#  $V$-filtrations and minimal exponents for local complete intersections

Qianyu Chen  Department of Mathematics, University of Michigan, 530 Church
Street, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>,  Bradley Dirks  Department
of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA
<EMAIL_ADDRESS>,  Mircea Mustaţă  Department of Mathematics,
University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA
<EMAIL_ADDRESS>and  Sebastián Olano  Department of Mathematics, University
of Toronto, 40 St. George St., Toronto, Ontario, CANADA , M5S 2E4
<EMAIL_ADDRESS>
###### Abstract.

We define and study a notion of _minimal exponent_ for a local complete
intersection subscheme $Z$ of a smooth complex algebraic variety $X$,
extending the invariant defined by Saito in the case of hypersurfaces. Our
definition is in terms of the Kashiwara-Malgrange $V$-filtration associated to
$Z$. We show that the minimal exponent describes how far the Hodge filtration
and order filtration agree on the local cohomology
$\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$, where $r$ is the codimension of $Z$ in
$X$. We also study its relation to the Bernstein-Sato polynomial of $Z$. Our
main result describes the minimal exponent of a higher codimension subscheme
in terms of the invariant associated to a suitable hypersurface; this allows
proving the main properties of this invariant by reduction to the codimension
$1$ case. A key ingredient for our main result is a description of the
Kashiwara-Malgrange $V$-filtration associated to _any_ ideal
$(f_{1},\ldots,f_{r})$ in terms of the microlocal $V$-filtration associated to
the hypersurface defined by $\sum_{i=1}^{r}f_{i}y_{i}$.

###### 2020 Mathematics Subject Classification:

14F10, 14B05, 14J17

Q.C. was partially supported by NSF grant DMS-1952399, B.D. was partly
supported by NSF grant DMS-2001132 and NSF-MSPRF grant DMS-2303070, and M.M.
was partially supported by NSF grants DMS-2301463 and DMS-1952399.

##  1\. 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

$F_{k}\mathcal{H}^{1}_{Z}(\mathcal{O}_{X})\subseteq
O_{k}\mathcal{H}_{Z}^{1}(\mathcal{O}_{X}):=\mathcal{O}_{X}\big{(}(k+1)Z\big{)}/\mathcal{O}_{X}$

and Saito showed in [Saito-MLCT] that

(1)
$F_{k}\mathcal{H}^{1}_{Z}(\mathcal{O}_{X})=O_{k}\mathcal{H}^{1}(\mathcal{O}_{X})\,\,\text{for
all}\,\,k\leq p\quad\text{if and only if}\quad\widetilde{\alpha}(Z)\geq p+1.$

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

$B_{\mathbf{f}}:=\iota_{+}\mathcal{O}_{X}=\bigoplus_{\beta\in{\mathbf{Z}}_{\geq
0}^{d}}\mathcal{O}_{X}\partial_{t}^{\beta}\delta_{\mathbf{f}},$

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

$\widetilde{B}_{g}=\bigoplus_{j\in{\mathbf{Z}}}\mathcal{O}_{X}\partial_{t}^{j}\delta_{g}.$

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

$\widetilde{\alpha}(Z)=\sup\\{\gamma>0\mid\delta_{g}\in
V^{\gamma}\widetilde{B}_{g}\\}$

(see [Saito-MLCT, (1.3.4)]). 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

(2) $\widetilde{\alpha}(Z)\geq q+\gamma\quad\text{if and only
if}\quad\partial_{t}^{q}\delta_{g}\in V^{\gamma}B_{g}.$

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

(3)
$\widetilde{\alpha}(Z)=\left\\{\begin{array}[]{cl}\sup\\{\gamma>0\mid\delta_{\mathbf{f}}\in
V^{\gamma}B_{{\bf f}}\\},&\text{if}\,\,\delta_{\mathbf{f}}\not\in
V^{r}B_{\mathbf{f}};\\\\[5.69054pt]
\sup\\{r-1+q+\gamma\mid\partial_{t}^{\beta}\delta_{\mathbf{f}}\in
V^{r-1+\gamma}B_{{\bf f}}\,\,\text{for}\,\,|\beta|\leq
q\\},&\text{if}\,\,\delta_{\mathbf{f}}\in
V^{r}B_{\mathbf{f}},\end{array}\right.$

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

$\operatorname{lct}(X,Z)=\min\\{\widetilde{\alpha}(Z),r\\}.$

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
$U=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$.

######  Theorem 1.1.

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

$\widetilde{\alpha}(Z)=\widetilde{\alpha}(g|_{U}).$

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 $X$.

######  Theorem 1.2.

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 $r$.

  1. i)

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

$\widetilde{\alpha}_{x}(H,Z_{H})\leq\widetilde{\alpha}_{x}(X,Z).$

  2. ii)

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:

    1. ${\rm ii_{1})}$

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

$\big{\\{}x\in
Z\mid\widetilde{\alpha}_{x}(X_{\mu(x)},Z_{\mu(x)})\geq\alpha\\}$

is open in $Z$.

    2. ${\rm ii_{2})}$

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

$\widetilde{\alpha}_{x}(X_{t},Z_{t})=\widetilde{\alpha}_{x}(X,Z).$

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 $\alpha\in{\mathbf{Q}}_{>0}$.

  3. iii)

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

$\widetilde{\alpha}_{x}(Z)\leq\frac{n}{k}.$

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

$\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})=\mathcal{O}_{X}[1/f_{1}\cdots
f_{r}]/\sum_{i=1}^{r}\mathcal{O}_{X}[1/f_{1}\cdots\widehat{f_{i}}\cdots
f_{r}].$

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

$O_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=\big{\\{}u\in\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})\mid
I_{Z}^{k+1}u=0\big{\\}},$

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

$p(Z)={\rm sup}\big{\\{}k\geq 0\mid
F_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=O_{k}\mathcal{H}^{r}(\mathcal{O}_{X})\big{\\}},$

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

######  Theorem 1.3.

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

$p(Z)=\max\big{\\{}\lfloor\widetilde{\alpha}(Z)\rfloor-r,-1\big{\\}}.$

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
into 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:

######  Theorem 1.4.

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 pure codimension $r$, then for every $p\geq 0$, we
have

$F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=\left\\{\left[\sum_{|\alpha|\leq
p}\frac{\alpha_{1}!\cdots\alpha_{r}!h_{\alpha}}{f_{1}^{\alpha_{1}+1}\cdots
f_{r}^{\alpha_{r}+1}}\right]\mid\sum_{|\alpha|\leq
p}h_{\alpha}\partial_{t}^{\alpha}\delta_{\mathbf{f}}\in
V^{r}B_{\mathbf{f}}\right\\}.$

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
$b_{Z}(s)/(s+r)=1$).

######  Question 1.5.

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:

######  Theorem 1.6.

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

$\min\big{\\{}\widetilde{\alpha}(Z),r+1\big{\\}}=\min\big{\\{}\widetilde{\gamma}(Z),r+1\big{\\}}.$

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].

######  Corollary 1.7.

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.

We note that since the first version of this paper was written, a positive
answer to Question 1.5 has been given in [Dirks].

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 and to the anonymous referee
for the comments on a previous version of this article.

##  2\. Review of $V$-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

$\iota\colon X\hookrightarrow
W=X\times{\mathbf{A}}^{d},\quad\iota(x)=\big{(}x,f_{1}(x),\ldots,f_{d}(x)\big{)}$

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

$B_{\mathbf{f}}=\bigoplus_{\beta\in{\mathbf{Z}}_{\geq
0}^{d}}\mathcal{O}_{X}\partial_{t}^{\beta}\delta_{\mathbf{f}},$

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

(4) $D\cdot
h\partial_{t}^{\beta}\delta_{\mathbf{f}}=D(h)\partial_{t}^{\beta}\delta_{\mathbf{f}}-\sum_{i=1}^{d}D(f_{i})h\partial_{t}^{\beta+e_{i}}\delta_{\mathbf{f}}\quad\text{and}\quad
t_{i}\cdot
h\partial_{t}^{\beta}\delta_{\mathbf{f}}=f_{i}h\partial_{t}^{\beta}\delta_{\mathbf{f}}-\beta_{i}h\partial_{t}^{\beta-
e_{i}}\delta_{\mathbf{f}},$

where $e_{1},\ldots,e_{d}$ is the standard basis of ${\mathbf{Z}}^{d}$. We
will also consider on $B_{{\bf f}}$ the _Hodge filtration_ 111It 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

$F_{p}B_{{\bf f}}=\bigoplus_{|\beta|\leq
p}\mathcal{O}_{X}\partial_{t}^{\beta}\delta_{{\bf f}}.$

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

$B_{\mathbf{f}}^{+}=\bigoplus_{\beta\in{\mathbf{Z}}_{\geq
0}^{d}}\mathcal{O}_{X}[1/f_{1}\cdots
f_{d}]\partial_{t}^{\beta}\delta_{\mathbf{f}}.$

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

$Q_{\beta}(s_{1},\ldots,s_{d})=\prod_{i=1}^{d}Q_{\beta_{i}}(s_{i})\in{\mathbf{C}}[s_{1},\ldots,s_{d}]\quad\text{for}\quad\beta=(\beta_{1},\ldots,\beta_{d})\in{\mathbf{Z}}_{\geq
0}^{d}.$

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

(5)
$\sum_{\beta}h_{\beta}\partial_{t}^{\beta}\delta_{\mathbf{f}}=\sum_{\beta}h_{\beta}\tfrac{Q_{\beta}(s_{1},\ldots,s_{d})}{f_{1}^{\beta_{1}}\cdots
f_{d}^{\beta_{d}}}\delta_{\mathbf{f}},$

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

(6) $B_{\mathbf{f}}^{+}\simeq\mathcal{O}_{X}[1/f_{1}\cdots
f_{d},s_{1},\ldots,s_{d}]{\mathbf{f}}^{\mathbf{s}},$

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:

$D\cdot{\mathbf{f}}^{\mathbf{s}}=\sum_{i=1}^{d}\frac{s_{i}D(f_{i})}{f_{i}}{\mathbf{f}}^{\mathbf{s}}.$

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

$V^{m}\mathcal{R}=\bigoplus_{|\alpha|-|\beta|\geq
m}\mathcal{D}_{X}t^{\alpha}\partial_{t}^{\beta}$

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

$V^{0}\mathcal{R}=\mathcal{D}_{X}\langle t_{i},t_{i}\partial_{t_{j}}\mid
i,j\rangle\quad\text{and}\quad V^{1}\mathcal{R}=\sum_{i=1}^{d}t_{i}\cdot
V^{0}\mathcal{R}=\sum_{i=1}^{d}V^{0}\mathcal{R}\cdot t_{i}.$

The $V$-filtration on $B_{{\bf f}}$ is a decreasing, exhaustive filtration
$(V^{\gamma}B_{\mathbf{f}})_{\gamma}$ indexed by rational numbers, which is
_discrete and left-continuous 222This means that there is a positive integer
$\ell$ such that $V^{\gamma}B_{\mathbf{f}}$ has constant value for all
$\gamma$ in an interval of the form
$\left(\tfrac{i-1}{\ell},\tfrac{i}{\ell}\right]$, with $i\in{\mathbf{Z}}$._
and satisfies the following properties:

  1. i)

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

  2. ii)

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

  3. iii)

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

  4. iv)

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.

######  Remark 2.1.

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

$\\{h\in\mathcal{O}_{X}\mid h\delta_{{\bf f}}\in V^{\gamma}B_{{\bf
f}}\\}=\mathcal{J}(\mathfrak{a}^{\gamma-\epsilon}),\quad\text{for}\quad
0<\epsilon\ll 1.$

In particular, we have

(7) $\delta_{{\bf f}}\in V^{\gamma}B_{{\bf f}}\quad\text{if and only
if}\quad\gamma\leq\operatorname{lct}(\mathfrak{a}),$

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

(8) $\big{\\{}b(s)\in{\mathbf{C}}[s]\mid b(s)u\in V^{1}\mathcal{R}\cdot
u\big{\\}}.$

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

(9) $V^{\gamma}B_{{\bf f}}=\big{\\{}u\in B_{{\bf f}}\mid\text{all roots
of}\,b_{u}(s)\,\text{are}\,\leq-\gamma\\}.$

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

(10) $\max\\{\lambda\in{\mathbf{Q}}\mid
b_{Z}(\lambda)=0\\}=-\operatorname{lct}(X,Z).$

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

$\widetilde{B}_{f}:=\bigoplus_{j\in{\mathbf{Z}}}\mathcal{O}_{X}\partial_{t}^{j}\delta_{f},$

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

(11) $D\cdot
h\partial_{t}^{j}\delta_{f}=D(h)\partial_{t}^{j}\delta_{f}-hD(f)\partial_{t}^{j+1}\delta_{f}\quad\text{and}\quad
t\cdot
h\partial_{t}^{j}\delta_{f}=fh\partial_{t}^{j}\delta_{f}-jh\partial_{t}^{j-1}\delta_{f}.$

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

$V^{m}\widetilde{\mathcal{R}}=\bigoplus_{i-j\geq
m}\mathcal{D}_{X}t^{i}\partial_{t}^{j},$

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

$V^{0}\widetilde{\mathcal{R}}=\mathcal{D}_{X}\langle
t,t\partial_{t},\partial_{t}^{-1}\rangle\quad\text{and}\quad
V^{j}\widetilde{\mathcal{R}}=\partial_{t}^{-j}\cdot
V^{0}\widetilde{\mathcal{R}}=V^{0}\widetilde{\mathcal{R}}\cdot\partial_{t}^{-j}\,\,\text{for
all}\,\,j\in{\mathbf{Z}}.$

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

$F_{p}\widetilde{B}_{f}=\bigoplus_{i\leq
p}\mathcal{O}_{X}\partial_{t}^{i}\delta_{f}.$

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

$V^{\gamma}\widetilde{B}_{f}=V^{\gamma}B_{f}\oplus\bigoplus_{j\geq
1}\mathcal{O}_{X}\partial_{t}^{-j}\delta_{f}\quad\text{for}\quad\gamma\leq 1$

and

$V^{\gamma}\widetilde{B}_{f}=\partial_{t}^{-j}V^{\gamma-j}\widetilde{B}_{f}\quad\text{for}\quad\gamma>1,$

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 $B_{f}$:

  1. i)

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

  2. ii)

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

  3. iii)

$s+\gamma$ 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

(12) $\partial_{t}^{j}\colon V^{\gamma}\widetilde{B}_{f}\simeq
V^{\gamma-j}\widetilde{B}_{f}$

(see [Saito_microlocal, Lemma 2.2]).

