The homotopy type of the Baily-Borel and allied compactifications

TL;DR

This paper analyzes the rational homotopy types of important complex-analytic compactifications, revealing their structure through simplicial sets and generalizing previous results on moduli space compactifications.

Contribution

It provides a unified description of the homotopy types of Baily-Borel, toroidal, and Deligne-Mumford compactifications using simplicial categories, extending known results.

Findings

Homotopy types described via simplicial sets attached to categories

Generalization of Charney-Lee's result on Baily-Borel compactification

Extension of the period map for Riemann surfaces

Abstract

A number of compactifications familiar in complex-analytic geometry, in particular, the Baily-Borel compactification and its toroidal variants, as well as the Deligne-Mumford compactifications, can be covered by open subsets whose nonempty intersections are Eilenberg-MacLane spaces. We exploit this fact to describe the (rational) homotopy type of these spaces and the natural maps between them in terms of the simplicial sets attached to certain categories. We thus generalize an old result of Charney-Lee on the Baily-Borel compactification of A_g and recover (and rephrase) a more recent one of Ebert-Giansiracusa on the Deligne-Mumford compactifications. We also describe an extension of the period map for Riemann surfaces (going from the Deligne-Mumford compactification to the Baily-Borel compactification of the moduli space of principally polarized varieties) in these terms.