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.

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.

Hi Ian. I'd be just as interested to find out there are combinatorial restrictions on foam with nonsimply-connected bubbles. But the actual question that came up for me is even more restricted: if you draw a cubic planar graph on the boundary of a ball, what kind of minimal surfaces, with Plateau singularities and without interior regions, can you fill inside?

Probably better use a flag $F$ that is transverse to $JF$, so that $HP^{n-1}$ meets the middle-dimensional Schubert varieties transversely.

Sounds like a fun variant of a Schubert calculus problem: how many 2-planes in a given Schubert variety are quaternionic, i.e. fixed by J? The answer is the coefficient of your fundamental class in the Schubert basis. For example there is a unique quaternionic line containing a given complex line, that gives a coefficient of "1" for one of the Schubert varieties.

I experimented with gluing two ideal triangulations of an n-gon together. This gives a family of about 8^n n-vertex triangulations of the sphere, and for this family the number of chromatic polynomials (starting with n = 4) is 2,3,5,8,13,>=23,>=42,>=80,>=181, maybe exponential growth with a smaller base than 8, not like I know how to prove it.

Thanks Ian. I'm not sure I get it. There are $3^v$ $3$-colorings of the graph with no edges, and some planar graphs have $0$. Are you saying that, if only a polynomial-sized chunk of the interval $[0,3^v]$ could occur as the number of 3-colorings of a graph on $v$ vertices, it might allow for too fast of an algorithm to determine that a graph was $3$-colorable?

Dragos, sorry for the long delay. You're right, the question's no good as there's no reason to think that $\rtimes$ should be closed under cones. Thanks!

Thanks Dragos. I'm not sure that $\langle D_1,D_2\rangle$ contains only those objects that are either extensions of $d_1$ by $d_2$ or that are extensions of $d_2$ by $d_1$. It could also contain an object that carries a filtration whose 1st, 3rd, 5th,... graded pieces belong to $D_1$ and whose 2nd, 4th, 6th,... graded pieces belong to $D_2$. I also don't think that $x$ and $y$ are $D$-equivalent only if there is either a $D$-equivalence $x \to y$ or a $D$-equivalence $y \to x$. But I still don't know whether $D_1 \rtimes D_2 = \langle D_1,D_2\rangle$, maybe my two objections cancel out.

Hi Semyon. How about chapter VIII, section 1 of Kashiwara and Schapira's "Sheaves on Manifolds"?