V-filtrations and minimal exponents for locally complete intersection singularities

Introduction

Let $X$ be a smooth, irreducible, complex algebraic variety. If $Z$ is a nonempty hypersurface in $X$, then the minimal exponent $\widetilde{\alpha}(Z)$ of $Z$ (written also as $\widetilde{\alpha}(f)$ if $Z$ is defined by $f\in\mathcal{O}_{X}(X)$) is an important invariant of the singularities of $Z$ introduced by Saito [Saito_microlocal]. When $Z$ has isolated singularities, it can be described via asymptotic expansions of integrals along vanishing cycles and it was studied extensively in the 80s, see for example [Varchenko], [Steenbrink], and [Loeser]; in this setting, it has been known as complex singularity index or Arnold exponent of $Z$. In general, it is defined as the negative of the largest root of the reduced Bernstein-Sato polynomial of $Z$ (with the convention that it is $\infty$ if this polynomial is $1$, which is the case if and only if $Z$ is smooth). By results of Kollár [Kollar] and Lichtin [Lichtin], it is known that the minimal exponent refines an important invariant of singularities in birational geometry, the log canonical threshold ${\rm lct}(X,Z)$; more precisely, we always have ${\rm lct}(X,Z)=\min\{\widetilde{\alpha}(Z),1\}$. Our main goal in this paper is to introduce and study a generalization of the minimal exponent to the case when $Z$ is locally a complete intersection in $X$.

Before giving the definition in the general context, we recall the connection between the minimal exponent of hypersurfaces and two important $\mathcal{D}$-module theoretic constructions associated to $Z$, the Hodge filtration on the local cohomology $\mathcal{H}_{Z}^{1}(\mathcal{O}_{X})$ of $\mathcal{O}_{X}$ along $Z$ and the Kashiwara-Malgrange $V$-filtration associated to $Z$. Recall that if $Z$ is any closed subscheme of $X$, the local cohomology sheaves $\mathcal{H}^{q}_{Z}(\mathcal{O}_{X})$ underlie mixed Hodge modules in the sense of Saito’s theory [Saito_MHM]. In particular, they carry a Hodge filtration: this is an increasing filtration by coherent $\mathcal{O}_{X}$-modules which is compatible with the order filtration on the sheaf $\mathcal{D}_{X}$ of differential operators on $X$. If $Z$ is a reduced hypersurface in $X$, then the only nonzero local cohomology is $\mathcal{H}^{1}_{Z}(\mathcal{O}_{X})=\mathcal{O}_{X}(*Z)/\mathcal{O}_{X}$ (where $\mathcal{O}_{X}(*Z)$ is the sheaf of rational functions with poles along $Z$). In this case it is known that for every $k\geq 0$ we have

Using a refinement of this result to a setting involving twists by rational multiples of $Z$, as well as properties of Hodge filtrations, it was shown in [MP1] that one can extend the known properties of the Arnold exponent to arbitrary hypersurface singularities.

The proof of (1) makes use of results about $V$-filtrations. Let us briefly recall this notion, due to Malgrange [Malgrange] and Kashiwara [Kashiwara], in the more general context that is relevant to this paper. Working locally, let us suppose that $Z$ is a closed subscheme of $X$ defined by the ideal generated by nonzero regular functions $f_{1},\ldots,f_{d}\in\mathcal{O}_{X}(X)$. If $\iota\colon X\hookrightarrow X\times{\mathbf{A}}^{d}$ is the graph embedding associated to ${\mathbf{f}}=(f_{1},\ldots,f_{d})$, that is, \iota(x)=\big{(}x,f_{1}(x),\ldots,f_{d}(x)\big{)}, then the $V$-filtration is a decreasing filtration $(V^{\gamma}B_{\mathbf{f}})_{\gamma\in{\mathbf{Q}}}$ on

indexed by rational numbers, and characterized by a few properties (for details, see Section 2). In the case of one function, the $V$-filtration plays an important role in the theory of mixed Hodge modules. We also recall that in the case when we only have one function $g\in\mathcal{O}_{X}(X)$, Saito introduced in [Saito_microlocal] a related filtration, the microlocal $V$-filtration $(V^{\gamma}\widetilde{B}_{g})_{\gamma\in{\mathbf{Q}}}$ on

If $Z$ is the hypersurface defined by $g$, then the minimal exponent $\widetilde{\alpha}(Z)$ is described as follows:

(see [Saito-MLCT, (1.3.8)]). In terms of the usual $V$-filtration, this says that if $q$ is a nonnegative integer and $\gamma\in(0,1]$ is a rational number, then

Suppose now that $Z$ is a closed subscheme of $X$ that is a local complete intersection, of pure codimension $r\geq 1$. We define the minimal exponent $\widetilde{\alpha}(Z)$ such that the analogue of formula of (2) holds in this setting. Working locally, we may assume that $Z$ is defined by the ideal generated by $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$. In this case, we put

where in the latter case, the supremum is over all nonnegative integers $q$ and all rational numbers $\gamma\in(0,1]$ with the property that $\partial_{t}^{\beta}\delta_{\mathbf{f}}\in V^{r-1+\gamma}B_{{\bf f}}$ for all $\beta=(\beta_{1},\ldots,\beta_{r})\in{\mathbf{Z}}_{\geq 0}^{r}$, with $\beta_{1}+\ldots+\beta_{r}\leq q$. In fact, the supremum in the definition is a maximum unless $\widetilde{\alpha}(Z)=\infty$ (which we show is the case if and only if $Z$ is smooth). We note that by [BMS, Theorem 1], which describes the multiplier ideals of $Z$ in terms of $V^{\bullet}B_{\mathbf{f}}$, we have

We note that $\widetilde{\alpha}(Z)$ does depend on the ambient variety and not just on $Z$. Whenever $X$ is not understood from the context, we write $\widetilde{\alpha}(X,Z)$ in order to avoid confusion. However, the dependence is easy to understand: the difference $\widetilde{\alpha}(X,Z)-\dim(X)$ only depends on $Z$ (see Proposition 4.14).

