Hypercube decomposition of perverse sheaves

Question

There is a well known theorem that says that the functor associating to a perverse sheaf $F$ on $X$ the data $(F|_U,\phi_f(F),can:\psi_f(F) \to \phi_f(F),var:\phi_f(F)\to \psi_f(F)(-1))$ where $U = X \setminus (f=0)$ is an equivalence of categories.

In dimension 1, this gives that a perverse sheaf on $(\mathbb C,0)$ is described by a quiver $\psi_z(F) = V_0 \leftrightarrows V_1 = \phi_z(F)$ .

There is an analog statement for perverse sheaves $(\mathbb C,0)^n$. My question is how do you define directly the vector spaces in the quiver representation. For example, in dimension 2, I think we should have $V_{00} = \psi_x\psi_y(F) = \psi_x\psi_y(F)$, $V_{10} = \phi_x\psi_y(F) = \psi_y\phi_x(F)$, $V_{01} = \psi_x\phi_y(F) = \phi_y\psi_x(F)$, $V_{11} = \phi_x\phi_y(F) = \phi_y\phi_x(F)$ but I'm not sure why these functors should commute. Is the key point here that we have a normal crossing divisor?

Also, what if we want to make things more canonical replacing nearby cycles $\psi_f(F)$ by the restriction of the Verdier specialisation $\nu_Z(F)|_{T_Z^0X}$ and $\phi_f(F)$ by the evanescent part of $\nu_Z(F)$? What would be the precise statement in this case?

Answer 1

In some sense there is no canonical way to extract these vector spaces. Algebraically this is because projective objects usually have automorphisms. Geometrically because these vector spaces are the stalks of local systems on a manifold with no natural base point. But the manifold and the local system are canonical.

Each of the 2^n coordinate subspaces of C^n has a conormal bundle in C^{2n}. This gives an arrangement of 2^n Lagrangian subspaces of the symplectic vector space C^2n, which are in general position (as general as possible for Lagrangian subspaces). The smooth part of this arrangement, the points that lie on exactly one of these Lagrangians, is a disjoint union of 2^n copies of (C-0)^n. The 2^n vector spaces in your hypercube are the stalks of local systems on these.

I am not sure what Verdier specialization is, but I bet it's exactly the construction of these local systems. In general, if Y is a smooth subvariety of X, then it is possible to extract from a perverse sheaf a local system on an open subset of the conormal variety to Y in X, by the following recipe:

1. Take nearby cycles for the deformation-to-the-normal-bundle family. This family has a C^*-action so taking nearby cycles is more canonical than usual--the monodromy action (var compose can) is trivial.

2. Take the Fourier transform. (As you know, I'm still pretty confused about this, but I think this has to be the topological version, or Fourier-Sato transform. In particular even with the C^*-action on the family, I don't think the sheaf you get at the end of 1. is C^*-equivariant.)

3. The result is a perverse sheaf on the conormal bundle to Y. This sheaf is locally constant on an open subset.

At one time S. Gelfand, MacPherson, and Vilonen had a project to understand the natural maps between the fibers of these local systems on different strata, i.e. the analogs of can and var. From a certain perspective the work of Nadler and Zaslow is a (not very concrete) solution to this problem.

In general, whether \psi_f \psi_g F = \psi_g \psi_f F depends on the blowup behavior of F with respect to the map (f,g) to C^2. For a counterexample, consider F = constant sheaf on C^2, f = x, g = xy. But if F is constructible with respect to a stratification that satisfies Thom's condition a_{f,g}--that is, the stratification is without blowups/sans eclatement with respect to (f,g)--then you are in good shape. (Although even here there is no canonical isomorphism between the two. They behave like stalks of a local system on an open subset of the base C^2.)

For sheaves without blowups I am not sure about references, but try Sabbah's "Morphismes analytiques stratifies sans eclatement et cycles evanescents," or from the etale point of view Illusie's recent note: http://www.math.u-psud.fr/~illusie/vanishing1b.pdf