Chern classes of Deligne-Mumford stacks and their coarse moduli spaces

TL;DR

This paper establishes a relationship between Chern-Schwartz-MacPherson classes and Chern classes of inertia stacks for Deligne-Mumford stacks and their coarse moduli spaces, with implications for stringy and orbifold invariants.

Contribution

It proves that CSM classes of certain algebraic varieties coincide with pushforwards of Chern classes of inertia stacks, linking geometric and stringy invariants.

Findings

CSM class of X equals pushforward of c(T_{I extX})

Stringy Chern class of X equals pushforward of c(T_{II extX})

Results have implications for stringy/orbifold Hodge numbers

Abstract

Let $X$ be a complex projective algebraic variety with Gorenstein quotient singularities and $\X$ 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 $c(T_{I\X})$ of the inertia stack $I\X$. We also show that the stringy Chern class $c_{str}(X)$ of $X$, whenever is defined, coincides with the pushforward to $X$ of the total Chern class $c(T_{II\X})$ of the double inertia stack $II\X$. Some consequences concerning stringy/orbifold Hodge numbers are deduced.