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.

So thanks for your help!

  • 1
    $\begingroup$ 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. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.