# intersection complex for quotient singularities

Let $X$ be a projective variety over a field of characteristic zero and assume that $X$ has finite quotient singularities, that is, $X$ is a union of affine open subsets of the form $Y/G$, where $G$ is a finite group acting on $Y$.

Is it true that the intersection complex $IC_X$ is $\mathbb{Q}_X[\mathrm{dim X}]$?

I know this is true for non-singular $X$ and I would like to know if it extends to finite quotient singularities. I have no doubt that this is well-known but I was unable to find quickly a reference.

• I don't know a good reference off the top of my head, but yes the formula you want is true because $X$ is a rational homology manifold. – Donu Arapura Jan 20 '14 at 1:44
Proposition A1. iii) of the the article M. Brion: Rational smoothness and fixed points of torus actions implies that if X has finite quotient singularities, then X is rationally smooth. That is, for any points $x \in X$, $H^i_{\{x\}}(X,\mathbb{C}_X)=\mathbb{C}$ if $i=2 \mathrm{dim} X$ and 0 otherwise. Therefore, Proposition 8.2.21. of R. Hotta, K. Takeuchi, T. Taniksaki: D-Modules, Perverse Sheaves, and Representation Theory gives the positive answer to the question with complex coefficients, and everything goes through with the rationals as well.