The Deligne–Mumford and the Incidence Variety Compactifications of the Strata of $\Omega {ℳ}_g$

Gendron, Quentin

Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1169-1240.

L’objectif central de cet article est de construire et d’étudier une compactification raisonnable des strates des différentielles abéliennes. L’ingrédient principal pour l’étude de cette compactification des strates est une généralisation des techniques de plomberie cylindrique aux différentielles. Cette compactification nous permet de calculer la dimension de Kodaira de certaines de ces strates. Un autre résultat digne d’intérêt est le calcul de la dimension de la projection des strates dans l’espace des modules des surfaces de Riemann. Enfin nous étudions certains problèmes liés à la parité des strates au bord, les composantes hyperelliptiques ainsi que la strate minimale en genre trois.

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of the odd minimal stratum in genus three.

Classification : 14H15, 30F30, 14E99, 14H45
Keywords: Abelian differentials, Riemann surfaces, Moduli spaces, Strata, Compactification, Kodaira dimension
Mot clés : Différentielles abéliennes, Surfaces de Riemann, Espace des modules, Strates, Compactifications, Dimension de Kodaira

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