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

$\big{\\{}b(s)\in{\mathbf{C}}[s]\mid b(s)u\in
V^{1}\widetilde{\mathcal{R}}\cdot u\big{\\}}$

(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:

(13) $V^{\gamma}\widetilde{B}_{f}=\big{\\{}u\in\widetilde{B}_{f}\mid\text{all
roots of}\,\,\widetilde{b}_{u}(s)\,\,\text{are}\,\leq-\gamma\\}.$

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

$\widetilde{b}_{f}(s)=b_{f}(s)/(s+1)$

(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

(14) $\widetilde{\alpha}(H)={\rm sup}\\{\gamma\in{\mathbf{Q}}_{\geq
0}\mid\delta\in V^{\gamma}\widetilde{B}_{f}\\}.$

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

(15) $\widetilde{\alpha}(H)\geq q+\gamma\quad\text{if and only
if}\quad\partial_{t}^{q}\delta_{f}\in V^{\gamma}B_{f}$

(see [Saito-MLCT]).

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

$\widetilde{\alpha}_{x}(f):=\max_{U\ni x}\widetilde{\alpha}(f|_{U}),$

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

$\widetilde{\alpha}(f)=\min_{x\in H}\widetilde{\alpha}_{x}(f).$

##  3\. General $V$-filtrations via microlocal $V$-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 $0$ 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 ${\rm deg}(\partial_{z})=-1$.

We can write

$\widetilde{B}_{g}=\bigoplus_{j\in{\mathbf{Z}}}\bigoplus_{\alpha\in{\mathbf{Z}}_{\geq
0}^{d}}\mathcal{O}_{X}y^{\alpha}\partial_{z}^{j}\delta_{g}.$

If for every $m\in{\mathbf{Z}}$ we put

$\widetilde{B}_{g}^{(m)}=\bigoplus_{\alpha\in{\mathbf{Z}}_{\geq
0}^{d}}\mathcal{O}_{X}y^{\alpha}\partial_{z}^{|\alpha|-m}\delta_{g},$

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
$\theta_{y}:=\sum_{i=1}^{d}y_{i}\partial_{y_{i}}\in\mathcal{D}_{Y}$.

######  Lemma 3.1.

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

###### Proof.

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

$\theta_{y}\cdot u=\sum_{i=1}^{d}\alpha_{i}y_{i}hy^{\alpha-
e_{i}}\partial_{z}^{|\alpha|-m}\delta_{g}-\sum_{i=1}^{d}y_{i}f_{i}hy^{\alpha}\partial_{z}^{|\alpha|-m+1}\delta_{g}=|\alpha|hy^{\alpha}\partial_{z}^{|\alpha|-m}\delta_{g}-ghy^{\alpha}\partial_{z}^{|\alpha|-m+1}\delta_{g}.$

On the other hand, we have

$su=-\partial_{z}zu=-ghy^{\alpha}\partial_{z}^{|\alpha|-m+1}\delta_{g}+\big{(}|\alpha|-m\big{)}hy^{\alpha}\partial_{z}^{|\alpha|-m}\delta_{g},$

and the formula in the lemma follows. ∎

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

$V^{\gamma}\widetilde{B}_{g}=\bigoplus_{m\in{\mathbf{Z}}}V^{\gamma}\widetilde{B}_{g}^{(m)},\quad\text{where}\quad
V^{\gamma}\widetilde{B}_{g}^{(m)}=V^{\gamma}\widetilde{B}_{g}\cap\widetilde{B}_{g}^{(m)}.$

We get a corresponding decomposition

${\rm Gr}_{V}^{\gamma}(\widetilde{B}_{g})=\bigoplus_{m\in{\mathbf{Z}}}{\rm
Gr}_{V}^{\gamma}(\widetilde{B}_{g}^{(m)}),\quad\text{where}\quad{\rm
Gr}_{V}^{\gamma}(\widetilde{B}_{g}^{(m)})=V^{\gamma}\widetilde{B}_{g}^{(m)}/V^{>\gamma}\widetilde{B}_{g}^{(m)}.$

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

(16)
$V^{\gamma}\widetilde{B}_{g}^{(m)}=\partial_{z}^{-m}V^{\gamma-m}\widetilde{B}_{g}^{(0)}\quad\text{for
all}\quad\gamma\in{\mathbf{Q}},m\in{\mathbf{Z}}.$

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

(17)
$\varphi(y^{\alpha}\partial_{z}^{j}\delta_{g})=\partial_{t}^{\alpha}\delta_{{\bf
f}}\quad\text{for all}\quad\alpha\in{\mathbf{Z}}_{\geq
0}^{d},j\in{\mathbf{Z}}.$

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 $\varphi$.

######  Proposition 3.2.

With the above notation, the following hold:

  1. i)

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

  2. ii)

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

  3. iii)

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

  4. iv)

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

  5. v)

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

###### Proof.

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

$\varphi(Du)=\varphi\big{(}D(h)y^{\alpha}\partial_{z}^{j}\delta_{g}-hD(g)y^{\alpha}\partial_{z}^{j+1}\delta_{g}\big{)}=\varphi\big{(}D(h)y^{\alpha}\partial_{z}^{j}\delta_{g}-\sum_{i=1}^{d}hD(f_{i})y^{\alpha+e_{i}}\partial_{z}^{j+1}\delta_{g}\big{)}$
$=D(h)\partial_{t}^{\alpha}\delta_{{\bf
f}}-\sum_{i=1}^{d}hD(f_{i})\partial_{t}^{\alpha+e_{i}}\delta_{{\bf f}}=D\cdot
h\partial_{t}^{\alpha}\delta_{{\bf f}}=D\varphi(u).$

This completes the proof of i).

The assertions in ii) and iii) follow directly from the 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

$\varphi(\partial_{y_{i}}u)=\varphi\big{(}h\alpha_{i}y^{\alpha-
e_{i}}\partial_{z}^{j}\delta_{g}-f_{i}hy^{\alpha}\partial_{z}^{j+1}\delta_{g}\big{)}$
$=\alpha_{i}h\partial_{t}^{\alpha-e_{i}}\delta_{{\bf
f}}-f_{i}h\partial_{t}^{\alpha}\delta_{{\bf f}}=-t_{i}\cdot
h\partial_{t}^{\alpha}\delta_{{\bf f}}=-t_{i}\varphi(u).$

We thus obtain the assertion in iv).

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

$\varphi(su)=\varphi\big{(}(\theta_{y}-m)u\big{)}=\sum_{i=1}^{d}\varphi(y_{i}\partial_{y_{i}}u)-m\varphi(u)=-\sum_{i=1}^{d}\partial_{t_{i}}t_{i}\varphi(u)-m\varphi(u)=(s-m)\varphi(u).$

∎

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.

######  Theorem 3.3.

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

$V^{\gamma}\widetilde{B}_{g}^{(0)}=\varphi_{0}^{-1}(V^{\gamma}B_{{\bf f}}).$

###### Proof.

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

${\mathcal{R}}=\mathcal{D}_{X}\langle
t_{1},\ldots,t_{d},\partial_{t_{1}},\ldots,\partial_{t_{d}}\rangle\quad\text{and}\quad\widetilde{\mathcal{R}}=\mathcal{D}_{Y}\langle
z,\partial_{z},\partial_{z}^{-1}\rangle.$

Let’s prove first the inclusion

(18) $V^{\gamma}B_{{\bf f}}\subseteq W^{\gamma}B_{{\bf
f}}:=\varphi_{0}(V^{\gamma}\widetilde{B}_{g}^{(0)})\quad\text{for
all}\quad\gamma\in{\mathbf{Q}}.$

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

(19) $t_{i}\cdot W^{\gamma}B_{{\bf f}}\subseteq W^{\gamma+1}B_{{\bf
f}}\quad\text{for all}\quad\gamma\in{\mathbf{Q}},1\leq i\leq d.$

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

$t_{i}\cdot\varphi_{0}(u)=\varphi_{0}(-\partial_{z}^{-1}\partial_{y_{i}}u)\in
W^{\gamma+1}B_{{\bf f}}.$

We also have

(20) $\partial_{t_{i}}\cdot W^{\gamma}B_{{\bf f}}\subseteq W^{\gamma-1}B_{{\bf
f}}\quad\text{for all}\quad\gamma\in{\mathbf{Q}},1\leq i\leq d.$

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

$\partial_{t_{i}}\cdot\varphi_{0}(u)=\varphi_{0}(\partial_{z}y_{i}u)\in
W^{\gamma-1}B_{{\bf f}}.$

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

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

(21) $s+\gamma\quad\text{is nilpotent on}\quad{\rm Gr}_{W}^{\gamma}(B_{{\bf
f}}).$

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

(22) $\frac{V^{\gamma}B_{{\bf f}}\cap W^{\gamma^{\prime}}B_{{\bf
f}}}{(V^{>\gamma}B_{{\bf f}}\cap W^{\gamma^{\prime}}B_{{\bf
f}})+(V^{\gamma}B_{{\bf f}}\cap W^{>\gamma^{\prime}}B_{{\bf f}})}$

(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 $0$. We deduce that

(23) $V^{\gamma}B_{{\bf f}}\subseteq W^{\gamma}B_{{\bf f}}+V^{>\gamma}B_{{\bf
f}}\quad\text{for all}\quad\gamma\in{\mathbf{Q}}.$

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 $0$ 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

(24) $V^{\gamma}B_{{\bf f}}\subseteq W^{\gamma}B_{{\bf
f}}+V^{\gamma^{\prime}}B_{{\bf f}}.$

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

$V^{\gamma+q}B_{{\bf f}}\subseteq V^{q-q_{0}}{\mathcal{R}}\cdot
V^{\gamma+q_{0}}B_{{\bf f}}.$

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

$V^{\gamma}B_{{\bf f}}\subseteq W^{\gamma}B_{{\bf f}}+V^{\gamma+q}B_{{\bf
f}}\subseteq W^{\gamma}B_{{\bf f}}+V^{q-q_{0}}{\mathcal{R}}\cdot
W^{\beta}B_{{\bf f}}\subseteq W^{\gamma}B_{{\bf f}}+W^{\beta+q-q_{0}}B_{{\bf
f}}\subseteq W^{\gamma}B_{{\bf f}},$

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

$\varphi_{0}(V^{\gamma}\widetilde{B}_{g}^{(0)})\subseteq V^{\gamma}B_{{\bf
f}}\quad\text{for all}\quad\gamma\in{\mathbf{Q}}.$

In fact, we will prove the equivalent statement that

(25) $V^{\gamma}\widetilde{B}_{g}\subseteq
U^{\gamma}\widetilde{B}_{g}:=\bigoplus_{m\in{\mathbf{Z}}}\partial_{z}^{-m}\varphi_{0}^{-1}(V^{\gamma-m}B_{{\bf
f}})\quad\text{for all}\quad\gamma\in{\mathbf{Q}}.$

It is clear from the definition that $U^{\bullet}\widetilde{B}_{g}$ is an
exhaustive, decreasing filtration indexed by rational numbers (since the
$V$-filtration on $B_{{\bf f}}$ has these properties). Moreover, if $\ell$ is
a positive integer such that $V^{\gamma}B_{{\bf f}}$ is constant for $\gamma$
in each interval of the form $\left(\tfrac{i-1}{\ell},\tfrac{i}{\ell}\right]$,
with $i\in{\mathbf{Z}}$, then it follows from the definition that
$U^{\gamma}\widetilde{B}_{g}$ is constant for $\gamma$ in such an interval.
Therefore $U^{\bullet}\widetilde{B}_{g}$ is discrete and left continuous.

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

$\partial_{z}^{j}\cdot U^{\gamma}\widetilde{B}_{g}\subseteq
U^{\gamma-j}\widetilde{B}_{g}\quad\text{for
all}\quad\gamma\in{\mathbf{Q}},j\in{\mathbf{Z}}.$

We also have

(26) $z\cdot U^{\gamma}\widetilde{B}_{g}\subseteq
U^{\gamma+1}\widetilde{B}_{g}\quad\text{for all}\quad\gamma\in{\mathbf{Q}}.$

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

$z\partial_{z}^{-m}u=\partial_{z}^{-m}(zu+m\partial_{z}^{-1}u)=\partial_{z}^{-m-1}(\partial_{z}zu+mu)=\partial_{z}^{-m-1}(-su+mu).$

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

$\varphi_{0}(-su+mu)=-(s-m)\varphi_{0}(u)\in V^{\gamma-m}B_{{\bf f}},$

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

$\varphi_{0}\big{(}(s+\gamma-m)^{N}u\big{)}=(s+\gamma-m)^{N}\varphi_{0}(u)\in
V^{>\gamma-m}B_{{\bf f}}$

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

$(s+\gamma)^{N}\partial_{z}^{-m}u=\partial_{z}^{-m}(s+\gamma-m)^{N}u\in
U^{>\gamma}\widetilde{B}_{g}\quad\text{for}\quad N\gg 0.$

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

(27) $\frac{V^{\gamma}\widetilde{B}_{g}\cap
U^{\gamma^{\prime}}\widetilde{B}_{g}}{(V^{>\gamma}\widetilde{B}_{g}\cap
U^{\gamma^{\prime}}\widetilde{B}_{g})+(V^{\gamma}\widetilde{B}_{g}+U^{>\gamma^{\prime}}\widetilde{B}_{g})}=0\quad\text{for}\quad\gamma\neq\gamma^{\prime}$

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

(28) $V^{\gamma}\widetilde{B}_{g}\subseteq
U^{\gamma}\widetilde{B}_{g}+V^{\gamma^{\prime}}\widetilde{B}_{g}.$

Let us fix now $\gamma\in{\mathbf{Q}}$. Since $V^{0}\widetilde{B}_{g}$ is a
finitely generated $V^{0}\widetilde{\mathcal{R}}$-module and each
$U^{\beta}\widetilde{B}_{g}$ is a $V^{0}\widetilde{\mathcal{R}}$-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

$V^{\gamma}\widetilde{B}_{g}\subseteq
U^{\gamma}\widetilde{B}_{g}+V^{\gamma^{\prime}}\widetilde{B}_{g}=U^{\gamma}\widetilde{B}_{g}+\partial_{z}^{-\gamma^{\prime}}\cdot
V^{0}\widetilde{B}_{g}$ $\subseteq
U^{\gamma}\widetilde{B}_{g}+\partial_{z}^{-\gamma^{\prime}}\cdot
U^{\beta}\widetilde{B}_{g}\subseteq
U^{\gamma}\widetilde{B}_{g}+U^{\gamma^{\prime}+\beta}\widetilde{B}_{g}\subseteq
U^{\gamma}\widetilde{B}_{g}.$

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 $g$).