In order to prove the basic properties of the minimal exponent for local complete intersections, we describe it as the minimal exponent of a hypersurface. Arguing locally, we may again assume that $Z$ has pure codimension $r$ in $X$ and it is defined in $X$ by the ideal generated by $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$. We consider $Y=X\times{\mathbf{A}}^{r}$, with coordinates $y_{1},\ldots,y_{r}$ on ${\mathbf{A}}^{r}$, and let U=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\{0\}\big{)}.

With the above notation, if $g=\sum_{i=1}^{r}f_{i}y_{i}\in\mathcal{O}_{Y}(Y)$, then

The proof of Theorem 1.1 relies on a general result of independent interest describing the $V$-filtration associated to $f_{1},\ldots,f_{d}\in\mathcal{O}_{X}(X)$ (without any complete intersection assumption) in terms of the microlocal $V$-filtration associated to $g=\sum_{i=1}^{d}f_{i}y_{i}\in\mathcal{O}_{X}(X)[y_{1},\ldots,y_{d}]$; see Theorem 3.3 for the precise statement. Another application of this connection is a relation between $b$-functions corresponding to $f_{1},\ldots,f_{d}$ and microlocal $b$-functions corresponding to $g$. This greatly extends the main result of [Mustata], which says that the Bernstein-Sato polynomial $b_{Z}(s)$ of $Z$ is equal to the reduced Bernstein-Sato polynomial $b_{g}(s)/(s+1)$ of $g$.

The description in Theorem 1.1, together with the results on minimal exponents of hypersurfaces from [MP1], allow us to obtain similar results for local complete intersections. In order to state these, it is convenient to use a local version of the minimal exponent. If $Z$ is a local complete intersection in $X$ as above and $x\in Z$ is a point, then we put $\widetilde{\alpha}_{x}(Z):=\max_{V\ni x}\widetilde{\alpha}(V,Z\cap V)$, where the maximum is over the open neighborhoods $V$ of $x$ in $X$.

Let $X$ be a smooth, irreducible, $n$-dimensional complex algebraic variety and let $Z$ be a local complete intersection closed subscheme of $X$, of pure codimension $r$.

If $H$ is a smooth hypersurface in $X$ that contains no irreducible component of $Z$ and $Z_{H}=Z\cap H\hookrightarrow H$, then for every $x\in Z_{H}$, we have

Given a smooth morphism $\mu\colon X\to T$ such that for every $t\in T$, $Z_{t}:=Z\cap\mu^{-1}(t)\hookrightarrow X_{t}=\mu^{-1}(t)$ has pure codimension $r$, then the following hold:

For every $\alpha\in{\mathbf{Q}}_{>0}$, the set

There is an open subset $T_{0}$ of $T$ such that for every $t\in T_{0}$ and $x\in Z_{t}$, we have

In particular, the set \big{\{}\widetilde{\alpha}_{x}(X_{\mu(x)},Z_{\mu(x)})\mid x\in Z\big{\}} is finite. Moreover, if $s\colon T\to X$ is a section of $\mu$ such that $s(T)\subseteq Z$, then the set \big{\{}t\in T\mid\widetilde{\alpha}_{s(t)}(X_{t},Z_{t})\geq\alpha\big{\}} is open in $T$ for every $\alpha\in{\mathbf{Q}}_{>0}$.

If $x\in Z$ is a point defined by the ideal $\mathfrak{m}_{x}$ and the ideal defining $Z$ at $x$ is contained in $\mathfrak{m}_{x}^{k}$, for some $k\geq 2$, then

Another main result of the paper says that the minimal exponent controls the behavior of the Hodge filtration on local cohomology. Recall that if $Z$ is locally a complete intersection of pure codimension $r$, the only nontrivial local cohomology of the structure sheaf is $\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$, and if $Z$ is defined by $f_{1},\ldots,f_{r}$, then

The Hodge filtration on this mixed Hodge module was studied in [MP2]. There is another natural filtration, the order filtration (or Ext filtration) given by

where $I_{Z}$ is the ideal defining $Z$. For every $k\geq 0$ we have $F_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\subseteq O_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ and if equality holds for $k=p$, then it holds for all $k$, with $0\leq k\leq p$. The singularity level $p(Z)$ of the Hodge filtration on $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is

with the convention that this is $-1$ if the above set is empty. With this notation, we prove

If $X$ is a smooth, irreducible, complex algebraic variety and $Z$ is a local complete intersection closed subscheme of $X$, of pure codimension $r$, then

In particular, by combining Theorems 1.3 and 1.2, we see that the invariant $p(Z)$ satisfies analogous properties to those in Theorem 1.2. This was already shown in [MP2, Section 9] by different methods. For an application of Theorem 1.3 to an Inversion-of-Adjunction type statement, see Corollary 5.2. The main ingredient in the proof of Theorem 1.3 is the description of the Hodge filtration on $\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$ in terms of the $V$-filtration on $B_{\mathbf{f}}$. This relies on the interplay between the Hodge filtration and the $V$-filtration for filtered $\mathcal{D}$-modules that underlie mixed Hodge modules. In the case of one function, this is built in the definition of Hodge modules. However, the case of several functions is more subtle and has only recently been elucidated in [CD]. Using these results, we give the following description of the Hodge filtration on local cohomology:

If $X$ is a smooth, irreducible, complex algebraic variety and $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$ define a complete intersection closed subscheme $Z$ of codimension $r$, then for every $p\geq 0$, we have

