Computation of vanishing cycles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:28:35Z http://mathoverflow.net/feeds/question/38116 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38116/computation-of-vanishing-cycles Computation of vanishing cycles YBL 2010-09-08T23:54:02Z 2012-12-19T19:55:31Z <p>Here's the problem I'm looking at: </p> <p>$F$ 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 $$\begin{array}{ccc} \psi_{z_1}\psi_{z_2}(F) &amp; \leftrightarrows &amp; \psi_{z_1}\phi_{z_2}(F) \cr \uparrow \downarrow &amp; &amp; \uparrow \downarrow \cr \phi_{z_1}\psi_{z_2}(F) &amp; \leftrightarrows &amp; \phi_{z_1}\phi_{z_2}(F) \end{array}$$<br> where arrows are the canonical and variation maps.</p> <p>Consider $f(z_1,z_2) = z_1+z_2$ (or any curve passing through $(0,0)$ transverse to the axises). </p> <p>How can we compute $$can: \psi_f(F) \leftrightarrows \phi_f(F) : var$$ in terms of these data? </p> http://mathoverflow.net/questions/38116/computation-of-vanishing-cycles/116812#116812 Answer by YBL for Computation of vanishing cycles YBL 2012-12-19T19:47:49Z 2012-12-19T19:55:31Z <p>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. </p> <p>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. </p> <p>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. </p> <p>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). </p> <p>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. </p> <p>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. </p>