######  Proposition 3.4.

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

$\widetilde{b}_{u}(s-m)=b_{\varphi(u)}(s).$

###### Proof.

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

(29) $\widetilde{b}_{u}(s)u=Pu.$

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

(30) $\varphi\big{(}\widetilde{b}_{u}(s)u)=\varphi(Pu).$

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

$\sum_{i=1}^{d}\varphi(\partial_{y_{i}}\partial_{z}^{-1}Q_{i}u)=-\sum_{i=1}^{d}t_{i}\cdot\varphi(Q_{i}u).$

Note that for every $i$, we have

$Q_{i}\in\sum_{\alpha,\beta,a,b}\mathcal{D}_{X}y^{\alpha}\partial_{y}^{\beta}z^{a}\partial_{z}^{b},$

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

$\varphi(Q_{i}u)\in\sum_{|\alpha|\leq|\beta|}\mathcal{D}_{X}[s]\cdot\partial_{t}^{\alpha}t^{\beta}\cdot\varphi(u)\subseteq
V^{0}{\mathcal{R}}\cdot\varphi(u).$

We thus conclude that

$\widetilde{b}_{u}(s-m)\varphi(u)\in\sum_{i=1}^{d}t_{i}\cdot
V^{0}{\mathcal{R}}\cdot\varphi(u)\subseteq V^{1}{\mathcal{R}}\cdot\varphi(u),$

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

(31) $b_{\varphi(u)}(s)\quad\text{divides}\quad\widetilde{b}_{u}(s-m).$

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

$b_{\varphi(u)}(s)\varphi(u)=T\varphi(u).$

We can write

$T=\sum_{|\alpha|\geq|\beta|+1}T_{\alpha,\beta}t^{\alpha}\partial_{t}^{\beta},\quad\text{with}\quad
T_{\alpha,\beta}\in\mathcal{D}_{X}.$

Let us consider

$\widetilde{T}=\sum_{|\alpha|\geq|\beta|+1}T_{\alpha,\beta}(-y)^{\alpha}\partial_{y}^{\beta}\partial_{z}^{|\beta|-|\alpha|}\in
V^{1}\widetilde{\mathcal{R}},$

of degree $0$, so $\widetilde{T}u\in\widetilde{B}_{g}^{(m)}$. Using
Proposition 3.2, we see that

$\varphi\big{(}b_{\varphi(u)}(s+m)u\big{)}=b_{\varphi(u)}(s)\varphi(u)=T\varphi(u)=\varphi(\widetilde{T}u).$

Since the restriction of $\varphi$ to $\widetilde{B}_{g}^{(m)}$ is injective,
we conclude that

$b_{\varphi(u)}(s+m)u=\widetilde{T}u.$

We deduce using the definition of $\widetilde{b}_{u}$ that

(32) $\widetilde{b}_{u}(s)\quad\text{divides}\quad b_{\varphi(u)}(s+m).$

By combining (31) and (32), we see that $\widetilde{b}_{u}(s-m)$ and
$b_{\varphi(u)}(s)$ are monic polynomials that divide each other, hence they
are equal. ∎

######  Remark 3.5.

Note that if we apply the above proposition for
$u=\delta_{g}\in\widetilde{B}_{g}^{(0)}$, then we recover the fact that the
Bernstein-Sato polynomial of the ideal $(f_{1},\ldots,f_{d})$ coincides with
the microlocal $b$-function of $\delta_{g}$ (which, as we have mentioned in
Section 2, is equal to $b_{g}(s)/(s+1)$). This is the main result in
[Mustata].

##  4\. The minimal exponent of a local complete intersection subscheme

Our goal in this section is to define and study the minimal exponent of a
local complete intersection subscheme. Let $X$ be a smooth, irreducible
complex algebraic variety and $Z$ a (nonempty) proper closed subscheme of $X$.

######  Remark 4.1.

We note that in general we have $\operatorname{lct}(X,Z)\leq{\rm
codim}_{X}(Z)$. Indeed, the inclusion $Z_{\rm red}\hookrightarrow Z$ implies
$\operatorname{lct}(X,Z)\leq\operatorname{lct}(X,Z_{\rm red})$, hence we may
assume that $Z$ is reduced. If $U$ is an open subset of $X$ such that $U\cap
Z$ is smooth and irreducible of codimension $r$ in $U$, then
$\operatorname{lct}(X,Z)\leq\operatorname{lct}(U,Z\cap U)=r$.

######  Remark 4.2.

Note also that if $Z$ is Cohen-Macaulay, of pure codimension $r$, and
$\operatorname{lct}(X,Z)=r$, then $Z$ is reduced. Indeed, if this is not the
case, then $Z$ is not generically reduced (being Cohen-Macaulay). It follows
that we have an irreducible component $Z_{0}$ of $Z$ such that the local ring
$\mathcal{O}_{Z,Z_{0}}$ is not a field; therefore the embedding dimension $m$
of $\mathcal{O}_{Z,Z_{0}}$ is positive. After possibly replacing $X$ by a
suitable open subset that intersects $Z_{0}$ nontrivially, we may assume that
$Z$ is irreducible, $Z_{0}$ is smooth, and there is a smooth, irreducible
subvariety $W$ of $X$ of dimension $\dim(Z_{0})+m=n-r+m$ such that $Z$ is
contained in $W$ and, in fact, the ideal defining $Z$ in $W$ is contained in
the ideal $I_{Z_{0}/W}^{2}$, where $I_{Z_{0}/W}$ is the ideal defining $Z_{0}$
in $W$. By considering the exceptional divisor on the blow-up of $W$ along
$Z_{0}$ and the description of $\operatorname{lct}(W,Z)$ in terms of log
resolutions (see [Lazarsfeld, Example 9.3.16]), it follows easily that
$\operatorname{lct}(W,Z)\leq{\rm codim}_{W}(Z_{0})/2=m/2$. On the other hand,
we have $\operatorname{lct}(X,Z)=\operatorname{lct}(W,Z)+{\rm codim}_{X}(W)$
(see for example [Mustata0, Proposition 2.6]). Therefore we have

$\operatorname{lct}(X,Z)=r-m+\tfrac{m}{2}=r-\tfrac{m}{2}\leq r-\tfrac{1}{2}.$

Suppose now that $Z$ is a local complete intersection, of pure codimension
$r\geq 1$ in $X$. We first consider the case when $Z$ is globally a complete
intersection, that is, there are $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$
such that $Z$ is defined by the ideal generated by $f_{1},\ldots,f_{r}$. In
this case we consider the $V$-filtration on $B_{{\bf f}}$. Note that by (7)
and Remark 4.1, we have $\delta_{{\bf f}}\not\in V^{\gamma}B_{{\bf f}}$ for
$\gamma>r$.

######  Definition 4.3.

We define the minimal exponent $\widetilde{\alpha}(Z)$, as explained in the
introduction, by the formula

(33)
$\widetilde{\alpha}(Z)=\left\\{\begin{array}[]{cl}\sup\\{\gamma>0\mid\delta_{\mathbf{f}}\in
V^{\gamma}B_{{\bf f}}\\},&\text{if}\,\,\delta_{\mathbf{f}}\not\in
V^{r}B_{\mathbf{f}};\\\\[5.69054pt] \sup\\{r-1+q+\gamma\mid F_{q}B_{{\bf
f}}\subseteq V^{r-1+\gamma}B_{{\bf f}}\\},&\text{if}\,\,\delta_{\mathbf{f}}\in
V^{r}B_{\mathbf{f}},\end{array}\right.$

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
$\beta_{1}+\ldots+\beta_{r}\leq q$. We note that the value of
$\widetilde{\alpha}(Z)$ does not depend just on $Z$, but also on $X$ (for the
precise way in which it depends on $X$, see Proposition 4.14 below). Because
of this, whenever the ambient variety is not clear from the context, we write
$\widetilde{\alpha}(X,Z)$ instead of $\widetilde{\alpha}(Z)$.

######  Remark 4.4.

Since the $V$-filtration is left continuous, the supremum in the definition is
a maximum, unless $\widetilde{\alpha}(Z)=\infty$ (which happens if and only if
$Z$ is smooth, see Remark 4.15 below).

######  Remark 4.5.

It follows from the definition and (7) that

(34) $\operatorname{lct}(X,Z)=\min\big{\\{}\widetilde{\alpha}(Z),r\big{\\}}.$

######  Remark 4.6.

If $q_{1}$ and $q_{2}$ are nonnegative integers and
$\gamma_{1},\gamma_{2}\in(0,1]$ are rational numbers such that
$q_{1}+\gamma_{1}\geq q_{2}+\gamma_{2}$, and if $F_{q_{1}}B_{{\bf f}}\subseteq
V^{r-1+\gamma_{1}}B_{{\bf f}}$, then $F_{q_{2}}B_{{\bf f}}\subseteq
V^{r-1+\gamma_{2}}B_{{\bf f}}$. Indeed, this is clear if $q_{1}=q_{2}$, and if
this is not the case, then our hypothesis implies $q_{2}=q_{1}-1$ and it is
enough to show that if $u=\partial_{t}^{\beta}\delta_{{\bf f}}$, with
$|\beta|\leq q_{1}-1$, then $u\in V^{r}B_{{\bf f}}$. The assumption implies
$\partial_{t_{i}}u\in V^{r-1}B_{{\bf f}}$ for all $i$ and thus
$t_{i}\partial_{t_{i}}u\in V^{r}B_{{\bf f}}$. We conclude that

$(s+r)u=(-\partial_{t_{1}}t_{1}-\ldots-\partial_{t_{r}}t_{r}+r)u=-\sum_{i=1}^{r}t_{i}\partial_{t_{i}}u\in
V^{r}B_{{\bf f}}.$

For every $\gamma\neq r$, since $s+\gamma$ is nilpotent on ${\rm
Gr}_{V}^{\gamma}(B_{{\bf f}})$, it follows that $s+r$ is invertible on this
graded piece. Since $(s+r)u\in V^{r}B_{{\bf f}}$, using the discreteness of
the $V$-filtration, we conclude that $u\in V^{r}B_{{\bf f}}$.

The same argument shows that in order to have $F_{q}B_{{\bf f}}\subseteq
V^{r-1+\gamma}B_{{\bf f}}$, it is enough to require
$\partial_{t}^{\beta}\delta_{{\bf f}}\in V^{r-1+\gamma}B_{{\bf f}}$ for all
$\beta$ with $|\beta|=q$.

######  Remark 4.7.

Suppose that $U_{1},\ldots,U_{N}$ are open subsets of $X$ such that all $Z\cap
U_{i}$ are nonempty and $Z\subseteq U_{1}\cup\ldots\cup U_{N}$. Since
$\partial_{t}^{\beta}\delta_{{\bf f}}\in V^{\gamma}B_{{\bf f}}$ if and only if
the same containment holds on each $U_{i}$ (note that the condition
automatically holds over $X\smallsetminus Z$ by Remark 2.1), it follows using
also the assertion in Remark 4.6 that

$\widetilde{\alpha}(X,Z)=\min_{1\leq i\leq N}\widetilde{\alpha}(U_{i},Z\cap
U_{i}).$

######  Remark 4.8.

The definition of $\widetilde{\alpha}(Z)$ does not depend on the choice of
$f_{1},\ldots,f_{r}$. By taking an affine open cover of $X$ and using Remark
4.7, we see that it is enough to prove this assertion when $X$ is affine.
Suppose now that we have regular functions $f_{1},\ldots,f_{r}$ and
$g_{1},\ldots,g_{r}$ such that $(f_{1},\ldots,f_{r})=(g_{1},\ldots,g_{r})$.
The condition $\delta_{{\bf f}}\in V^{\gamma}B_{{\bf f}}$ is equivalent to
$\operatorname{lct}(X,Z)\geq\gamma$, hence it is independent of the choice of
generators for the ideal. We thus only need to show that if
$q\in{\mathbf{Z}}_{\geq 0}$ and $\gamma\in(0,1]$ is a rational number, then
$F_{q}B_{{\bf f}}\subseteq V^{r-1+\gamma}B_{{\bf f}}$ if and only if
$F_{q}B_{\mathbf{g}}\subseteq V^{r-1+\gamma}B_{\mathbf{g}}$.