One interesting question that remains open is the precise relation between $\widetilde{\alpha}(Z)$ and the Bernstein-Sato polynomial of $Z$. We recall that for an arbitrary closed subscheme of the smooth variety $X$, one can define a Bernstein-Sato polynomial $b_{Z}(s)\in{\mathbf{Q}}[s]$, extending the classical notion from the case of hypersurfaces (see [BMS]). As in the classical case, all its roots are negative rational numbers, with the largest root being $-\operatorname{lct}(X,Z)$. It is easy to see that if $Z$ is a (nonempty) local complete intersection of pure codimension $r$, then $(s+r)$ divides $b_{Z}(s)$, see Proposition 6.1 below. By analogy with the definition of the minimal exponent in the case of hypersurfaces, we define $\widetilde{\gamma}(Z)$ to be the negative of the largest root of $b_{Z}(s)/(s+r)$ (with the convention that this is infinite if $b_{Z}(s)/(s+r)=1$).

If $Z$ is locally a complete intersection in the smooth irreducible variety $X$, of pure codimension $r$, do we have $\widetilde{\alpha}(Z)=\widetilde{\gamma}(Z)$?

Note that in light of Theorem 1.3, a positive answer to Question 1.5 would provide a positive answer to [MP2, Conjecture 9.11], relating the Hodge filtration on $\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$ and the invariant $\widetilde{\gamma}(Z)$. We can prove the following relation between the two invariants:

With the notation in Question 1.5, we have $\widetilde{\alpha}(Z)\geq\widetilde{\gamma}(Z)$ and

We recall that by [BMS, Theorem 4], under the assumptions of Theorem 1.6, the subscheme $Z$ has rational singularities if and only if $\widetilde{\gamma}(Z)>r$. By combining Theorems 1.3 and 1.6, we obtain the following result, which gives a positive answer to [MP2, Conjecture 8.4].

If $X$ is a smooth, irreducible variety and $Z$ is a local complete intersection closed subscheme of $X$, of pure codimension $r$, then $Z$ has rational singularities if and only if $\widetilde{\alpha}(Z)>r$. In particular, if $F_{1}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=O_{1}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$, then $Z$ has rational singularities.

Outline of the paper. In Section 2 we review the basic facts about $V$-filtrations and $b$-functions. The following section is devoted to the result relating the $V$-filtration associated to $f_{1},\ldots,f_{d}$ and the microlocal $V$-filtration associated to $\sum_{i=1}^{d}f_{i}y_{i}$. In Section 4 we introduce the minimal exponent of a local complete intersection subscheme, prove the description in Theorem 1.1, as well as various general properties of this invariant, including the ones in Theorem 1.2. In Section 5 we relate the minimal exponent to the Hodge filtration on local cohomology, proving Theorems 1.3 and 1.4. Finally, in the last section we discuss the connection with the Bernstein-Sato polynomial and prove Theorem 1.6.

Acknowledgments. We would like to thank Mihnea Popa and Christian Schnell for many discussions related to the subject of this work. We are also grateful to Karl Schwede for providing some useful references.

Review of V𝑉V-filtrations

In this section we recall the definition and some basic properties of $V$-filtrations. For details, we refer to [Kashiwara], [BMS, Section 1], and [Saito-MHP, Section 3.1]. Let $X$ be a fixed smooth, irreducible, complex algebraic variety. Recall that $\mathcal{D}_{X}$ denotes the sheaf of differential operators on $X$. In this paper all $\mathcal{D}$-modules will be left $\mathcal{D}$-modules. For general facts about $\mathcal{D}$-modules, we refer to [HTT].

Given nonzero regular functions $f_{1},\ldots,f_{d}\in\mathcal{O}_{X}(X)$, we denote by $\mathfrak{a}\subseteq\mathcal{O}_{X}$ the ideal $(f_{1},\ldots,f_{d})$ and by $Z$ the closed subscheme of $X$ defined by $\mathfrak{a}$. We consider the graph embedding

and the $\mathcal{D}$-module theoretic push-forward $B_{\mathbf{f}}=\iota_{+}(\mathcal{O}_{X})$ (we denote by ${\mathbf{f}}$ the $d$-tuple $(f_{1},\ldots,f_{d})$). We denote the standard coordinates on ${\mathbf{A}}^{d}$ by $t_{1},\ldots,t_{d}$ and use multi-index notation, so for $\beta=(\beta_{1},\ldots,\beta_{d})\in{\mathbf{Z}}_{\geq 0}^{d}$, we put $t^{\beta}=t_{1}^{\beta_{1}}\cdots t_{d}^{\beta_{d}}$ and $\partial_{t}^{\beta}=\partial_{t_{1}}^{\beta_{1}}\cdots\partial_{t_{d}}^{\beta_{d}}$. We also put $\beta!=\prod_{i}\beta_{i}!$ and $|\beta|=\sum_{i}\beta_{i}$. Finally, we consider $s_{i}=-\partial_{t_{i}}t_{i}$ for $1\leq i\leq d$ and $s=\sum_{i=1}^{d}s_{i}$.

