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Orbifold cohomology for global quotients

2003; Duke University Press; Volume: 117; Issue: 2 Linguagem: Inglês

10.1215/s0012-7094-03-11721-4

1547-7398

Barbara Fantechi, Lothar Göttsche,

Homotopy and Cohomology in Algebraic Topology

Let $X$ be an orbifold that is a global quotient of a manifold $Y$ by a finite group $G$. We construct a noncommutative ring $H\sp \ast(Y, G)$ with a $G$-action such that $H\sp*(Y, G)\sp G$ is the orbifold cohomology ring of $X$ defined by W. Chen and Y. Ruan [CR]. When $Y=S\sp n$, with $S$ a surface with trivial canonical class and $G = \mathfrak {S}\sb n$, we prove that (a small modification of) the orbifold cohomology of $X$ is naturally isomorphic to the cohomology ring of the Hilbert scheme $S\sp {[n]}$, computed by M. Lehn and C. Sorger [LS2].