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Let $Y$ be a smooth algebraic variety over $\mathbb{C}$, let $X = Y \times \mathbb{C}$ and let $f: X \to \mathbb{C}$ be the projection. Let $M$ be a (not necessarily regular) holonomic $D_X$-module that is specializable along $f^{-1}(0)$.

I know of two procedures for computing the nearby cycles module $\Psi_{f} M$.

  1. Compute the $V$-filtration $V_\bullet M$ and compute $\Psi_{f} = \text{gr}^V_0 M$ as in Kashiwara's Vanishing Cycle Sheaves and Holonomic Systems of Differential Equations.

  2. Find a certain lattice $L_0 \subset f^s M[f^{-1},s]$ and compute $\Psi_{f} = L_0/tL_0$ as in Section 5 of Ginzburg's Characteristic Varieties and Vanishing Cycles

In Kashiwara's paper he gives an algorithm for taking an arbitrary good $V_\bullet D_X$ filtration $F_\bullet M$ and modifying it to produce $V_\bullet M$.

Q1: In the specific example I am interested in, it seems as though I can pick $F_\bullet M$ in such a way that $\text{gr}^V_0 M$ and $\text{gr}^F_0 M$ have the same characteristic cycle. Is there some general principle at work here?

This would follow from general facts about lattices if I could prove that I can construct a $D_X[s]$-lattice $L_F$ from $F_\bullet M$ such that $L_F/tL_F \cong \text{gr}^F_0 M$. This motivates the following question:

Q2: What is the precise relationship between the two procedures for computing $\Psi_f M$?

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  • $\begingroup$ I don't think I can answer Q2 precisely, but my understanding is that this is usual connection between filtrations and deformations using Rees modules. That is, I think that Ginzburg's $L_0$ is the Rees module of the V-filtration (suitably interpreted). $\endgroup$
    – Ben Webster
    Jul 26, 2015 at 3:45

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