It is convenient to consider $B_{\mathbf{f}}$ as an $\mathcal{R}$-module on $X$, where $\mathcal{R}=\mathcal{D}_{X}\langle t_{1},\ldots,t_{d},\partial_{t_{1}},\ldots\partial_{t_{d}}\rangle$. The general description of $\mathcal{D}$-module push-forward via closed immersions gives

where the actions of $\mathcal{O}_{X}$ and $\partial_{t_{i}}$ are the obvious ones, while the actions of $D\in{\rm Der}_{{\mathbf{C}}}(\mathcal{O}_{X})$ and of the $t_{i}$ are given by

where $e_{1},\ldots,e_{d}$ is the standard basis of ${\mathbf{Z}}^{d}$. We will also consider on $B_{{\bf f}}$ the Hodge filtrationIt is often the case that one shifts this filtration so that what we denote by $F_{p}B_{{\bf f}}$ is considered to be $F_{p+d}B_{{\bf f}}$. given by

It is sometimes convenient to consider the larger $\mathcal{R}$-module $B_{\mathbf{f}}^{+}$ corresponding to the push-forward \iota_{+}\big{(}\mathcal{O}_{X}[1/f_{1}\cdots f_{d}]\big{)}, namely

We may also write the elements of $B_{\mathbf{f}}^{+}$ in terms of the operators $s_{1},\ldots,s_{d}$, as follows. Let us put $Q_{m}(x)=(-1)^{m}m!{x\choose m}=x(x-1)\cdots(x-m+1)\in{\mathbf{C}}[x]$ for $m\in{\mathbf{Z}}_{\geq 0}$ and

It follows from Lemma 7.1 that we can write $\partial_{t_{i}}^{\beta_{i}}=Q_{\beta_{i}}(s_{i})t_{i}^{-\beta_{i}}$ and since $t_{i}^{-\beta_{i}}\delta_{\mathbf{f}}=\tfrac{1}{f_{i}^{\beta_{i}}}\delta_{\mathbf{f}}$ in $B_{\mathbf{f}}^{+}$, we have

where $h_{\beta}\in\mathcal{O}_{X}[1/f_{1}\cdots f_{d}]$ for all $\beta\in{\mathbf{Z}}_{\geq 0}^{d}$ (with only finitely many nonzero).

Since the polynomials $Q_{\beta}(s_{1},\ldots,s_{d})$, with $\beta=(\beta_{1},\ldots,\beta_{d})$ running over ${\mathbf{Z}}^{d}_{\geq 0}$, give a basis of ${\mathbf{C}}[s_{1},\ldots,s_{d}]$ over ${\mathbf{C}}$, it is easy to see that if we put ${\mathbf{f}}^{\mathbf{s}}=f_{1}^{s_{1}}\cdots f_{d}^{s_{d}}$, then we have an isomorphism of ${\mathcal{R}}$-modules

that maps $\partial_{t}^{\beta}\delta_{\mathbf{f}}$ to $\tfrac{Q_{\beta}(s_{1},\ldots,s_{r})}{f_{1}^{\beta_{1}}\cdots f_{r}^{\beta_{r}}}{\mathbf{f}}^{\mathbf{s}}$. Note that a derivation $D\in{\rm Der}_{{\mathbf{C}}}(\mathcal{O}_{X})$ acts on ${\mathbf{f}}^{\mathbf{s}}$ in the expected way:

We also note that the action of $t_{i}$ on the left-hand side of (6) corresponds on the right-hand side to the automorphism that maps $s_{i}$ to $s_{i}+1$, and similarly, the action of $s_{i}=-\partial_{t_{i}}t_{i}$ on the left-hand side of (6) corresponds on the right-hand side to multiplication by $s_{i}$. We sometimes tacitly use this isomorphism to denote an element of $B_{\mathbf{f}}^{+}$ by $P(s_{1},\ldots,s_{d}){\mathbf{f}}^{\mathbf{s}}$, for some $P\in\mathcal{O}_{X}[1/f_{1}\cdots,f_{d},s_{1},\ldots,s_{d}]$. Note that it follows from (5) that if we write $P(s_{1},\ldots,s_{d})=\sum_{\beta}g_{\beta}Q_{\beta}(s_{1},\ldots,s_{d})$, with $g_{\beta}\in\mathcal{O}_{X}[1/f_{1}\cdots f_{d}]$, then $P(s_{1},\ldots,s_{d}){\mathbf{f}}^{\mathbf{s}}\in B_{\mathbf{f}}$ if and only if $g_{\beta}\in\mathcal{O}_{X}\cdot\tfrac{1}{f_{1}^{\beta_{1}}\cdots f_{d}^{\beta_{d}}}$ for all $\beta\in{\mathbf{Z}}_{\geq 0}^{d}$.

We now turn to the $V$-filtration. On $\mathcal{R}$ we consider the decreasing filtration

for $m\in{\mathbf{Z}}$. It is then clear that

Every $V^{\gamma}B_{{\bf f}}$ is a coherent $V^{0}\mathcal{R}$-submodule of $B_{{\bf f}}$.

$t_{i}\cdot V^{\gamma}B_{{\bf f}}\subseteq V^{\gamma+1}B_{{\bf f}}$ and $\partial_{t_{i}}\cdot V^{\gamma}B_{{\bf f}}\subseteq V^{\gamma-1}B_{{\bf f}}$ for all $i\leq d$ and $\gamma\in{\mathbf{Q}}$.

$V^{1}\mathcal{R}\cdot V^{\gamma}B_{{\bf f}}=V^{\gamma+1}B_{{\bf f}}$ if $\gamma\gg 0$.

The action of $s+\gamma$ on ${\rm Gr}_{V}^{\gamma}(B_{{\bf f}})$ is nilpotent for all $\gamma\in{\mathbf{Q}}$.

Here we put ${\rm Gr}_{V}^{\gamma}(B_{{\bf f}})=V^{\gamma}B_{{\bf f}}/V^{>\gamma}B_{{\bf f}}$, where $V^{>\gamma}B_{{\bf f}}=\bigcup_{\beta>\gamma}V^{\beta}B_{{\bf f}}$.

By the theory of Kashiwara [Kashiwara], extending a result of Malgrange [Malgrange], there is a unique such $V$-filtration. Uniqueness follows by easy arguments, while existence is a deeper statement.

We note that for every $\gamma\in{\mathbf{Q}}$, we have $V^{\gamma}B_{{\bf f}}|_{X\smallsetminus Z}=B_{{\bf f}}|_{X\smallsetminus Z}$.