Let us write $g_{i}=\sum_{j}a_{i,j}f_{j}$ for $1\leq i\leq r$. Note that
$D={\rm det}(a_{i,j})$ does not vanish at any point in $Z$. After replacing
$X$ by the complement of the zero-locus of $D$, we may assume that $D$ is
invertible (see Remark 4.7). In this case

$u\colon X\times{\mathbf{A}}^{r}\to X\times{\mathbf{A}}^{r},\quad
u(x,t_{1},\ldots,t_{r})=\left(x,\sum_{j}a_{1,j}t_{j},\ldots,\sum_{j}a_{r,j}t_{j}\right)$

is an isomorphism such that $u(X\times\\{0\\})=X\times\\{0\\}$ and
$B_{\mathbf{g}}=u_{+}B_{{\bf f}}$. We thus have an isomorphism $u^{*}$ of
${\mathcal{R}}=\mathcal{D}_{X}\langle
t_{1},\ldots,t_{r},\partial_{t_{1}},\ldots,\partial_{t_{r}}\rangle$ that keeps
$\mathcal{D}_{X}$ fixed and maps each $t_{i}$ to a linear form in
$t_{1},\ldots,t_{r}$ and an isomorphism of ${\mathcal{R}}$-modules $\tau\colon
B_{\mathbf{g}}\to B_{{\bf f}}$ (where we view $B_{{\bf f}}$ as an
${\mathcal{R}}$-module via $u^{*}$). This clearly has the property that
$\tau(F_{q}B_{\mathbf{g}})=F_{q}B_{{\bf f}}$ for every $p$ and using the
uniqueness of the $V$-filtration, we see that
$\tau(V^{\gamma}B_{\mathbf{g}})=V^{\gamma}B_{{\bf f}}$ for all
$\gamma\in{\mathbf{Q}}$. It is then clear that we have $F_{q}B_{{\bf
f}}\subseteq V^{r-1+\gamma}B_{{\bf f}}$ if and only if
$F_{q}B_{\mathbf{g}}\subseteq V^{r-1+\gamma}B_{\mathbf{g}}$.

Suppose now that $Z$ is an arbitrary local complete intersection closed
subscheme of $X$, of pure codimension $r\geq 1$. We can find open subsets
$U_{1},\ldots,U_{N}$ of $X$ with $Z\subseteq\bigcup_{i=1}^{N}U_{i}$ such that
each $Z\cap U_{i}$ is nonempty and defined in $U_{i}$ by an ideal generated by
$r$ regular functions on $U_{i}$. In particular, each
$\widetilde{\alpha}(U_{i},Z\cap U_{i})$ is well-defined.

######  Definition 4.9.

With the above notation, the _minimal exponent_ of $Z$ is

$\widetilde{\alpha}(Z)=\widetilde{\alpha}(X,Z):=\min_{1\leq i\leq
N}\widetilde{\alpha}(U_{i},Z\cap U_{i}).$

######  Remark 4.10.

It is easy to see, using Remark 4.7, that the definition is independent of the
choice of open subsets $U_{1},\ldots,U_{N}$. Moreover, given any open subsets
$V_{1},\ldots,V_{m}$ of $X$, with $Z\subseteq\bigcup_{j=1}^{m}V_{j}$ such that
each $Z\cap V_{j}$ is nonempty, we have

$\widetilde{\alpha}(X,Z)=\min_{1\leq j\leq m}\widetilde{\alpha}(V_{j},Z\cap
V_{j}).$

######  Remark 4.11.

In the case of hypersurfaces (that is, $r=1$), we recover the usual definition
of the minimal exponent by (15).

######  Proposition 4.12.

If $\pi\colon Y\to X$ is a surjective smooth morphism of smooth, irreducible
varieties, and $Z$ is a local complete intersection closed subscheme of $X$,
of pure codimension $r$, then
$\widetilde{\alpha}(X,Z)=\widetilde{\alpha}\big{(}Y,\pi^{-1}(Z)\big{)}$.

###### Proof.

We may and will assume that $Z$ is defined in $X$ by the ideal generated by
$f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$ and let $g_{i}=f_{i}\circ\pi$ for
$1\leq i\leq r$. Using the fact that $\pi$ is smooth, it is then
straightforward to see that we have an isomorphism

$B_{\mathbf{g}}\simeq\pi^{*}B_{{\bf f}}$

such that for every $p\in{\mathbf{Z}}_{\geq 0}$ and every
$\alpha\in{\mathbf{Q}}$, we get

$F_{p}B_{\mathbf{g}}\simeq\pi^{*}F_{p}B_{{\bf f}}\quad\text{and}\quad
V^{\alpha}B_{\mathbf{g}}\simeq\pi^{*}V^{\alpha}B_{{\bf f}}.$

The assertion in the proposition then follows directly from the definition of
the minimal exponent. ∎

Our next goal is to describe the minimal exponent of $Z$ via the minimal
exponent of a hypersurface. Suppose that $Z$ is a nonempty closed subscheme of
$X$, of pure codimension $r\geq 1$, whose ideal is generated by
$f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$. We put
$g=\sum_{i=1}^{r}f_{i}y_{i}\in\mathcal{O}_{X}(X)[y_{1},\ldots,y_{r}]$. Let
$U=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}\subseteq
Y=X\times{\mathbf{A}}^{r}$. We will freely use the notation in Section 3. The
following is the key observation:

######  Lemma 4.13.

If $\gamma\in{\mathbf{Q}}$ and $\alpha\in{\mathbf{Z}}_{\geq_{0}}^{r}$ are such
that $y^{\alpha}\partial_{z}^{|\alpha|}\delta_{g}\in
V^{\gamma}\widetilde{B}_{g}\smallsetminus V^{>\gamma}\widetilde{B}_{g}$ and
$y^{\alpha}\partial_{z}^{|\alpha|}\delta_{g}|_{U}\in
V^{>\gamma}\widetilde{B}_{g}|_{U}$, then $\gamma\geq r$ and
$\gamma\in{\mathbf{Z}}$.

###### Proof.

By assumption, if $u$ is the class of
$y^{\alpha}\partial_{z}^{|\alpha|}\delta_{g}$ in ${\rm
Gr}_{V}^{\gamma}(\widetilde{B}_{g})$, then $u\neq 0$, but there is $N$ such
that $(y_{1},\ldots,y_{r})^{N}u=0$. This implies that there is
$\beta\in{\mathbf{Z}}_{\geq 0}$ such that $v=y^{\beta}u\neq 0$, but
$(y_{1},\ldots,y_{r})v=0$. Note that $u\in{\rm
Gr}_{V}^{\gamma}(\widetilde{B}^{(0)}_{g})$, hence $v\in{\rm
Gr}_{V}^{\gamma}(\widetilde{B}^{(m)}_{g})$, where $m=|\beta|\geq 0$. By Lemma
3.1, we have $(\theta_{y}-s)v=mv$.

On the other hand, since $y_{i}v=0$ for all $i$, we have

$\theta_{y}v=\sum_{i=1}^{r}y_{i}\partial_{y_{i}}u_{i}=-rv+\sum_{i=1}^{r}\partial_{y_{i}}y_{i}v=-rv,$

hence $(s+m+r)v=\theta_{y}v+rv=0$. By definition of the $V$-filtration,
$s+\gamma$ is nilpotent on ${\rm Gr}^{\gamma}_{V}(\widetilde{B}_{g})$, and
thus $s+\lambda$ is invertible on ${\rm Gr}^{\gamma}_{V}(\widetilde{B}_{g})$
for every $\lambda\neq\gamma$. Since $v\neq 0$, we conclude that
$\gamma=m+r\geq r$, which completes the proof. ∎

We can prove now the equality
$\widetilde{\alpha}(Z)=\widetilde{\alpha}(g|_{U})$.

###### Proof of Theorem 1.1.

It is enough to show that for every $\beta\in{\mathbf{Q}}_{>0}$, we have
$\widetilde{\alpha}(Z)\geq\beta$ if and only if
$\widetilde{\alpha}(g|_{U})\geq\beta$. We treat separately the cases when
$\beta\leq r$ and when $\beta>r$.

If $\beta\leq r$, then by definition we have $\widetilde{\alpha}(Z)\geq\beta$
if and only if $\delta_{{\bf f}}\in V^{\beta}B_{{\bf f}}$. By Theorem 3.3,
this is equivalent to $\delta_{g}\in V^{\beta}\widetilde{B}_{g}$. On the other
hand, it follows from (14) that $\widetilde{\alpha}(g|_{U})\geq\beta$ if and
only if $\delta_{g}\in V^{\beta}\widetilde{B}_{g}$ on $U$. It is clear that if
$\delta_{g}\in V^{\beta}\widetilde{B}_{g}$ then this also holds after
restricting to $U$ and we need to show that the converse holds. Arguing by
contradiction, let us assume that $\delta_{g}\in V^{\beta}\widetilde{B}_{g}$
on $U$, but $\delta_{g}\not\in V^{\beta}\widetilde{B}_{g}$. In this case, let
$\beta^{\prime}=\max\\{\gamma\in{\mathbf{Q}}_{\geq 0}\mid\delta_{g}\in
V^{\gamma}\widetilde{B}_{g}\\}<\beta\leq r$. By assumption, we have
$\delta_{g}\in V^{>\beta^{\prime}}\widetilde{B}_{g}$ on $U$, hence Lemma 4.13
implies $\beta^{\prime}\geq r$, a contradiction.

If $\beta>r$, let us write $\beta=r-1+q+\gamma$, where $q$ is a positive
integer and $\gamma\in(0,1]$ is a rational number. By definition, we have
$\widetilde{\alpha}(Z)\geq\beta$ if and only if
$\partial_{t}^{\alpha}\delta_{{\bf f}}\in V^{r-1+\gamma}B_{{\bf f}}$ for every
$\alpha\in{\mathbf{Z}}_{\geq 0}^{r}$ with $|\alpha|\leq q$. By Theorem 3.3,
this holds if and only if $y^{\alpha}\partial_{z}^{|\alpha|}\delta_{g}\in
V^{r-1+\gamma}\widetilde{B}_{g}$ for all such $\alpha$. On the other hand, it
follows from (14) that $\widetilde{\alpha}(g|_{U})\geq\beta$ if and only if
$\delta_{g}\in V^{\beta}\widetilde{B}_{g}$ on $U$.

Note that $U=U_{1}\cup\ldots\cup U_{r}$, where $U_{i}$ is the complement of
the zero-locus of $y_{i}$. We thus see that if
$y_{i}^{q}\partial_{z}^{q}\delta_{g}\in V^{r-1+\gamma}\widetilde{B}_{g}$, we
have $\partial_{z}^{q}\delta_{g}\in V^{r-1+\gamma}\widetilde{B}_{g}$ on
$U_{i}$, and thus $\delta_{g}\in V^{\beta}\widetilde{B}_{g}$ on $U_{i}$ by
(12). We conclude that if $\widetilde{\alpha}(Z)\geq\beta$, then
$\delta_{g}\in V^{\beta}\widetilde{B}_{g}$ on $U$ and thus
$\widetilde{\alpha}(g|_{U})\geq\beta$.

In order to prove the converse, we argue by contradiction: we assume that
$\delta_{g}\in V^{\beta}\widetilde{B}_{g}$ on $U$, but there is
$\alpha\in{\mathbf{Z}}_{\geq 0}$ with $|\alpha|\leq q$ such that
$y^{\alpha}\partial_{z}^{|\alpha|}\delta_{g}\not\in
V^{r-1+\gamma}\widetilde{B}_{g}$. Let

$\beta^{\prime}=\max\\{\eta\in{\mathbf{Q}}_{\geq 0}\mid
y^{\alpha}\partial_{z}^{|\alpha|}\delta_{g}\in V^{\eta}\widetilde{B}_{g}\\}.$

Note that $\beta^{\prime}<r-1+\gamma\leq r$. On the other hand, since
$\delta_{g}\in V^{\beta}\widetilde{B}_{g}$ on $U$, we have
$y^{\alpha}\partial_{z}^{|\alpha|}\delta_{g}\in
V^{\beta-|\alpha|}\widetilde{B}_{g}\subseteq
V^{>\beta^{\prime}}\widetilde{B}_{g}$ on $U$. Applying Lemma 4.13, we get
$\beta^{\prime}\geq r$, a contradiction. ∎

######  Proposition 4.14.

If $Z$ is a local complete intersection scheme of pure dimension and
$Z\hookrightarrow X$ is a closed embedding, where $X$ is a smooth, irreducible
variety, then $\widetilde{\alpha}(X,Z)-\dim(X)$ does not depend on $X$, but
only on $Z$.

###### Proof.

Let us consider two embeddings $i\colon Z\hookrightarrow X$ and
$i^{\prime}\colon Z\hookrightarrow X^{\prime}$, where both $X$ and
$X^{\prime}$ are smooth irreducible varieties. After comparing both these
embeddings with the diagonal embedding $(i,i^{\prime})\colon Z\hookrightarrow
X\times X^{\prime}$, we see that we may assume that there is a smooth morphism
$p\colon X^{\prime}\to X$ such that $p\circ i^{\prime}=i$. Given any point
$x\in Z$, we can choose regular systems of parameters $x_{1},\ldots,x_{n}$ in
$\mathcal{O}_{X,i(x)}$ and
$p^{*}(x_{1}),\ldots,p^{*}(x_{n}),y_{1},\ldots,y_{m}$ in
$\mathcal{O}_{X^{\prime},i^{\prime}(x)}$ such that $y_{1},\ldots,y_{m}$ vanish
along $i^{\prime}(Z)$. In a suitable neighborhood of $i^{\prime}(x)$, we get
an étale morphism $X^{\prime}\to X\times{\mathbf{A}}^{m}$, given by
$(p,y_{1},\ldots,y_{m})$ that maps $i^{\prime}(Z)$ inside $X\times\\{0\\}$.
After taking a suitable open cover of $Z$ and using the invariance of the
minimal exponent under étale morphisms (see Proposition 4.12), we see that it
is enough to prove that if $X^{\prime}=X\times{\mathbf{A}}^{m}$ and
$i^{\prime}=(i,0)$, then
$\widetilde{\alpha}(X^{\prime},Z)=\widetilde{\alpha}(X,Z)+m$. Of course,
arguing by induction on $m$, we see that it is enough to treat the case $m=1$.

