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Recall that for a sheaf $F$ on an analytic manifold $X$ the micro-support consists of those $\omega\in T^*X$ for which there exists a $C^1$ function $f$ defined around $\pi(\omega)$ with $f(\pi(\omega))=0$ and $df(\pi(\omega))=\omega$ such that $R\Gamma_{\{x\mid f(x)>0\}}F\not=0$ around $\pi(\omega)$.

Now there is an obvious "higher order" version of this definition, where we replace $T^*X$ by higher order jet bundles of $X$, and rather then remembering only the $1$-jet (i.e. $df$) we remember the higher order jet of $f$ (which now has to also be $C^r$ where $r$ is the order of the jet bundle that we are considering).

Has this notion shown up anywhere?

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    $\begingroup$ Part of the point of the microsupport definition is that for a generic function (one whose derivative intersects the microsupport transversely at isolated points, say) whether there exists vanishing cycles or not depends only on the 1-jet (I think - at least in the complex analogue this is true). So this would be mostly determined by the microsupport. $\endgroup$
    – Will Sawin
    Sep 12, 2021 at 13:39
  • $\begingroup$ @WillSawin Are you saying these higher micro-support wouldn't contain much new information in general, or just that they do not give additional information about the vanishing cycles? $\endgroup$
    – user2520938
    Sep 12, 2021 at 15:40
  • $\begingroup$ I don't know how much new information they would contain. I think they would express this new information in a somewhat awkward way. The usual micro-support is a nice mathematical object (a Lagrangian). The higher microsupport would be, I think, the inverse image of the usual microsupport, minus a positive codimension subspace, plus an even higher codimension subspace. It seems harder to make the geometric structure of that useful in an argument. $\endgroup$
    – Will Sawin
    Sep 12, 2021 at 17:29
  • $\begingroup$ @WillSawin I was thinking of the following: Let $g$ be an analytic function and consider the proper direct image of the constant sheaf on $X\setminus \{g=0\}$. Then it seems to me that the usual micro-support would be the conormal bundle to $\{g=0\}$, while these higher order versions would be the image of the prolongation of $g$ to the jet bundle intersected with $\{g=0\}$. So in this case the higher order micro support can be recovered from the usual microsupport via prolongation, but this gives a more point-wise approach. Moreover, I think that if $g$ is singular prolongation is trick $\endgroup$
    – user2520938
    Sep 12, 2021 at 17:35

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