We recall that the $V$-filtration on $B_{{\bf f}}$ induces on $\mathcal{O}_{X}\simeq\mathcal{O}_{X}\delta_{{\bf f}}$ the filtration by the multiplier ideals of $\mathfrak{a}$ (for the definition and basic properties of multiplier ideals, we refer to [Lazarsfeld, Section 9]). More precisely, it follows from [BMS, Theorem 1] that for every $\gamma\in{\mathbf{Q}}_{>0}$, we have

where $\operatorname{lct}(\mathfrak{a})$ is the log canonical threshold of $\mathfrak{a}$, characterized as $\min\{\lambda>0\mid\mathcal{J}(\mathfrak{a}^{\lambda})\neq\mathcal{O}_{X}\}$ (we also denote this by $\operatorname{lct}(X,Z)$).

The existence of $V$-filtrations is closely related to the existence of $b$-functions. Recall that for every $u\in B_{{\bf f}}$, the $b$-function $b_{u}(s)$ of $u$ is the monic generator of the ideal

We note that the condition in (8) is equivalent to $b(s)V^{0}\mathcal{R}\cdot u\subseteq V^{1}\mathcal{R}\cdot u$: this follows easily from Lemmas 7.3 and 7.4 in the Appendix and the fact that for all $i$ and $j$, we have $t_{i}\partial_{t_{j}}\cdot V^{0}\mathcal{R}\subseteq V^{0}\mathcal{R}$ and $t_{i}\cdot V^{0}\mathcal{R}\subseteq V^{1}\mathcal{R}$. It follows from the results in [Kashiwara] (see also [BMS]) that for every $u\in B_{{\bf f}}$, the ideal (8) is nonzero and thus $b_{u}(s)$ is well-defined. Moreover, all its roots are rational. The $V$-filtration can then be described as

In particular, for $u=\delta_{\mathbf{f}}\in B_{\mathbf{f}}$, the $b$-function $b_{u}(s)$ is the Bernstein-Sato polynomial of the ideal $\mathfrak{a}$, introduced and studied in [BMS]; this only depends on $\mathfrak{a}$ (not on the choice of $f_{1},\ldots,f_{d}$) and we denote it by $b_{Z}(s)$. In the case $d=1$ and $f=f_{1}$, this is the $b$-function of a hypersurface, introduced independently by Bernstein and Sato; we also denote it by $b_{f}(s)$. Note that for any $d$, it follows from (7) and (9) that

Suppose now that $d=1$, so we have only one nonzero regular function, that we denote $f$. In this case, Saito introduced in [Saito_microlocal] the microlocal $V$-filtration associated to $f$, defined as follows. Instead of $B_{f}$, we consider

which is a left module over $\widetilde{\mathcal{R}}=\mathcal{D}_{X}\langle t,\partial_{t},\partial_{t}^{-1}\rangle$. Note that the relation $[\partial_{t},t]=1$ implies $[\partial_{t}^{-1},t]=-\partial_{t}^{-2}$. The action of $\mathcal{O}_{X}$ and of $\partial_{t}$, $\partial_{t}^{-1}$ on $\widetilde{B}_{f}$ are the obvious ones, while the action of derivations and of $t$ are given by the following analogue of (4): for every $D\in{\rm Der}_{{\mathbf{C}}}(\mathcal{O}_{X})$, $h\in\mathcal{O}_{X}$, and $j\in{\mathbf{Z}}$, we have

The $V$-filtration on $\widetilde{\mathcal{R}}$ is defined as before: for $m\in{\mathbf{Z}}$, we have

where this time $i\in{\mathbf{Z}}_{\geq 0}$ and $j\in{\mathbf{Z}}$. It is easy to see that

The Hodge filtration on $\widetilde{B}_{f}$ is given by

On the other hand, the microlocal $V$-filtration on $\widetilde{B}_{f}$ is given by

where $j\in{\mathbf{Z}}$ is such that $0<\gamma-j\leq 1$. The following properties follow easily from the properties of the $V$-filtration on $B_{f}$:

$V^{1}\widetilde{\mathcal{R}}\cdot V^{\gamma}\widetilde{B}_{f}\subseteq V^{\gamma+1}\widetilde{B}_{f}$ for all $\gamma\in{\mathbf{Q}}$.

Every $V^{\gamma}\widetilde{B}_{f}$ is a finitely generated $V^{0}\widetilde{\mathcal{R}}$-module and it generates $\widetilde{B}_{f}$ over $\widetilde{\mathcal{R}}$.

$s+\gamma$ is nilpotent on ${\rm Gr}_{V}^{\gamma}(\widetilde{B}_{f})$ for every $\gamma\in{\mathbf{Q}}$.

A useful property of the microlocal $V$-filtration is that for every $j\in{\mathbf{Z}}$ and every $\gamma\in{\mathbf{Q}}$, multiplication by $\partial_{t}^{j}$ gives an isomorphism

Given $u\in\widetilde{B}_{f}$, the microlocal $b$-function $\widetilde{b}_{u}(s)\in{\mathbf{C}}[s]$ is the monic generator of the ideal

(the fact that this ideal is nonzero and all roots of $\widetilde{b}_{u}(s)$ are rational, follows from the fact that $V^{\gamma}\widetilde{B}_{f}\subseteq V^{1}\widetilde{\mathcal{R}}\cdot u$ for $\gamma\gg 0$). We have the following analogue of (9) describing the microlocal $V$-filtration in terms of microlocal $b$-functions:

An important example is that when $u=\delta_{f}\in\widetilde{B}_{f}$, when we write $\widetilde{b}_{f}(s)$ for $\widetilde{b}_{\delta_{f}}(s)$. If $f$ is not invertible, then it is easy to see that $b_{f}(s)=b_{\delta_{f}}(s)$ is divisible by $(s+1)$, and in fact we have

(see [Saito_microlocal, Proposition 0.3]).