We may and will assume that $X$ is affine, and the ideal defining $Z$ in $X$
is generated by a regular sequence $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$.
Of course, in this case the ideal defining $Z$ in $X^{\prime}$ is
$(f_{1},\ldots,f_{r},z)$, where $z$ denotes the coordinate on
${\mathbf{A}}^{1}$. Using the description of the minimal exponent in Theorem
1.1, we see that it is enough to show that if we put
$U=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$ and
$U^{\prime}=X^{\prime}\times\big{(}{\mathbf{A}}^{r+1}\smallsetminus\\{0\\}\big{)}$,
and

$g=\sum_{i=1}^{r}f_{i}y_{i}\quad\text{and}\quad
g^{\prime}=zy_{r+1}+\sum_{i=1}^{r}f_{i}y_{i},$

then
$\widetilde{\alpha}(g^{\prime}|_{U^{\prime}})=\widetilde{\alpha}(g|_{U})$.
Note that we can write $U^{\prime}=U^{\prime}_{1}\cup U^{\prime}_{2}$, where
$U^{\prime}_{2}$ is given by $y_{r+1}\neq 0$ and $U^{\prime}_{1}=U\times{\rm
Spec}\big{(}{\mathbf{C}}[y_{r+1},z]$). Note that the hypersurface defined by
$g^{\prime}$ in $U^{\prime}_{2}$ is smooth, while
$g^{\prime}|_{U^{\prime}_{1}}=g|_{U_{1}}+zy_{r+1}$, hence

$\widetilde{\alpha}(g^{\prime}|_{U^{\prime}})=\widetilde{\alpha}(g^{\prime}|_{U^{\prime}_{1}})=\widetilde{\alpha}(g|_{U_{1}})+1,$

where the second equality follows from the Thom-Sebastiani theorem for minimal
exponents (see [Saito_microlocal, Theorem 0.8]). ∎

We next use the description of the minimal exponent in Theorem 1.1 to prove
some basic properties of this invariant. Until the end of this section, we
assume that $X$ is a smooth, irreducible, $n$-dimensional variety and $Z$ is a
closed subscheme of $X$ that is locally a complete intersection, of pure
codimension $r$. Recall that if $f\in\mathcal{O}_{X}(X)$ is nonzero and $x\in
X$, then the multiplicity ${\rm mult}_{x}(f)$ is the largest $d$ such that
$f\in\mathfrak{m}_{x}^{d}$, where $\mathfrak{m}_{x}$ is the ideal defining
$x$. Before introducing a local version of the minimal exponent, we make the
following

######  Remark 4.15.

We have $\widetilde{\alpha}(Z)<\infty$ if and only if $Z$ is singular (and in
this case we have333For a sharper estimate, see Remark 4.21 below.
$\widetilde{\alpha}(Z)\leq\tfrac{n+r}{2}$). In order to see this, we may and
will assume that $Z$ is defined by the ideal generated by
$f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$, and let
$g=\sum_{i=1}^{r}f_{i}y_{i}$. We use the fact that by Theorem 1.1, we have
$\widetilde{\alpha}(Z)=\widetilde{\alpha}(g|_{U})$, where
$U=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$. If $Z$ is
smooth, we may assume that $f_{1},\ldots,f_{r}$ are part of a system of
coordinates $f_{1},\ldots,f_{n}$ on $X$ (that is, $df_{1},\ldots,df_{n}$
trivialize $\Omega_{X}$). In this case it is easy to see444For a more general
statement, see Lemma 4.22 below. that the singular locus of the hypersurface
defined by $g$ is contained in $X\times\\{0\\}$ and thus
$\widetilde{\alpha}(g|_{U})=\infty$. Conversely, if $Z$ is singular, then we
may and will assume that ${\rm mult}_{x}(f_{1})\geq 2$ for some $x\in Z$. If
$p=(1,0,\ldots,0)\in{\mathbf{A}}^{r}$, then $(x,p)\in U$ and ${\rm
mult}_{(x,p)}(g)\geq 2$, hence $\widetilde{\alpha}(g|_{U})\leq\tfrac{n+r}{2}$
by [MP1, Theorem E(3)].

Suppose now that $x\in Z$ is a fixed point. We define the _minimal exponent
of_ $Z$ _at_ $x$ as follows: it is clear from the definition that if
$V^{\prime}\subseteq V$ are open neighborhoods of $x$, then
$\widetilde{\alpha}(V^{\prime},Z\cap V^{\prime})\geq\widetilde{\alpha}(V,Z\cap
V)$. Moreover, there is $V$ such that for all $V^{\prime}$ as above, the
inequality is an equality. Indeed, otherwise we have a decreasing sequence of
open neighborhoods $V_{i}$ of $x$ such that
$\big{(}\widetilde{\alpha}(V_{i},Z\cap V_{i})\big{)}_{i}$ is a strictly
increasing sequence. If $\lim_{i\to\infty}\widetilde{\alpha}(V_{i},Z\cap
V_{i})<\infty$, then we easily get a contradiction using the discreteness of
the $V$-filtration. On the other hand, if
$\lim_{i\to\infty}\widetilde{\alpha}(V_{i},Z\cap V_{i})=\infty$, then it
follows from Remark 4.15 that $x\in Z$ is a smooth point, hence it is enough
to take $V$ to be a neighborhood of $x$ such that $V\cap Z$ is smooth.

######  Definition 4.16.

Given a point $x\in Z$, we put

$\widetilde{\alpha}_{x}(Z):=\max_{V\ni x}\widetilde{\alpha}(V,Z\cap V),$

where the maximum is over all open neighborhoods $V$ of $x$ in $X$. Note that
this maximum exists by the previous discussion. Moreover, it follows from
Remark 4.15 that $\widetilde{\alpha}_{x}(Z)=\infty$ if and only if $x$ is a
smooth point of $Z$. As before, if the ambient space is not clear from the
context, we write $\widetilde{\alpha}_{x}(X,Z)$ instead of
$\widetilde{\alpha}_{x}(Z)$.

######  Remark 4.17.

It is a consequence of the definition of the minimal exponent, of the
discreteness of the $V$-filtration, and of Remark 4.15, that the set

$\big{\\{}\widetilde{\alpha}_{x}(Z)\mid x\in Z\\}$

is a finite set. It is then clear that

$\widetilde{\alpha}(Z)=\min\big{\\{}\widetilde{\alpha}_{x}(Z)\mid x\in
Z\big{\\}}.$

Furthermore, for every $\gamma\in{\mathbf{Q}}_{>0}$, the set $\big{\\{}x\in
Z\mid\widetilde{\alpha}_{x}(Z)\geq\gamma\big{\\}}$ is open in $Z$.

Slightly more generally, if $X$ is a smooth, but possibly disconnected
variety, and $Z$ is a local complete intersection closed subscheme of $X$,
with both $X$ and $Z$ pure dimensional, then we can define
$\widetilde{\alpha}_{x}(Z)$ for every $x\in Z$ by restricting to the connected
component of $X$ that contains $x$. This is useful, for instance, in the
setting of Theorem 1.2, when we do not assume that the fibers of $\mu$ are
connected.

Before we prove the main properties of the minimal exponent in general, we
need to handle one such property in the special case of hypersurfaces.

######  Lemma 4.18.

The assertion in Theorem 1.2ii) holds when $r=1$.

We note that in the presence of a section $s\colon T\to X$, the openness
assertion in the lemma is [MP1, Theorem E(2)]. However, for our purpose it
will be important to have the stronger openness assertion, that does not make
reference to a section, since this is the one that will allow us to handle
arbitrary codimension. The proof follows closely the approach in [MP1] (which
in turn was modeled on the approach to prove the semicontinuity of log
canonical thresholds via multiplier ideals, see [Lazarsfeld, Example 9.5.41]).
However, we include a detailed proof for the benefit of the reader.

The proof of assertion ${\rm ii_{1})}$ makes use of the notion of _Hodge
ideals_ for ${\mathbf{Q}}$-divisors, introduced and studied in [MP3]. For
every hypersurface $Z$ in a smooth variety $X$, every nonnegative integer $p$,
and every positive rational number $\alpha$, the corresponding Hodge ideal is
denoted by $I_{p}(\alpha Z)$. For $p=0$, this is just the multiplier ideal
$\mathcal{J}\big{(}(\alpha-\epsilon)Z\big{)}$, where $0<\epsilon\ll 1$, see
[MP3, Proposition 9.1] (since $Z$ is a hypersurface, we follow the traditional
notation to write $\mathcal{J}(\lambda Z)$ for what we denoted before by
$\mathcal{J}(\mathfrak{a}^{\lambda})$, where $\mathfrak{a}$ is the ideal
defining $Z$). Moreover, it was shown in [MP3] that many basic properties of
multiplier ideals admit extensions to Hodge ideals.

Recall that by definition of the log canonical threshold, we have
$\operatorname{lct}(X,Z)\geq\alpha$ if and only if
$\mathcal{J}\big{(}(\alpha-\epsilon)Z\big{)}=\mathcal{O}_{X}$ for
$0<\epsilon\ll 1$. Similarly, it was shown in [MP1, Corollary C] that if $Z$
is reduced, $p$ is a nonnegative integer, and
$\alpha\in{\mathbf{Q}}\cap(0,1]$, then $\widetilde{\alpha}(Z)\geq p+\alpha$ if
and only if $I_{p}(\alpha Z)=\mathcal{O}_{X}$. On the other hand, if $Z$ is
not reduced, then we automatically have $\operatorname{lct}(X,Z)<1$, hence
$\widetilde{\alpha}(Z)=\operatorname{lct}(X,Z)$ can be characterized using
multiplier ideals.

###### Proof of Lemma 4.18.

We first note that for every $t\in T$, the fiber $X_{t}$ is a smooth
subvariety of the smooth variety $X$ that contains no component of the
hypersurface $Z$. Locally around any $x\in X_{t}$, we can write $X_{t}$ as a
transverse intersection of $\dim(T)$ smooth hypersurfaces in $X$, so that
successively applying [MP1, Theorem E(1)] to restrict to each of these smooth
hypersurfaces, we obtain

(35) $\widetilde{\alpha}_{x}(X_{t},Z_{t})\leq\widetilde{\alpha}_{x}(X,Z).$

We next show that there is a nonempty open subset $T_{0}$ of $T$ such that for
every $t\in T_{0}$ and every $x\in X_{t}$, the inequality in (35) is an
equality, thus proving the assertion in ${\rm ii_{2})}$ in our setting. One
way to see this is by using the characterization of the minimal exponent in
terms of Hodge ideals and multiplier ideals and the fact that there is an open
subset $T_{0}$ such that

$\mathcal{J}(\lambda Z_{t})=\mathcal{J}(\lambda
Z)\cdot\mathcal{O}_{Z_{t}}\quad\text{for all}\quad\lambda>0,\,\,t\in T_{0}$

(see [Lazarsfeld, Theorem 9.5.35]) and, assuming that $Z$ is reduced and thus
$Z_{t}$ is also reduced for general $t\in T$, a similar formula holds for
Hodge ideals

$I_{p}(\lambda Z_{t})=I_{p}(\lambda Z)\cdot\mathcal{O}_{Z_{t}}\quad\text{for
all}\quad p\in{\mathbf{Z}}_{\geq 0},\,\,\lambda\in{\mathbf{Q}}_{>0},\,\,t\in
T_{0}$

(see the last assertion in [MP3, Theorem 13.1]). Alternatively, one can use
the characterization of the minimal exponent in terms of the $V$-filtration in
(14) and the results concerning the behavior of the $V$-filtration with
respect to non-characteristic restriction in [DMST].

We next prove the assertion in ${\rm ii_{1})}$. For every $\alpha$, let

$W_{\alpha}=\big{\\{}x\in
Z\mid\widetilde{\alpha}_{x}(X_{\mu(x)},Z_{\mu(x)})\geq\alpha\big{\\}}.$

We first note that the assertion in ${\rm ii_{1})}$ makes sense also when $T$
is not assumed to be a smooth variety, but just a (reduced, but not
necessarily irreducible) algebraic variety. However, in order to have the
statement for such varieties of dimension $n$, it is enough to prove it for
smooth, irreducible, $n$-dimensional varieties. Indeed, using resolution of
singularities, we can find a proper surjective morphism $g\colon T^{\prime}\to
T$, with $T^{\prime}$ $n$-dimensional and smooth (but possibly disconnected).
Consider the Cartesian diagram

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{T^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{T}$

and let $Z^{\prime}=h^{*}(Z)$. The assertion follows by noting that

$W^{\prime}_{\alpha}:=\big{\\{}x\in
Z^{\prime}\mid\widetilde{\alpha}_{x}(X^{\prime}_{f(x)},Z^{\prime}_{f(x)})\geq\alpha\big{\\}}=h^{-1}(W_{\alpha}),$

and thus $Z\smallsetminus W_{\alpha}=h(Z^{\prime}\smallsetminus
W^{\prime}_{\alpha})$ is closed in $Z$ if $W^{\prime}_{\alpha}$ is open in
$Z^{\prime}$.

We now prove that $W_{\alpha}$ is open in $Z$ by induction on $\dim(T)$. The
assertion is clear if $\dim(T)=0$, hence we may and will assume that
$\dim(T)\geq 1$. We first show that the subset $W_{\alpha}\subseteq Z$ is
constructible. Indeed, note first that if $T_{0}\subseteq T$ is a nonempty
open subset that satisfies condition ${\rm ii_{2})}$, then

$W_{\alpha}\cap\mu^{-1}(T_{0})=\big{\\{}x\in\mu^{-1}(T_{0})\cap
Z\mid\widetilde{\alpha}_{x}(X,Z)\geq\alpha\big{\\}}$

is open in $Z\cap\mu^{-1}(T_{0})$. On the other hand, we can apply the
induction hypothesis to the morphism $\mu^{-1}(T\smallsetminus T_{0})\to
T\smallsetminus T_{0}$ and the hypersurface $Z\cap\mu^{-1}(T\smallsetminus
T_{0})$ to conclude that $W_{\alpha}\cap\mu^{-1}(T\smallsetminus T_{0})$ is
open in $\mu^{-1}(T\smallsetminus T_{0})$ (as we have discussed, the fact that
$T\smallsetminus T_{0}$ might not be smooth is not an issue). Therefore
$W_{\alpha}$ is constructible.

