# Computation of vanishing cycles

Here's the problem I'm looking at:

$F$ is a perverse sheaf (or a regular holonomic D-module, or even a mixed hodge module) on $\mathbb{C}^2$ stratified by $z_1 = 0$, $z_2=0$. It can be caracterized by a quiver $$\begin{array}{ccc} \psi_{z_1}\psi_{z_2}(F) & \leftrightarrows & \psi_{z_1}\phi_{z_2}(F) \cr \uparrow \downarrow & & \uparrow \downarrow \cr \phi_{z_1}\psi_{z_2}(F) & \leftrightarrows & \phi_{z_1}\phi_{z_2}(F) \end{array}$$
where arrows are the canonical and variation maps.

Consider $f(z_1,z_2) = z_1+z_2$ (or any curve passing through $(0,0)$ transverse to the axises).

How can we compute $$can: \psi_f(F) \leftrightarrows \phi_f(F) : var$$ in terms of these data?

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Describing perverse sheaves on f = 0 in Deligne style, as representations of $\bullet \leftrightarrows \bullet$, I get $\psi_f F = [\psi \psi F \leftrightarrows \psi \phi F \oplus \phi\psi F]$. That's a nice-looking answer but I didn't arrive at it in a nice way. –  David Treumann Sep 9 '10 at 16:58
That is the obvious answer together with $\phi_fF = \phi \phi F$ but I can't find a nice proof. And the determination of can and var seems even trickier. –  YBL Sep 9 '10 at 18:00
This is quite frustrating as this is the simplest non trivial situation I can think of. –  YBL Sep 9 '10 at 18:01
Great question! I also thought a bit about this without arriving at a rigorous or even elegant proof. It would be great if this question was answered! –  Jan Weidner Dec 6 '12 at 15:50

Here's the bottom line. If $F$ is a perverse sheaf or a D-module, there is an isomorphism between $\psi_fF$ and the complex $[\psi \psi F \to \psi \phi F \oplus \phi \psi F]$. The reason why we can't get to it in a nice way is that it is really non canonical.

To see it, consider a weight-filtrered perverse sheaf $(F,W)$ or an F-filtered D-module, the natual filtration on $\psi_f F$ does not correspond to the natural convolution filtration on $[\psi \psi F \to \psi \phi F \oplus \phi \psi F]$. Also the isomorphisms you get for perverse sheaves on one side and D-modules on the other side are not compatible with the Riemann-Hilbert correspondance. Non trival periods appear because integrals over the fibers $\int_{z_1+z_2 = 1} z_1^a z_2^b = B(a+1,b+1)$ are related to Euler's Beta function.

What is canonical (for $F$ bi-monodromic) is the isomorphism $\psi_f F = i_1^*Rf_*F$ between nearby cycles and the fiber over 1 of the direct image. The right way to approach the problem is to consider the additive convolution on the affine line as a new fundamental operation distinct from the tensor product.

In this way one can prove a general Sebastiani-Thom theorem: in a neighborhood of $f(x) = g(y) = 0$, the vanishing cycles of $\phi_{f\oplus g}(F\boxtimes G)$ is the additive convolution of $\phi_f(F)$ and $\phi_g(G)$ (where vanishing cycles are interpreted as monodromic sheaves on the normal cones with 0 fiber on the zero sections). If $F$ and $G$ are perverse sheaves (resp. D-modules) then the additive convolution is non canonically isomorphic to the tensor product (this was known to Deligne).

One has a similar result for nearby cycles and we can describe the canonical and variation morphism at $f(x) = g(y) = 0$. But I havent found how to describe the vertical monodromy of the nearby cycles yet.

PS: I should mention that when I finally understood all this over a year ago. I sent a preprint to Claude Sabbah who informed me than M. Saito had proved a similar result more than 10 years ago but never published it. I still hope I will find the time to write things down properly and publish my version as I find the problem is both elementary, very deep and interesting.

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