Under the same assumption that $f$ is not invertible, the negative of the largest root of $b_{f}(s)/(s+1)$ is the minimal exponent $\widetilde{\alpha}(f)$, that we also write as $\widetilde{\alpha}(H)$ if $H$ is the hypersurface defined by $f$. Here we make the convention that $\widetilde{\alpha}(f)=\infty$ if $b_{f}(s)/(s+1)=1$. Note that by (10), we have \min\big{\{}\widetilde{\alpha}(H),1\big{\}}=\operatorname{lct}(X,H). We also recall that by a result of Saito (see [Saito-B, Theorem 0.4]), we have $\widetilde{\alpha}(H)>1$ if and only if $H$ has rational singularities. For a discussion of minimal exponents and basic properties, see [MP1, Section 6]. One property that is very relevant for us is its connection with the $V$-filtration: it follows from (13) and the fact that $b_{f}(s)/(s+1)=\widetilde{b}_{f}(s)$ that

In terms of the $V$-filtration on $B_{f}$, this is equivalent to the fact that for every nonnegative integer $q$ and every rational number $\gamma\in(0,1]$, we have

We will also make use of a local version of the minimal exponent of hypersurfaces. If $f\in\mathcal{O}_{X}(X)$ and $H$ are as above and $x\in H$, then

where the maximum is over all open neighborhoods $U$ of $x$. With this notation, we have

General V𝑉V-filtrations via microlocal V𝑉V-filtrations along hypersurfaces

Let $X$ be a smooth, irreducible, complex algebraic variety. In this section we consider nonzero regular functions $f_{1},\ldots,f_{d}\in\mathcal{O}_{X}(X)$ and let $g=\sum_{i=1}^{d}f_{i}y_{i}\in\mathcal{O}_{Y}(Y)$, where $Y=X\times{\mathbf{A}}^{d}$ and we denote by $y_{1},\ldots,y_{d}$ the standard coordinates on ${\mathbf{A}}^{d}$. Our goal is to relate the $V$-filtration on $B_{{\bf f}}=\bigoplus_{\alpha\in{\mathbf{Z}}_{\geq 0}^{d}}\mathcal{O}_{X}\partial_{t}^{\alpha}\delta_{{\bf f}}$ and the microlocal $V$-filtration on $\widetilde{B}_{g}=\bigoplus_{j\in{\mathbf{Z}}}\mathcal{O}_{Y}\partial_{z}^{j}\delta_{g}$ (note that we denote by $z$ the extra variable that acts on $\widetilde{B}_{g}$ in order to avoid confusion with the variables $t_{1},\ldots,t_{d}$ that act on $B_{{\bf f}}$).

Note that $g$ is homogeneous of degree $1$ with respect to the grading on $\mathcal{O}_{Y}$ such that $\mathcal{O}_{X}$ lies in degree and ${\rm deg}(y_{i})=1$ for all $i$. We have a corresponding grading on $\mathcal{D}_{Y}$ such that ${\rm deg}(\partial_{y_{i}})=-1$ for all $i$. Furthermore, on $\mathcal{D}_{Y}\langle z,\partial_{z},\partial_{z}^{-1}\rangle$ we have a grading such that ${\rm deg}(z)=1={\rm deg}(\partial_{z}^{-1})$ and ${\rm deg}(\partial_{z})=-1$.

then it follows easily from the formulas (11) and the fact that $g$ is homogeneous of degree $1$ that the decomposition $\widetilde{B}_{g}=\bigoplus_{m\in{\mathbf{Z}}}\widetilde{B}_{g}^{(m)}$ makes $\widetilde{B}_{g}$ a graded $\mathcal{D}_{Y}\langle z,\partial_{z},\partial_{z}^{-1}\rangle$-module. Let $\theta_{y}:=\sum_{i=1}^{d}y_{i}\partial_{y_{i}}\in\mathcal{D}_{Y}$.

For every $m\in{\mathbf{Z}}$ and every $u\in\widetilde{B}_{g}^{(m)}$, we have $(\theta_{y}-s)u=mu$.

We may and will assume that $u=hy^{\alpha}\partial_{z}^{|\alpha|-m}\delta_{g}$, for some $h\in\mathcal{O}_{X}$. On one hand, we have

Note that $V^{\gamma}\widetilde{B}_{g}$ is preserved by the action of $\theta_{y}-s$ for every $\gamma\in{\mathbf{Q}}$. By Lemma 3.1, the decomposition $\widetilde{B}_{g}=\bigoplus_{m\in{\mathbf{Z}}}\widetilde{B}_{g}^{(m)}$ is an eigenspace decomposition with respect to the endomorphism $\theta_{y}-s$. We deduce that we get an induced decomposition

Finally, we note that it follows from (12) that

We now define the map that will allow us to compare the $V$-filtration on $B_{{\bf f}}$ with the microlocal $V$-filtration on $\widetilde{B}_{g}$. Let $\varphi\colon\widetilde{B}_{g}\to B_{{\bf f}}$ be the unique $\mathcal{O}_{X}$-linear map such that

It is clear from the definition that for every $m\in{\mathbf{Z}}$, $\varphi$ induces an isomorphism of $\mathcal{O}_{X}$-modules $\widetilde{B}_{g}^{(m)}\simeq B_{{\bf f}}$. We collect in the following proposition some basic properties of $\varphi$.

With the above notation, the following hold:

The map $\varphi$ is $\mathcal{D}_{X}$-linear.

We have $\varphi(\partial_{z}u)=\varphi(u)=\varphi(\partial_{z}^{-1}u)$ for every $u\in\widetilde{B}_{g}$.

We have $\varphi(y_{i}u)=\partial_{t_{i}}\varphi(u)$ for every $u\in\widetilde{B}_{g}$ and $1\leq i\leq d$.

