# Where should I look for computing the intersection homology of projective varieties?

I'm learning about intersection cohomology topologically through MacPherson's "New York Times Article". This is a very nice guide which gives a nice idea on how to use these methods for low-dimensional topological spaces, I'm not sure how I can compute the intersection homology for higher dimensional varieties. For example, if I take the quintic threefold $$\textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w,u]}{(x^5 + y^5 + z^2wu^2 + x^3y^2)} \right)$$ It has jacobian $$\text{Jac}(f) = [5x^4 + 3x^2y^2, 5y^4 + 2x^3y, 2zwu^2, z^2u^2, 2z^2wu]$$ hence ther are the singular loci along the rational curves $$x,y,z = 0, \text{ } x,y,u=0$$ We can form a normal stratification on this variety with these stratum, but now I'm not sure how to actually compute the intersection cohomology groups from here.

• FYI, Greg Friedman's draft book Singular Intersection Homology is at faculty.tcu.edu/gfriedman/IHbook.pdf Jul 21 '16 at 20:53
• I don't really know this area - but since you haven't gotten an answer yet - my impression is that people usually do these calculations on algebraic varieties by calculating with D-modules and using the Riemann-Hilbert correspondence. Jul 22 '16 at 18:22
• @AlexanderWoo Thanks for the tip. I'll try and figure out how to find d-modules giving simple objects under the RH-correspondence then. Jul 22 '16 at 19:36
• I think Alex has it wrong: to use Riemann-Hilbert to compute IH, you would want a good handle on the Riemann-Hilbert correspondent of the IC sheaf. But not much is known about those "IC D-modules" in general. We have their generators but almost never their relations. Jul 25 '16 at 18:40
• The singularities on your 3-fold might look like (C^2/subgroup of SU(2)) x C locally. In that case it is "rationally smooth", and intersection homology = regular homology with Q-coefficients. Jul 25 '16 at 18:45