Compactification (mathematics)

Abstract: I. Preliminaries.- II. Degeneration of Polarized Abelian Varieties.- III. Mumford's Construction.- IV. Toroidal Compactification of Ag.- V. Modular Forms and the Minimal Compactification.- VI. Eichler Integrals in Several Variables.- VII. Hecke Operators and Frobenii.- Glossary of Notations.- An Analytic Construction of Degenerating Abelian Varieties over Complete Rings.- David Mumford.

Abstract: The Stone-fiech compactification PX has been a topic of increasing study since its introduction in 1937. The algebraic content of this research is collected in the 1960 textbook, Rings of continuous Functions, by L. Gillman and M. Jerison. Here we take a more purely topological viewpoint of the Stone-Cech compactification and attempt to collect the most important results which have emerged since Rings of Continuous Functions. The construction of pX is described in an historical perspective. The theory of Boolean algebras is developed and used as a tool, primarily in a detailed investigation of 0 IN and p 3N\IN. The relationships between a space X and its "growth" PX\X are examined, including the non-homogeneity of $X\X, the cellularity of pX\X, and mappings of pX to PX\X. The Glicksberg product theorem which characterizes the products such that 0(x X ) = x (PX^) and related results are cc cc presented. Finally, the Stone-cech compactification is studied in a categorical context.