We have $\varphi(\partial_{y_{i}}u)=-t_{i}\varphi(u)$ for every $u\in\widetilde{B}_{g}$ and $1\leq i\leq d$.

We have $\varphi(su)=(s-m)\varphi(u)$ for every $u\in\widetilde{B}^{(m)}_{g}$, where $m\in{\mathbf{Z}}$.

For i), since $\varphi$ is $\mathcal{O}_{X}$-linear by definition, it is enough to show that $\varphi(Du)=D\varphi(u)$ for every $D\in{\rm Der}_{{\mathbf{C}}}(\mathcal{O}_{X})$ and every $u\in\widetilde{B}_{g}$. We may and will assume that $u=hy^{\alpha}\partial_{z}^{j}\delta_{g}$ for some $h\in\mathcal{O}_{X}$, $\alpha\in{\mathbf{Z}}_{\geq 0}^{d}$ and $j\in{\mathbf{Z}}$. In this case we have

The assertions in ii) and iii) follow directly from definition. In order to prove iv), we may assume that $u=hy^{\alpha}\partial_{z}^{j}\delta_{g}$ for some $h\in\mathcal{O}_{X}$, $\alpha\in{\mathbf{Z}}_{\geq 0}^{d}$ and $j\in{\mathbf{Z}}$. We then have

Finally, in order to prove v), we note that by Lemma 3.1, if $u\in\widetilde{B}^{(m)}_{g}$, then $su=(\theta_{y}-m)u$, hence using iii) and iv), we have

We now come to the main result of this section. Let us denote by $\varphi_{0}$ the restriction of $\varphi$ to $\widetilde{B}_{g}^{(0)}$. Note that since $\varphi_{0}$ is bijective, the assertion in the next theorem together with (16) say that the $V$-filtration on $B_{{\bf f}}$ and the microlocal $V$-filtration on $\widetilde{B}_{g}$ determine each other.

With the above notation, for every $\gamma\in{\mathbf{Q}}$, we have

The argument is similar to that proving the uniqueness of $V$-filtrations (see for example [Saito-MHP, Lemme 3.1.2]). Recall that we write

Note that by definition $W^{\bullet}B_{{\bf f}}$ is an exhaustive, decreasing filtration indexed by rational numbers, which is discrete and left continuous (since the microlocal filtration on $\widetilde{B}_{g}$ has these properties) and Proposition 3.2i) implies that each $W^{\gamma}B_{{\bf f}}$ is a $\mathcal{D}_{X}$-submodule of $B_{{\bf f}}$. This filtration also satisfies

Indeed, note that if $u\in V^{\gamma}\widetilde{B}_{g}^{(0)}$, then $-\partial_{z}^{-1}\partial_{y_{i}}u\in V^{\gamma+1}\widetilde{B}_{g}^{(0)}$ and it follows from properties ii) and iv) in Proposition 3.2 that

Indeed, if $u\in V^{\gamma}\widetilde{B}_{g}^{(0)}$, then $\partial_{z}y_{i}u\in V^{\gamma-1}\widetilde{B}_{g}^{(0)}$, and it follows from properties ii) and iii) in Proposition 3.2 that

In particular, we see that each $W^{\gamma}B$ is a $V^{0}{\mathcal{R}}$-submodule of $B_{{\bf f}}$.

Furthermore, for every $\gamma\in{\mathbf{Q}}$, we have

Indeed, assertion v) in Proposition 3.2 gives $\varphi(su)=s\varphi(u)$ for every $u\in\widetilde{B}_{g}^{(0)}$ and we know that $s+\gamma$ is nilpotent on ${\rm Gr}^{\gamma}_{V}(\widetilde{B}_{g})$.

We can now prove the inclusion (18). If $\gamma$, $\gamma^{\prime}$ are distinct rational numbers, then both $s+\gamma$ and $s+\gamma^{\prime}$ are nilpotent on

(this follows from (21) and the fact that $s+\gamma$ is nilpotent on ${\rm Gr}_{V}^{\gamma}(B_{{\bf f}})$ by definition of the $V$-filtration on $B_{{\bf f}}$). This implies that the quotient in (22) is . We deduce that

Indeed, if $u\in V^{\gamma}B_{{\bf f}}$, since $W^{\bullet}B_{{\bf f}}$ is exhaustive, there is $\gamma^{\prime}$ such that $u\in W^{\gamma^{\prime}}B_{{\bf f}}$. If $\gamma^{\prime}\geq\gamma$, then we are done. Suppose now that $\gamma^{\prime}<\gamma$. The fact that the quotient in (22) is implies that we can write $u=u_{1}+u_{2}$, with $u_{1}\in V^{>\gamma}B_{{\bf f}}\cap W^{\gamma^{\prime}}B_{{\bf f}}$ and $u_{2}\in V^{\gamma}B_{{\bf f}}\cap W^{>\gamma^{\prime}}B_{{\bf f}}$. Note that $u$ lies in the right-hand side of (23) if and only if $u_{2}$ does. Also, we have $u_{2}\in V^{\gamma}B_{{\bf f}}\cap W^{\gamma^{\prime\prime}}B_{{\bf f}}$ for some $\gamma^{\prime\prime}>\gamma^{\prime}$. We can repeat the argument with $u$ replaced by $u_{2}$; since $W^{\bullet}B_{{\bf f}}$ is discrete, we see that after finitely many steps we conclude that $u\in W^{\gamma}B_{{\bf f}}+V^{>\gamma}B_{{\bf f}}$.

