Homology, Homotopy and Applications

Volume 23 (2021)

Number 2

The homotopy type of the Baily–Borel and allied compactifications

Pages: 95 – 119

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a6

Authors

Jiaming Chen (Institut de Mathématiques de Jussieu, Paris Rive Gauche, Université de Paris, France)

Eduard Looijenga (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China; and Mathematisch Instituut, Universiteit Utrecht, The Netherlands)

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 classified by their fundamental groups.We exploit this fact to define a ‘stacky homotopy type’ for these spaces as the homotopy type of a small category. We thus generalize an old result of Charney–Lee on the Baily–Borel compactification of $\mathcal{A}_g$ 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.

Keywords

homotopy stack, Satake compactification, toric compactification, Deligne–Mumford compactification

2010 Mathematics Subject Classification

14F35, 55R35, 55U10

Received 14 June 2020

Accepted 10 October 2020

Published 21 April 2021