Since $W_{\alpha}$ is constructible, in order to prove that it is open in $Z$,
it is enough to show that if $W\subseteq Z$ is an irreducible locally closed
subvariety of $Z$, of positive dimension, and $x\in W$ is such that
$W\smallsetminus\\{x\\}\subseteq Z\smallsetminus W_{\alpha}$, then $x\not\in
W_{\alpha}$. Of course, we may assume that $W$ dominates $T$, since otherwise
we are done by induction. Arguing by contradiction, let us assume that $x\in
W_{\alpha}$. In this case it follows from (35) that
$\widetilde{\alpha}_{x}(X,Z)\geq\alpha$ and thus there is an open neighborhood
$V$ of $x$ in $Z$ such that $\widetilde{\alpha}_{y}(X,Z)\geq\alpha$ for all
$y\in V$. On the other hand, if $T_{0}\subseteq T$ is a nonempty open subset
that satisfies property ${\rm ii_{2})}$, then $W\cap\mu^{-1}(T_{0})\cap V$
contains some $y\neq x$. In this case we have

$\widetilde{\alpha}_{y}(X_{\mu(y)},Z_{\mu(y)})=\widetilde{\alpha}_{y}(X,Z)\geq\alpha,$

hence $y\in W_{\alpha}$, a contradiction. This completes the proof of ${\rm
ii_{1})}$.

The finiteness of the set

$\big{\\{}\widetilde{\alpha}_{x}(X_{\mu(x)},Z_{\mu(x)})\mid x\in Z\big{\\}}$

now follows easily by induction on $\dim(T)$, using the fact that if $T_{0}$
satisfies the condition in ${\rm ii_{2})}$, then

$\big{\\{}\widetilde{\alpha}_{x}(X_{\mu(x)},Z_{\mu(x)})\mid
x\in\mu^{-1}(T_{0})\cap
Z\big{\\}}\subseteq\big{\\{}\widetilde{\alpha}_{x}(X,Z)\mid
x\in\mu^{-1}(T_{0})\cap Z\big{\\}}$

and the right-hand side is clearly finite. Finally, if $s\colon T\to X$ is a
section of $\pi$ such that $s(T)\subseteq Z$, then

$\big{\\{}t\in
T\mid\widetilde{\alpha}_{s(t)}(X_{t},Z_{t})\geq\alpha\big{\\}}=s^{-1}\left(\big{\\{}x\in
Z\mid\widetilde{\alpha}_{x}(X_{\mu(x)},Z_{\mu(x)})\geq\alpha\big{\\}}\right),$

and thus it is open in $T$. This completes the proof of the lemma. ∎

We can now prove the properties of the minimal exponent in arbitrary
codimension.

###### Proof of Theorem 1.2.

Since all assertions are local with respect to $X$, we may and will assume
that $Z$ is defined by the ideal generated by
$f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$. Let
$g=\sum_{i=1}^{r}f_{i}y_{i}\in\mathcal{O}_{X}(X)[y_{1},\ldots,y_{r}]$ and let
$Z^{\prime}$ be the hypersurface defined by $g$ in
$U=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$. Note that
$Z\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}\subseteq
Z^{\prime}$. The plan is to use Theorem 1.1 to reduce to the case of
hypersurfaces. The only subtlety is that while the results concern local
minimal exponents, the description provided by Theorem 1.1 is not of a local
nature. However, we will go around this issue using the homogeneity of $g$ in
$y_{1},\ldots,y_{r}$. More precisely, we have the following

######  Claim 4.19.

If $X_{0}\subseteq Z$ is a subset such that
$\widetilde{\alpha}_{(x,\lambda)}(g)\geq\gamma$ for all $(x,\lambda)\in
X_{0}\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$, then after
replacing $X$ by an open neighborhood of $X_{0}$, we may assume that
$\widetilde{\alpha}(g|_{U})\geq\gamma$.

Indeed, consider the canonical projection $\pi\colon U\to
X\times{\mathbf{P}}^{r-1}$. Since $g$ is homogeneous with respect to
$y_{1},\ldots,y_{r}$, it follows that the set

$W:=\big{\\{}(x,\lambda)\in
Z^{\prime}\mid\widetilde{\alpha}_{(x,\lambda)}(g)<\gamma\\}$

is equal to $\pi^{-1}(W^{\prime})$, for some subset $W^{\prime}\subseteq
Z\times{\mathbf{P}}^{r-1}$. Since $W$ is closed in $U$, we see that
$W^{\prime}$ is closed in $Z\times{\mathbf{P}}^{r-1}$. By assumption,
$W^{\prime}\cap(X_{0}\times{\mathbf{P}}^{r-1})=\emptyset$. It follows that if
$F\subseteq X$ is the projection of $W^{\prime}$, then after replacing $X$ by
$X\smallsetminus F$, which is an open neighborhood of $X_{0}$, we have
$\widetilde{\alpha}(g|_{U})\geq\gamma$. This proves the above claim.

Let’s begin with the proof of i). Note that the hypothesis implies that
$Z_{H}$ is a complete intersection in $H$, of pure codimension $r$, defined by
the ideal generated by $f_{1}|_{H},\ldots,f_{r}|_{H}$. Let
$U_{H}=H\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$. If
$\gamma=\widetilde{\alpha}_{x}(H,Z_{H})$, then after replacing $X$ by a
suitable open neighborhood of $x$, we may assume that
$\widetilde{\alpha}(H,Z_{H})=\gamma$, hence
$\widetilde{\alpha}(g|_{U_{H}})=\gamma$ by Theorem 1.1. In this case, it
follows from [MP1, Theorem E(1)] that
$\widetilde{\alpha}_{(z,\lambda)}(g)\geq\gamma$ for every $z\in Z_{H}$ and
every $\lambda\in{\mathbf{A}}^{r}\smallsetminus\\{0\\}$. We deduce using Claim
4.19 that after possibly replacing $X$ by a neighborhood of $Z_{H}$, we have
$\widetilde{\alpha}(g|_{U})\geq\lambda$. Another application of Theorem 1.1
gives $\widetilde{\alpha}(X,Z)\geq\gamma$, which completes the proof of i).

We next prove ii). Arguing as in the proof of Lemma 4.18, it is
straightforward to see that if ${\rm ii_{1})}$ and ${\rm ii_{2})}$ hold, then
the other two assertions hold as well. Let us prove first ${\rm ii_{1})}$. We
need to show that for every $x\in Z$, there is an open neighborhood $U_{x}$ of
$x$ such that

(36)
$\widetilde{\alpha}_{z}(X_{\mu(z)},Z_{\mu(z)})\geq\alpha:=\widetilde{\alpha}_{x}(X_{\mu(x)},Z_{\mu(x)})\quad\text{for
every}\quad z\in U_{x}\cap Z.$

Let $\varphi$ be the composition
$U=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}\to
X\overset{\mu}{\longrightarrow}T$. For every $t\in T$, we denote by $g_{t}$
the restriction of $g$ to $X_{t}\times{\mathbf{A}}^{r}$. After possibly
replacing $X$ by an open neighborhood of $x$, we may and will assume that
$\widetilde{\alpha}(X_{\mu(x)},Z_{\mu(x)})\geq\alpha$. By Theorem 1.1, we have

(37) $\widetilde{\alpha}(g_{\mu(x)}|_{U_{\mu(x)}})\geq\alpha.$

Applying Lemma 4.18 for the smooth morphism $\varphi$ and the hypersurface
defined by $g$ in $U$, we see that the set

$V_{\alpha}:=\big{\\{}(z,\lambda)\in
Z\times({\mathbf{A}}^{r}\smallsetminus\\{0\\})\mid\widetilde{\alpha}_{(z,\lambda)}(g_{\mu(z)}|_{U_{\mu(z)}})\geq\alpha\big{\\}}$

is open in $Z\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$. Note
that by (37), we have

(38)
$Z_{\mu(x)}\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}\subseteq
V_{\alpha}.$

Arguing as in the proof of Claim 4.19, we see that after possibly replacing
$X$ by an open neighborhood of $Z_{\mu(x)}$, we may assume that
$\widetilde{\alpha}(g_{t})\geq\alpha$ for all $t\in T$ (indeed, $V_{\alpha}$
is the inverse image of an open subset $W\subseteq Z\times{\mathbf{P}}^{r-1}$
and we may take the open neighborhood of $Z_{\mu(x)}$ to be the complement in
$X$ of the projection of $(X\times{\mathbf{P}}^{r-1})\smallsetminus W$ onto
the first component). In this case, Theorem 1.1 gives
$\widetilde{\alpha}(X_{t},Z_{t})\geq\alpha$ for all $t\in T$. Thus ${\rm
ii_{1})}$ holds.

Keeping the same notation, we now prove ${\rm ii_{2})}$. Applying Lemma 4.18
for $\varphi$ and the hypersurface $Z^{\prime}$ defined by $g$, we see that
there is an open subset $T_{0}$ of $T$ such that for every $t\in T_{0}$ and
every $x\in Z^{\prime}_{t}$, we have

(39)
$\widetilde{\alpha}_{(x,\lambda)}(g_{t})=\widetilde{\alpha}_{(x,\lambda)}(g)\quad\text{for
all}\quad\lambda\in{\mathbf{A}}^{r}\smallsetminus\\{0\\}.$

It is enough to show that for every $t\in T_{0}$ and every $x\in Z_{t}$, we
have

(40) $\widetilde{\alpha}_{x}(X_{t},Z_{t})=\widetilde{\alpha}_{x}(X,Z).$

We fix such $t$ and $x$ and note that we may replace $X$ by any open
neighborhood of $x$. The inequality “$\leq$” in (40) always holds by part i)
of the theorem, so we only need to prove that
$\widetilde{\alpha}_{x}(X_{t},Z_{t})\geq\alpha_{0}:=\widetilde{\alpha}_{x}(X,Z)$.
After possibly replacing $X$ by a suitable neighborhood of $x$, we may and
will assume that $\widetilde{\alpha}(X,Z)=\alpha_{0}$, hence Theorem 1.1 gives
$\widetilde{\alpha}(g|_{U})=\alpha_{0}$. By (39), we thus have
$\widetilde{\alpha}_{(x^{\prime},\lambda)}(g_{t})\geq\alpha_{0}$ for every
$x^{\prime}\in Z_{t}$, and $\lambda\in{\mathbf{A}}^{r}\smallsetminus\\{0\\}$,
and another application of Theorem 1.1 gives
$\widetilde{\alpha}_{x}(X_{t},Z_{t})\geq\widetilde{\alpha}(X_{t},Z_{t})\geq\alpha_{0}$.
This completes the proof of ${\rm ii_{2})}$.

Let us prove the inequality in iii). For
$\lambda=(\lambda_{1},\ldots,\lambda_{r})\in{\mathbf{A}}^{r}\smallsetminus\\{0\\}$,
we put $g_{\lambda}=\sum_{i=1}^{r}\lambda_{i}f_{i}$. It follows from the
assertion in ${\rm ii_{2})}$ that if $\lambda$ is general, then
$\widetilde{\alpha}(g_{\lambda})\geq\widetilde{\alpha}(g|_{U})=\widetilde{\alpha}(Z)$,
where the equality follows from Theorem 1.1. Since ${\rm
mult}_{x}(g_{\lambda})\geq k$, it follows from [MP1, Theorem E(3)] that
$\widetilde{\alpha}_{x}(g_{\lambda})\leq\tfrac{n}{k}$, and thus
$\widetilde{\alpha}(Z)\leq\tfrac{n}{k}$. Since the same argument applies to
any open neighborhood of $x$, we get
$\widetilde{\alpha}_{x}(Z)\leq\tfrac{n}{k}$. ∎

######  Remark 4.20.

Note that the assertion in Theorem 1.2${\rm ii_{1})}$ makes sense when $T$ is
any (reduced, but not necessarily irreducible) variety. Moreover, the
assertion in the general case can be easily reduced to the case when $T$ is
smooth using resolution of singularities, as explained in the proof of Lemma
4.18. Furthermore, the same argument implies that in this case, too, the set
$\big{\\{}\widetilde{\alpha}_{x}(X_{\mu(x)},Z_{\mu(x)})\mid x\in Z\big{\\}}$
is finite.

######  Remark 4.21.

We can now see that if $X$ is a smooth, irreducible variety and $Z$ is a local
complete intersection closed subscheme of $X$, of pure dimension, then for
every singular point $x\in Z$, we have

$\widetilde{\alpha}_{x}(X,Z)\leq\dim(X)-\frac{1}{2}{\rm embdim}_{x}(Z),$

where ${\rm embdim}_{x}(Z)=\dim_{{\mathbf{C}}}T_{x}Z$. Indeed, note first that
if $d={\rm embdim}_{x}(Z)$, then after possibly replacing $X$ by an open
neighborhood of $x$, we have a closed embedding $Z\hookrightarrow X^{\prime}$,
where $X^{\prime}$ is smooth, irreducible, of dimension $d$. Since $x\in Z$ is
a singular point, the ideal defining $Z$ in $X^{\prime}$ is contained in
$\mathfrak{m}_{x}^{2}$, where $\mathfrak{m}_{x}$ is the ideal defining $x$,
hence Theorem 1.2iii) gives
$\widetilde{\alpha}_{x}(X^{\prime},Z)\leq\tfrac{d}{2}$. In this case
Proposition 4.14 implies that

$\widetilde{\alpha}_{x}(X,Z)=\dim(X)-d+\widetilde{\alpha}_{x}(X^{\prime},Z)\leq\dim(X)-\frac{d}{2}.$

Our next goal is to give one nontrivial computation of minimal exponent when
$r>1$. Before doing this, we give an easy lemma describing the singular locus
of the hypersurface that we associate to a complete inersection subscheme. We
assume that we have global coordinates $x_{1},\ldots,x_{n}$ on the smooth
variety $X$ (that is, $dx_{1},\ldots,dx_{n}$ trivialize $\Omega_{X}$) and let
$\partial_{x_{1}},\ldots,\partial_{x_{n}}$ be the corresponding derivations.
As usual, we suppose that we have $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$
that define a closed subscheme $Z$ of $X$, of pure codimension $r$, and
consider $Y=X\times{\mathbf{A}}^{r}$,
$U=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$, and
$g=\sum_{i=1}^{r}f_{i}y_{i}\in\mathcal{O}_{Y}(Y)$. We denote by $J_{f}^{t}$
the transpose matrix of the Jacobian matrix
$\big{(}\partial_{x_{j}}(f_{i})\big{)}_{i,j}$.