Using the fact that the filtration $V^{\bullet}B_{{\bf f}}$ is discrete, we deduce from (23) that for every $\gamma$, $\gamma^{\prime}\in{\mathbf{Q}}$, we have

We next note that given $\gamma\in{\mathbf{Q}}$, it follows from property iii) in the definition of the $V$-filtration on $B_{{\bf f}}$ that there is an integer $q_{0}$ such that for every integer $q\geq q_{0}$, we have

On the other hand, since $V^{\gamma+q_{0}}B_{{\bf f}}$ is a finitely generated $V^{0}{\mathcal{R}}$-module and $W^{\bullet}B_{{\bf f}}$ is exhaustive, there is $\beta$ such that $V^{\gamma+q_{0}}B_{{\bf f}}\subseteq W^{\beta}B_{{\bf f}}$. By taking $q$ such that $q-q_{0}+\beta\geq\gamma$, we conclude that

where the first inclusion follows from (24) and the third one follows from (19) and (20). This completes the proof of (18).

In order to complete the proof of the theorem, it is enough to also show that

In fact, we will prove the equivalent statement that

Note next that by Proposition 3.2i), every $U^{\gamma}\widetilde{B}_{g}$ is a $\mathcal{D}_{X}$-submodule of $\widetilde{B}_{g}$. Moreover, it follows directly from the definition that

Indeed, if $u\in\varphi_{0}^{-1}(V^{\gamma-m}B_{{\bf f}})$ for some $m\in{\mathbf{Z}}$, then using the fact that $[z,\partial_{z}^{-m}]=m\partial_{z}^{-m-1}$ (see Lemma 7.1 in the Appendix), we have

Note that $-su+mu\in\widetilde{B}_{g}^{(0)}$ and using Proposition 3.2 we see that

hence $z\partial_{z}^{-m}u\in U^{\gamma+1}\widetilde{B}_{g}$, proving (26). In particular, we see that each $U^{\gamma}\widetilde{B}_{g}$ is a $V^{0}\widetilde{\mathcal{R}}$-module.

Finally, $s+\gamma$ is nilpotent on ${\rm Gr}_{U}^{\gamma}(\widetilde{B}_{g})$ for every $\gamma\in{\mathbf{Q}}$. Indeed, suppose that $u_{0}\in\varphi_{0}^{-1}(V^{\gamma-m}B_{{\bf f}})$. In this case

for $N\gg 0$, where the equality follows from Proposition 3.2v). Since $P(s)\partial_{z}^{-m}=\partial_{z}^{-m}P(s-m)$ for every $P\in{\mathbf{C}}[s]$ (see Lemma 7.3 in Appendix), it follows that

We can now prove the inclusion (25). Since the argument is very similar to that we used in the proof of (18), we omit some of the details. First, we see that

using the fact that both $s+\gamma$ and $s+\gamma^{\prime}$ are nilpotent on this quotient. As before, we use the fact that $U^{\bullet}\widetilde{B}_{g}$ is exhaustive and discrete and $V^{\bullet}\widetilde{B}_{g}$ is discrete to deduce from (27) that for every $\gamma,\gamma^{\prime}\in{\mathbf{Q}}$, we have

Let us fix now $\gamma\in{\mathbf{Q}}$. Since $V^{0}\widetilde{B}_{g}$ is a finitely generated $V^{0}\widetilde{R}$-module and each $U^{\beta}\widetilde{B}_{g}$ is a $V^{0}\widetilde{R}$-module, it follows that there is $\beta\in{\mathbf{Q}}$ such that $V^{0}\widetilde{B}_{g}\subseteq U^{\beta}\widetilde{B}_{g}$. If we take $\gamma^{\prime}\in{\mathbf{Z}}$ such that $\gamma^{\prime}+\beta\geq\gamma$, then using (28) and (12) we conclude that

This completes the proof of the theorem. ∎

We end this section with another application of the map $\varphi$, relating $b$-functions (with respect to $f_{1},\ldots,f_{d}$) to microlocal $b$-functions (with respect to $g$).

For every $m\in{\mathbf{Z}}$ and every $u\in\widetilde{B}_{g}^{(m)}$, we have

By definition of $\widetilde{b}_{u}(s)$, working locally on $X$, we can find $P\in V^{1}\widetilde{\mathcal{R}}=\sum_{a-b\geq 1}\mathcal{D}_{Y}z^{a}\partial_{z}^{b}$ such that

We may and will assume that ${\rm deg}(P)=0$. This implies that we can write $P=\sum_{i=1}^{d}\partial_{y_{i}}P_{i}$ for some $P_{1},\ldots,P_{d}$ of degree $1$. If we put $Q_{i}=\partial_{z}P_{i}$ for all $i$, then $P=\sum_{i=1}^{d}\partial_{y_{i}}\partial_{z}^{-1}Q_{i}$. Applying $\varphi$ to (29), we obtain

By Proposition 3.2v), the left-hand side of (30) is equal to $\widetilde{b}_{u}(s-m)\varphi(u)$. On the other hand, it follows from properties ii) and iv) in Proposition 3.2 that the right-hand side of (30) is equal to

where $a,b\in{\mathbf{Z}}$ are such that $a\geq b$ and $a\geq 0$, while $\alpha,\beta\in{\mathbf{Z}}^{d}$ are such that $|\alpha|\leq|\beta|$ (this follows from the condition on $a$ and $b$ and the fact that ${\rm deg}(Q_{i})=0$). We can write $z^{a}\partial_{z}^{b}=(z^{a}\partial_{z}^{a})\partial_{z}^{b-a}\in{\mathbf{C}}[s]\cdot\partial_{z}^{b-a}$ (see Lemma 7.3 in Appendix), hence using Proposition 3.2 we conclude that for every $i$, we have

hence by the definition of $b_{\varphi(u)}(s)$, we have

Going in the opposite direction, it follows from the definition of $b_{\varphi(u)}(s)$ that, working locally on $X$, there is $T\in V^{1}{\mathcal{R}}$ such that