Let $h:Y\to \mathbb C$ be a regular map and let $f:X\to \mathbb C$ be the restriction of $h$ to a closed subvariety $X\subset Y$. Both $X$ and $Y$ are assumed to be smooth. The maps $h,f$ induce nearly cycles complexes $\Psi_h\mathbb Q$ and $\Psi_f\mathbb Q$, which compute the cohomology of the Milnor fibers of the corresponding map. I am interested in the motivic version of these complexes, introduced by Denef and Loeser here. Letting $Y_0=h^{-1}(0)$ and $X_0=f^{-1}(0)$, the motivic nearby cycles are relative motivic classes $$[\psi_h]\in \mathcal M_{Y_0}^{\hat\mu},\qquad [\psi_f]\in\mathcal M_{X_0}^{\hat\mu}$$ living in the monodromic Grothendieck ring (varieties are endowed with a $\hat\mu$-action, where $\hat\mu$ is the inverse limit of the groups $\mu_n$ of $n$-th roots of unity).

Question. If $i:X_0\to Y_0$ is the inclusion and $i^\ast:\mathcal M_{Y_0}^{\hat\mu}\to \mathcal M_{X_0}^{\hat\mu}$ is the pullback map, is it true that $i^\ast[\psi_h]=[\psi_f]$? If not, does this become true when the motivic nearby cycles live in the subrings $\mathcal M_{Y_0}$ and $\mathcal M_{X_0}$ respectively?

The problem is I do not do know if the corresponding property holds for the non-motivic $\Psi$'s, so I do not have an actual guess. Thanks for any thought!

  • $\begingroup$ Have you looked in Sect 7-8 of Bittner's xxx.lanl.gov/pdf/math/0307033.pdf? She studied such relative questions there. $\endgroup$ – Balazs Apr 17 '15 at 8:58
  • $\begingroup$ The answer is no, already in the usual case of constructible sheaves. Take $X$ a point in $Y_0$, for instance, or if this looks too degenerate take $X$ a smooth curve meeting $Y_0$ in one point with non-trivial vanishing cycles. $\endgroup$ – Simon Pepin Lehalleur Apr 17 '15 at 12:22
  • $\begingroup$ @Balazs: Thanks, I did not know that paper and it is indeed helpful! $\endgroup$ – Brenin Apr 18 '15 at 0:34

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