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.

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    $\begingroup$ FYI, Greg Friedman's draft book Singular Intersection Homology is at faculty.tcu.edu/gfriedman/IHbook.pdf $\endgroup$ Jul 21 '16 at 20:53
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    $\begingroup$ 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. $\endgroup$ Jul 22 '16 at 18:22
  • $\begingroup$ @AlexanderWoo Thanks for the tip. I'll try and figure out how to find d-modules giving simple objects under the RH-correspondence then. $\endgroup$
    – 54321user
    Jul 22 '16 at 19:36
  • $\begingroup$ 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. $\endgroup$ Jul 25 '16 at 18:40
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    $\begingroup$ 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. $\endgroup$ Jul 25 '16 at 18:45

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