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Chern classes of Deligne-Mumford stacks and their coarse moduli spaces
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 1, February 2011
- pp. 29-38
- 10.1353/ajm.2011.0001
- Article
- Additional Information
Let $X$ be a complex projective algebraic variety with Gorenstein quotient
singularities and $\mathcal {X}$ a smooth Deligne-Mumford stack having $X$
as its coarse moduli space. We show that the CSM class $c^{SM}(X)$
coincides with the pushforward to $X$ of the total Chern class $c(T_{I
\mathcal{X}})$ of the inertia stack $I \mathcal {X}$. We also show that
the stringy Chern class $c_{str}(X)$ of $X$, whenever it is defined,
coincides with the pushforward to $X$ of the total Chern class $c(T_{II
\mathcal{X}})$ of the double inertia stack $II \mathcal {X}$. Some
consequences concerning stringy/orbifold Hodge numbers are deduced.