######  Lemma 4.22.

With the above notation, the singular locus of the hypersurface $V(g)$ defined
by $g$ in $Y$ is

$V(g)_{\rm sing}=\bigsqcup_{x\in Z}\\{x\\}\times W_{x},$

where $W_{x}={\rm Ker}\,J_{f}^{t}(x)$ is a linear subspace of
${\mathbf{A}}^{r}$ of dimension $\dim_{{\mathbf{C}}}(T_{x}Z)-\dim(Z)$, and
thus

$V(g|_{U})_{\rm sing}=\bigsqcup_{x\in Z_{\rm
sing}}\\{x\\}\times\big{(}W_{x}\smallsetminus\\{0\\}\big{)}.$

###### Proof.

The singular locus $V(g)_{\rm sing}$ is defined by the equations

$\partial_{y_{1}}(g)=\ldots=\partial_{y_{r}}(g)=\partial_{x_{1}}(g)=\dots=\partial_{x_{n}}(g)=0$

(note that these imply $g=0$ since $g$ is homogeneous of degree 1 in
$y_{1},\ldots,y_{r}$). The formula for $V(g)_{\rm sing}$ follows from the fact
that

$\partial_{y_{i}}(g)=f_{i}\quad\text{for}\quad 1\leq i\leq
r\quad\text{and}\quad\partial_{x_{j}}(g)=\sum_{i=1}^{r}y_{i}\partial_{x_{j}}(f_{i})\quad\text{for}\quad
1\leq j\leq n.$

We deduce the formula for $\dim(W_{x})$ from the fact that the rank of $J_{f}$
at $x\in Z$ is $n-\dim_{{\mathbf{C}}}(T_{x}Z)$ and $J_{f}$ and $J_{f}^{t}$
have the same rank. In particular, we see that $W_{x}\neq\\{0\\}$ if and only
if $x\in Z_{\rm sing}$, and we obtain the description of $V(g|_{U})_{\rm
sing}$. ∎

######  Example 4.23.

Let $f_{1},\dots,f_{r}\in{\mathbf{C}}[x_{1},\dots,x_{n}]$ be homogeneous
polynomials of degree $d\geq 2$ that define a smooth, irreducible variety of
codimension $r$ in ${\mathbf{P}}^{n-1}$. Therefore the subvariety
$Z=V(f_{1},\dots,f_{r})\subseteq{\mathbf{A}}^{n}$ is a complete intersection,
with a unique singular point at $0$. We will show that

$\widetilde{\alpha}(Z)=\frac{n}{d},$

generalizing the well-known formula for $r=1$.

Let $g=\sum_{i=1}^{r}f_{i}y_{i}$ and
$U={\mathbf{A}}^{n}\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$.
We denote by $B_{U}$, respectively $\widetilde{B}_{U}$, the
$\mathcal{D}$-modules on which we have the $V$-filtration (respectively, the
microlocal $V$-filtration) associated to $g|_{U}$ and we simply write
$\delta_{U}$ for $\delta_{g|_{U}}$. Note that it follows from Lemma 4.22 and
our assumption on $Z$ that the singular locus of $V(g|_{U})$ is equal to
$\\{0\\}\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$. Recall
that by (14), if $\lambda$ is such that $\delta_{U}\in
V^{\lambda}\widetilde{B}_{U}$ and its class $\overline{\delta_{U}}\in{\rm
Gr}_{V}^{\lambda}(\widetilde{B}_{U})$ is nonzero, then
$\lambda=\widetilde{\alpha}(g|_{U})$. We will show that
$\lambda=\tfrac{n}{d}$.

Recall that for every $\alpha\in[0,1)\cap{\mathbf{Q}}$, the filtered
$\mathcal{D}_{U}$-module $\big{(}{\rm Gr}_{V}^{\alpha}(B_{U}),F\big{)}$ is a
filtered direct summand of a mixed Hodge module. In particular, it is regular
and quasi-unipotent along every hypersurface in the sense of [Saito-MHP,
Section 3.2.1]. Moreover, its support is contained in the singular locus of
$V(g|_{U})$. On the other hand, we have an isomorphism of filtered
$\mathcal{D}_{U}$-modules

$\big{(}{\rm Gr}_{V}^{\alpha}(\widetilde{B}_{U}),F\big{)}\simeq\big{(}{\rm
Gr}_{V}^{\alpha}(B_{U}),F\big{)}$

by [Saito_microlocal, (2.1.4)]. Finally, if we write $\lambda=k+\alpha$, where
$k\in{\mathbf{Z}}$ and $\alpha\in[0,1)$, it follows from [Saito_microlocal,
(2.2.3)] that we have a filtered isomorphism

$\partial_{t}^{k}:\big{(}{\rm
Gr}_{V}^{\lambda}(\widetilde{B}_{U}),F\big{)}\to\big{(}{\rm
Gr}_{V}^{\alpha}(\widetilde{B}_{U}),F[k]\big{)},$

where $F[k]$ is the shifted filtration $F[k]_{p}=F_{p+k}$. We thus conclude
that $\big{(}{\rm Gr}^{\lambda}_{V}(\widetilde{B}_{U}),F\big{)}$ is regular
and quasi-unipotent along every hypersurface; moreover, its support is
contained in the subset defined by $(x_{1},\ldots,x_{n})$.

Note that by definition of the Hodge filtration on $\widetilde{B}_{U}$, we
have $\delta_{U}\in F_{0}\widetilde{B}_{U}$ and
$F_{-1}\widetilde{B}_{U}=\oplus_{i\leq-1}\mathcal{O}_{U}\partial_{t}^{i}\delta_{U}$.
Since $\delta_{U}\in V^{\lambda}\widetilde{B}_{U}$, it follows that
$F_{-1}\widetilde{B}_{U}\subseteq\sum_{i\leq-1}\partial_{t}^{i}\widetilde{B}_{U}\subseteq
V^{\lambda+1}\widetilde{B}_{U}$. We thus see that $F_{-1}{\rm
Gr}_{V}^{\lambda}(\widetilde{B}_{U})=0$.

Since $\big{(}{\rm Gr}^{\lambda}_{V}(\widetilde{B}_{U}),F\big{)}$ is regular
and quasi-unipotent along every hypersurface, with support contained in the
zero-locus of $x_{1},\ldots,x_{n}$, it follows from [Saito-MHP, Lemme 3.2.6]
that $x_{1},\ldots,x_{n}$ annihilate the first nonzero piece of the Hodge
filtration on ${\rm Gr}^{\lambda}_{V}(\widetilde{B}_{U})$. In particular, if
$\theta_{x}=\sum_{i=1}^{n}x_{i}\partial_{x_{i}}$, so that
$\theta_{x}+n=\sum_{i=1}^{n}\partial_{x_{i}}x_{i}$, we see that
$(\theta_{x}+n)\overline{\delta_{U}}=0$.

Note that by (4), we have

$\theta_{x}\delta_{U}=-\sum_{i=1}^{n}x_{i}\partial_{x_{i}}(g)\partial_{t}\delta_{U}=-dg\partial_{t}\delta_{U},$

where the second equality follows from the fact that $g$ is homogeneous of
degree $d$ with respect to $x_{1},\ldots,x_{n}$. On the other hand, using
again (4), we have

$s\delta_{U}=-\partial_{t}t\delta_{U}=-g\partial_{t}\delta_{U}.$

We thus conclude that

$(\theta_{x}+n)\overline{\delta_{U}}=(n+sd)\overline{\delta_{U}}=0.$

Since $s+\lambda$ is nilpotent on ${\rm Gr}^{\lambda}_{V}(\widetilde{B}_{U})$,
this implies $\lambda=\tfrac{n}{d}$.

##  5\. The minimal exponent and the Hodge filtration on local cohomology

In this section we give the proofs of Theorems 1.3 and 1.4 by making use of a
key result from [CD], saying that, under the assumptions of Theorem 1.4, we
have an isomorphism of filtered $\mathcal{D}_{X}$-modules

(41) $\sigma\colon
V^{r}(B_{\mathbf{f}})/(t_{1},\ldots,t_{r})V^{r-1}(B_{\mathbf{f}})\overset{\sim}{\longrightarrow}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X}).$

For us, it is important to have an explicit description of this isomorphism
and this is not easy to obtain from the proof in _loc. cit_. Because of this,
we proceed in a roundabout way: we first construct a nonzero morphism of
$\mathcal{D}_{X}$-modules as in (41) and then use the following lemma that
describes the endomorphisms of $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$.

######  Lemma 5.1.

Let $X$ be a smooth, irreducible complex algebraic variety. If $Z$ is a
connected, local complete intersection subscheme of $X$, of pure codimension
$r$, then the canonical map ${\mathbf{C}}\to{\rm
End}_{\mathcal{D}_{X}}\big{(}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\big{)}$ is
bijective.

###### Proof.

Let $X^{\rm an}$ denote the complex manifold corresponding to $X$. In this
proof we make use of some standard results on holonomic $\mathcal{D}$-modules.
In order to prove that every endomorphism of
$\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is given by multiplication with a
scalar, it is enough to prove that the same property holds for
${\mathbf{D}}_{X}\big{(}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\big{)}$, where
${\mathbf{D}}_{X}$ is the duality functor for holonomic
$\mathcal{D}_{X}$-modules (see [HTT, Chapter 2.6]). Moreover, by the Riemann-
Hilbert correspondence (see [HTT, Chapter 7]), it is enough to show that every
endomorphism of the perverse sheaf corresponding to
${\mathbf{D}}_{X}\big{(}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\big{)}$ is given
by multiplication with a scalar. For an arbitrary $Z$, this perverse sheaf is
${}^{p}\mathcal{H}^{0}\big{(}\underline{{\mathbf{C}}}_{Z^{\rm
an}}[n-r]\big{)}$, where $n=\dim(X)$. However, since $Z$ is locally a complete
intersection, the sheaf $\underline{{\mathbf{C}}}_{Z^{\rm an}}[n-r]$ is a
perverse sheaf (see [Dimca, Theorem 5.1.20]), and thus
${}^{p}\mathcal{H}^{0}\big{(}\underline{\mathbf{C}}_{Z^{\rm
an}}[n-r]\big{)}=\underline{\mathbf{C}}_{Z^{\rm an}}[n-r]$. The fact that
every endomorphism of $\underline{\mathbf{C}}_{Z^{\rm an}}$ is given by scalar
multiplication follows from the fact that $Z$ is connected. ∎

###### Proof of Theorem 1.4.

Since the assertion in the theorem can be checked locally on $X$, we may and
will assume that $Z$ is connected. The key point is to construct an explicit
filtered isomorphism as in (41). We first consider the morphism

$\tau\colon\mathcal{O}_{X}[1/f_{1}\cdots
f_{r},s_{1},\ldots,s_{r}]{\mathbf{f}}^{\mathbf{s}}\to\mathcal{O}_{X}[1/f_{1}\cdots
f_{r}]$

given by specializing to $s_{1}=\ldots=s_{r}=-1$, that is,

$\tau\big{(}P(s_{1},\ldots,s_{r}){\mathbf{f}}^{\mathbf{s}}\big{)}=P(-1,\ldots,-1)\tfrac{1}{f_{1}\cdots
f_{r}}.$

It is clear that this is a morphism of left $\mathcal{D}_{X}$-modules.

Via the isomorphism (6), we will identify $\tau$ with a morphism
$B_{\mathbf{f}}^{+}\to\mathcal{O}_{X}[1/f_{1}\cdots f_{r}]$. Using (5) and the
fact that $Q_{m}(-1)=m!$, we can describe $\tau$ by

(42)
$\tau\big{(}\sum_{\beta}h_{\beta}\partial_{t}^{\beta}\delta_{\mathbf{f}}\big{)}=\sum_{\beta}\tfrac{\beta_{1}!\cdots\beta_{r}!h_{\beta}}{f_{1}^{\beta_{1}+1}\cdots
f_{r}^{\beta_{r}+1}},$

where $h_{\beta}\in\mathcal{O}_{X}[1/f_{1}\cdots f_{r}]$ for all $\beta$.

