Jiaming Chen — Page 3

arXiv2016

The stable cohomology of the Satake compactification of $\mathcal{A}_g$

Charney and Lee have shown that the rational cohomology of the Satake-Baily-Borel compactification the moduli space of principally polarized abelian varieties of dimension g stabilizes as g grows and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebro-geometric proof of their theorem that also takes into account the mixed Hodge structure that is present here. We find the latter to be impure.

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arXiv2015

The homotopy type of the Baily-Borel and allied compactifications

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.

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