Removing chambers in Bruhat-Tits buildings

We introduce and study a family of countable groups constructed from Euclidean buildings by “removing” suitably chosen subsets of chambers.     

Cluster networks and Bruhat-Tits buildings

We consider a clustering procedure in the case where a family of metrics is used instead of a fixed metric. In this case, a classification network (a directed acyclic graph with nondirected cycles) is obtained instead of a classification tree. We discuss the relation to Bruhat-Tits buildings and introduce the notion of the dimension of a general cluster network.     

On compression of Bruhat-Tits buildings

Consider an affine Bruhat-Tits building Lat _n of type A_n−1 and the complex distance in Lat _n, i.e., the complete system of invariants of a pair of vertices of the building. An element of the Nazarov semigroup is a lattice in the duplicated p-adic space ℚ _p ⁿ ⊕ ℚ _p ⁿ . We investigate the behavior of the complex distance with respect to the natural action of the Nazarov semigroup on the building. Bibliography: 18 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 163–170.     

Compactifications of Bruhat-Tits buildings associated to linear representations

Let G be a connected semisimple group over a non-Archimedeanlocal field. For every faithful, geometrically irreducible linearrepresentation of G we define a compactification of the associatedBruhat–Tits building X(G). This yields a finite familyof compactifications of X(G) which contains the polyhedral compactification.In addition, this family can be seen as a non-Archimedean analogueof the family of Satake compactifications for symmetric spaces.     

A tropical view on Bruhat-Tits buildings and their compactifications

We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character of the representation.     

Compactification of the Bruhat-Tits building of PGL by seminorms

We construct a compactification of the Bruhat-Tits building X associated to the group PGL(V) which can be identified with the space of homothety classes of seminorms on V endowed with the topology of pointwise convergence. Then we define a continuous map from the projective space to which extends the reduction map from Drinfelds p-adic symmetric domain to the building X.Mathematics Subject Classification (2000): 20E42, 20G25in final form: 4 October 2003     

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.     

On compactification of metric spaces

Iff:X →X* is a homeomorphism of a metric separable spaceX into a compact metric spaceX* such thatf(X)=X*, then the pair (f,X*) is called a metric compactification ofX. An absoluteG _δ-space (F _σ-space)X is said to be of the first kind, if there exists a metric compactification (f,X*) ofX such that , whereG _i are sets open inX* and dim[Fr(G _i)]<dimX. (Fr(G _i) being the boundary ofG _i and dimX — the dimension ofX). An absoluteG _δ-space (F _σ-space), which is not of the first kind, is said to be of the second kind. In the present paper spaces which are both absoluteG _δ andF _σ-spaces of the second kind are constructed for any positive finite dimension, a problem related to one of A. Lelek in [11] is solved, and a sufficient condition onX is given under which dim [X* −f(X)]≧k, for any metric compactification (f,X*) ofX, wherek≦dimX is a given number. This research has been sponsored by the U.S. Navy through the Office of Naval Research under contract No. 62558-3315.     

A compactification of open varieties

In this paper we prove a general method to compactify certain open varieties by adding normal crossing divisors. This is done by showing that blowing up along an arrangement of subvarieties can be carried out. Important examples such as Ulyanov's configuration spaces and complements of arrangements of linear subspaces in projective spaces, etc., are covered. Intersection ring and (nonrecursive) Hodge polynomials are computed. Furthermore, some general structures arising from the blowup process are also described and studied.

     

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).