Chen–Ruan Cohomology of ${ℳ}_{1,n}$ and ${\overline{ℳ}}_{1,n}$

Pagani, Nicola

Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1469-1509.

Dans ce travail on calcule la cohomologie de Chen–Ruan de l’espace de modules des courbes lisses et stables de genre $1$ avec $n$ points marqués. Dans la première partie on étudie et on décrit les secteurs tordus de ${ℳ}_{1,n}$ et ${\overline{ℳ}}_{1,n}$, en tant que champs.

Dans la deuxième partie, on étudie la théorie d’intersection orbifold de ${\overline{ℳ}}_{1,n}$. On donne une définition possible de l’anneau tautologique orbifold en genre $1$, comme sous-anneau simultanément de la cohomologie de Chen–Ruan et de l’anneau de Chow orbifold.

In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$-pointed curves of genus $1$. In the first part of the paper we study and describe stack theoretically the twisted sectors of ${ℳ}_{1,n}$ and ${\overline{ℳ}}_{1,n}$. In the second part, we study the orbifold intersection theory of ${\overline{ℳ}}_{1,n}$. We suggest a definition for an orbifold tautological ring in genus $1$, which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

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