], and completes results by Harder and Looijenga-Rapoport [Eisenstein-Kohomologie arithmetischer Gruppen: Allgemeine Aspekte, Preprint, 1986; Proc. of Symp. in Pure Math., vol. 53, 1991, pp. 223-260]. The induced formula on the level of singular cohomology is equivalent to the theorem of Harris-Zucker on the Hodge structure of deleted neighbourhood cohomology of strata in toroidal compactifications [Inv. Math. 116 (1994) 243].     

A Stone-Čech ultrafuzzy compactification

It is shown that if (X, F) is a fuzzy topological space whose induced topology is Tychonoff, then (X, F) has a Stone-?ech ultra-fuzzy compactification.     

A compactification of the moduli space of polynomials

In this paper, we introduce a compactification of the moduli space of polynomial maps with a fixed degree such that the map from it to defined by using the elementary symmetric functions of multipliers at fixed points is a continuous surjection.

     

A compactification problem of J. De Groot

Recently, De Groot's conjecture that cmp X = def X holds for every separable and metrizable space X has been negatively resolved by Pol. In previous efforts to resolve De Groot's conjecture various functions like cmp have been introduced. A new inequality between two of these functions is established. Many examples which have been constructed so far in relation with the conjecture are obtained by attaching a locally compact space to a compact space. An upper bound for the compactness deficiency def of the resulting space is given.     

On compactification of schemes

For a Serre fibration with a fibre of theK(π, n)'s product type, obstructions to the section problem in each degree are defined by means of the Hirsch complex of fibration. This allows us to give the homotopy classification of sections (maps) as well as other applications. In particular, forG-bundles, these obstructions are related to theA _∞-module structure on the homology of the fibre and, consequently, some results in the fixed point theory are obtained.