It is clear from the definition that $\tau$ vanishes on
$\sum_{i=1}^{r}(s_{i}+1)B_{\mathbf{f}}^{+}$. Let us consider the image of
$t_{i}B_{\mathbf{f}}$. Note that by Lemma 7.4, if $\beta_{i}\geq 1$, then

$t_{i}Q_{\beta}(s_{1},\ldots,s_{r}){\mathbf{f}}^{\mathbf{s}}=(s_{i}+1)Q_{\beta^{\prime}}(s_{1},\ldots,s_{r})f_{i}\cdot{\mathbf{f}}^{\mathbf{s}}\in{\rm
Ker}(\tau),$

where $\beta^{\prime}=(\beta_{1},\ldots,\beta_{i}-1,\ldots,\beta_{r})$. On the
other hand, if
$\sum_{\beta}h_{\beta}\partial_{t}^{\beta}\delta_{\mathbf{f}}\in
B_{\mathbf{f}}$ (so $h_{\beta}\in\mathcal{O}_{X}$ for all $\beta$) and if
$\beta$ is such that $\beta_{i}=0$, then

$\tau(t_{i}h_{\beta}\partial_{t}^{\beta}\delta_{\mathbf{f}})=\tau(f_{i}h_{\beta}\partial_{t}^{\beta}\delta_{\mathbf{f}})=h_{\beta}\prod_{j\neq
i}\tfrac{\beta_{j}!}{f_{j}^{\beta_{j}+1}}\in\mathcal{O}_{X}[1/f_{1}\cdots\widehat{f_{i}}\cdots
f_{r}].$

We conclude that $\tau$ induces a morphism of $\mathcal{D}_{X}$-modules

$\overline{\tau}\colon V^{r}B_{\mathbf{f}}/\sum_{i=1}^{r}t_{i}\cdot
V^{r-1}B_{\mathbf{f}}\to\mathcal{O}_{X}[1/f_{1}\cdots
f_{r}]/\sum_{i=1}^{r}\mathcal{O}_{X}[1/f_{1}\cdots\widehat{f_{i}}\cdots
f_{r}]\simeq\mathcal{H}^{r}_{Z}(\mathcal{O}_{X}).$

Let us show that $\overline{\tau}$ is surjective. Given
$u=\tfrac{g}{(f_{1}\cdots f_{r})^{m}}\in\mathcal{O}_{X}[1/f_{1}\cdots f_{r}]$,
for some $g\in\mathcal{O}_{X}$ and some $m\geq 1$, we have $u=\tau(v)$, where
$v=\tfrac{g}{(m-1)!^{r}}\partial_{t}^{\beta}\delta_{\mathbf{f}}$, where
$\beta=(m-1,\ldots,m-1)$. By the properties of the $V$-filtration, we know
that we can find $\alpha_{1},\ldots,\alpha_{N}\in{\mathbf{Q}}$, with
$\alpha_{i}<r$ for all $i$, such that
$w:=(s+\alpha_{1})\cdots(s+\alpha_{N})v\in V^{r}B_{\mathbf{f}}$ (recall that
$s=s_{1}+\ldots+s_{r}$). Since we can find $p(s)$ and $q(s)$ such that
$p(s)(s+r)+q(s)\prod_{i=1}^{N}(s+\alpha_{i})=1$ and since
$(s+r)v\in\sum_{i=1}^{r}(s_{i}+1)B_{\mathbf{f}}\subseteq{\rm Ker}(\tau)$, it
follows that $u=\tau(v)=\tau\big{(}q(s)w\big{)}\in\tau(V^{r}B_{\mathbf{f}})$.
Therefore we see that $\overline{\tau}$ is surjective.

As we have already mentioned, it follows from [CD, Theorems 1.1 and 1.2(b)]
that we have the isomorphism of filtered $\mathcal{D}_{X}$-modules (41), where
the filtration on the left-hand side is induced by the Hodge filtration on
$B_{\mathbf{f}}$ and the filtration on the right-hand side is the Hodge
filtration that comes from the mixed Hodge module structure on
$\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$. We recall that our convention is that
the Hodge filtration on $B_{\mathbf{f}}$ is defined to be

$F_{q}B_{\mathbf{f}}=\bigoplus_{|\alpha|\leq
q}\mathcal{O}_{X}\cdot\partial^{\alpha}_{t}\delta_{\mathbf{f}},$

which differs by a shift by $r$ from the usual convention followed in [CD]. In
particular, $\overline{\tau}\circ\sigma^{-1}$ is a $\mathcal{D}_{X}$-linear
endomorphism of $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$; hence, by Lemma 5.1,
it is given by multiplication with some $\lambda\in{\mathbf{C}}$. Since the
morphism is nonzero (being surjective), it follows that $\lambda\neq 0$, hence
$\overline{\tau}$ is an isomorphism. Moreover, since
$\overline{\tau}=\lambda\sigma$, we deduce that $\overline{\tau}$ is a
filtered isomorphism, too. Therefore the assertion in the theorem follows from
the definition of the Hodge filtration on $B_{\mathbf{f}}$ and the description
of $\tau$ in (42). ∎

###### Proof of Theorem 1.3.

We shall prove the equivalent statement that $F_{k}B_{\mathbf{f}}\subseteq
V^{r}B_{\mathbf{f}}$ if and only if
$F_{k}\mathcal{H}^{r}_{Z}\mathcal{O}_{X}=O_{k}\mathcal{H}^{r}_{Z}\mathcal{O}_{X}$.
The “only if” part is clear: since the elements

$\left[\frac{1}{f^{\alpha_{1}+1}_{1}f_{2}^{\alpha_{2}+1}\cdots
f^{\alpha_{r}+1}_{r}}\right]\in\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$

with $\alpha_{1},\ldots,\alpha_{r}\geq 0$ and
$\alpha_{1}+\alpha_{2}+\cdots+\alpha_{r}\leq k$ generate
$O_{k}\mathcal{H}^{r}_{Z}\mathcal{O}_{X}$ (see for example [MP2, Lemma 9.2]),
the “only if” part follows from Theorem 1.4.

For the reverse implication, we use induction on $k$. Suppose first that
$k=0$. Since $\left[\tfrac{1}{f_{1}\cdots f_{r}}\right]\in
F_{0}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$, it follows from Theorem 1.4 that
locally on $X$, we can find $h\in\mathcal{O}_{X}$ such that
$h\delta_{\mathbf{f}}\in V^{r}B_{{\bf f}}$ and $(h-1)$ lies in the ideal
${\mathcal{I}}_{Z}$ defining $Z$. Therefore $F_{0}B_{{\bf f}}\subseteq
V^{r}B_{\mathbf{f}}$ at every point of $Z$ (and outside of $Z$, this is
automatic).

Suppose now we know the assertion for $k\geq 0$ and let us prove it for $k+1$.
Since
$F_{k+1}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=O_{k+1}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$,
it follows from [MP2, Lemma 9.3] that
$F_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=O_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$,
hence by the induction hypothesis we have $\partial^{\alpha}_{t}\delta\in
V^{r}B_{\mathbf{f}}$ for all $\alpha$ with $|\alpha|\leq k$. We need to show
that $\partial^{\alpha}_{t}\delta_{\mathbf{f}}\in V^{r}B_{\mathbf{f}}$ also
for all $\alpha$ with $|\alpha|=k+1$.

Since
$F_{k+1}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=O_{k+1}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$,
it follows from Theorem 1.4 that the map

$\sigma\colon F_{k+1}V^{r}B_{{\bf f}}\to{\rm
Gr}^{O}_{k+1}\big{(}\mathcal{H}^{r}_{Z}(\mathcal{O}_{Z})\big{)},\quad\sum_{|\beta|\leq
k+1}h_{\beta}\partial_{t}^{\beta}\delta_{{\bf
f}}\to\sum_{|\beta|=k+1}\left[\tfrac{\beta_{1}!\cdots\beta_{r}!h_{\beta}}{f_{1}^{\beta_{1}+1}\cdots
f_{r}^{\beta_{r}+1}}\right]$

is surjective. This implies that working locally on $X$, for every $\alpha$
with $|\alpha|=k+1$ we can find

$u_{\alpha}=\sum_{|\beta|\leq
k+1}h_{\alpha,\beta}\partial_{t}^{\beta}\delta_{{\bf f}}\in V^{r}B_{{\bf f}}$

that is mapped by $\sigma$ to
$v_{\alpha}=\left[\tfrac{\alpha_{1}!\alpha_{2}!\cdots\alpha_{r}!}{f_{1}^{\alpha_{1}+1}\cdots
f_{r}^{\alpha_{r}+1}}\right]\in{\rm
Gr}^{O}_{k+1}\big{(}\mathcal{H}^{r}_{Z}(\mathcal{O}_{Z})\big{)}$. Note that
since $Z$ is a complete intersection, ${\rm
Gr}^{O}_{k+1}\mathcal{H}^{r}_{Z}(\mathcal{O}_{Z})$ is a free
$\mathcal{O}_{Z}$-module, with a basis given by the $v_{\alpha}$, with
$|\alpha|=k+1$ (see for example [MP2, Lemmas 9.1, 9.2]). This implies that
$h_{\alpha,\alpha}-1$ and $h_{\alpha,\beta}$, for $\alpha\neq\beta$, lie in
$\mathcal{I}_{Z}$. It is then clear that for every $\alpha$ with
$|\alpha|=k+1$ we have $\partial_{t}^{\alpha}\delta_{{\bf f}}\in V^{r}B_{{\bf
f}}$ at every point of $Z$ (this holds trivially on the complement of $Z$).
This completes the proof of the induction step and thus the proof of the “if”
part. ∎

We obtain the following consequence to the characterization of local complete
intersection Du Bois singularities. We note that the “if” part is [Schwede,
Corollary 5.8] and the full equivalence in the case when $Z$ is normal is
[Kovacs, Theorem 3.6] (note that if $Z$ is normal and local complete
intersection, then $\operatorname{lct}(X,Z)=r$ if and only if $Z$ has log
canonical singularities by Inversion of Adjunction, see [EM, Corollary 3.2]).
A version of the “only if” implication also appears in [Doherty, Theorem 4.2].

######  Corollary 5.2.

If $X$ is a smooth, irreducible variety and $Z\hookrightarrow X$ is a local
complete intersection closed subscheme of pure codimension $r$, then $Z$ has
Du Bois singularities555We note that the condition of having Du Bois
singularities assumes, in particular, that $Z$ is reduced. if and only if
$\operatorname{lct}(X,Z)=r$.

###### Proof.

Recall first that $\operatorname{lct}(X,Z)\leq r$ and equality implies that
$Z$ is reduced: see Remarks 4.1 and 4.2. We may thus assume that $Z$ is
reduced. Since $Z$ is locally a complete intersection, it follows from [MP2,
Theorem C] that $Z$ has Du Bois singularities if and only if
$F_{0}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=O_{0}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$
and this condition is equivalent to $\operatorname{lct}(X,Z)\geq r$ by Theorem
1.3. ∎

##  6\. The minimal exponent and the Bernstein-Sato polynomial

Let $X$ be a smooth, irreducible, complex algebraic variety and $Z$ a proper
(nonempty) closed subscheme of $X$, defined by the ideal $\mathfrak{a}$. In
what follows we will make use of the notation and definitions introduced in
Section 2. Recall, in particular, that we discussed the Bernstein-Sato
polynomial $b_{Z}(s)$ in the case when $\mathfrak{a}$ is generated by nonzero
global regular functions $f_{1},\ldots,f_{d}$. The general case can be easily
reduced to this one (in fact, to the case when $X$ is affine) since for open
subsets $U_{1},\ldots,U_{N}$ of $X$ such that $Z\subseteq U_{1}\cup\ldots\cup
U_{N}$, we have

(43) $b_{Z}(s)={\rm LCM}\big{\\{}b_{Z\cap U_{i}}(s)\mid 1\leq i\leq
N\big{\\}}$

(with the convention that $b_{Z\cap U_{i}}(s)=1$ if $Z\cap U_{i}=\emptyset$).

######  Proposition 6.1.

If $Z$ is locally a complete intersection in $X$, of pure codimension $r$,
then $b_{Z}(-r)=0$.

###### Proof.

If $U$ is an open subset of $X$, then we deduce from (43) that $b_{Z\cap
U}(s)$ divides $b_{Z}(s)$. It follows that after replacing $X$ by a suitable
affine open neighborhood of a point in $Z$, we may assume that $X$ is affine
and the ideal $\mathfrak{a}$ defining $Z$ is generated by a regular sequence
$f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$. The condition in the definition of
$b_{Z}(s)$ can be reformulated as saying that $b_{Z}(s)$ is the monic
polynomial of minimal degree such that

(44) $b_{Z}(s)f_{1}^{s_{1}}\cdots
f_{r}^{s_{r}}\in\sum_{u\in{\mathbf{Z}}^{r},|u|=1}\mathcal{D}_{X}[s_{1},\ldots,s_{r}]\cdot\prod_{u_{i}<0}{{s_{i}}\choose{-u_{i}}}f_{1}^{s_{1}+u_{1}}\cdots
f_{r}^{s_{r}+u_{r}}$

(see [BMS, Section 2.10]). Here we use the isomorphism (6), that maps
$\delta_{{\bf f}}$ to $f_{1}^{s_{1}}\cdots f_{r}^{s_{r}}$, such that the
action of $-\partial_{t_{i}}t_{i}$ corresponds to the action of $s_{i}$ (so
that $s=s_{1}+\ldots+s_{r}$). Note also that ${{s_{i}}\choose{-u_{i}}}$
denotes the polynomial
$\tfrac{1}{(-u_{i})!}s_{i}(s_{i}-1)\cdots(s_{i}+u_{i}-1)$.

By making $s_{1}=\ldots=s_{r}=-1$ in (44), we obtain the following relation in
$\mathcal{O}_{X}[1/f_{1}\cdots f_{r}]$:

(45) $b_{Z}(-r)\frac{1}{f_{1}\cdots
f_{r}}\in\sum_{u\in{\mathbf{Z}}^{r},|u|=1}\mathcal{D}_{X}\cdot
f_{1}^{u_{1}-1}\cdots f_{r}^{u_{r}-1}.$

Note that if $u=(u_{1},\ldots,u_{r})$ is such that $|u|=1$, then there is $i$
such that $u_{i}\geq 1$, in which case

$\mathcal{D}_{X}\cdot f_{1}^{u_{1}-1}\cdots
f_{r}^{u_{r}-1}\subseteq\mathcal{D}_{X}\cdot\prod_{j\neq
i}f_{j}^{u_{j}-1}\subseteq\mathcal{O}_{X}[1/f_{1}\cdots\widehat{f_{i}}\cdots
f_{r}].$

Using (45), we thus conclude that

$b_{Z}(-r)\frac{1}{f_{1}\cdots
f_{r}}\in\sum_{i=1}^{r}\mathcal{O}_{X}[1/f_{1}\cdots\widehat{f_{i}}\cdots
f_{r}],$

hence the class of $b_{Z}(-r)\frac{1}{f_{1}\cdots f_{r}}$ in the local
cohomology $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is $0$. Since the class of
$\frac{1}{f_{1}\cdots f_{r}}$ in the local cohomology is nonzero, we conclude
that $b_{Z}(-r)=0$. ∎

From now on, for the rest of this section, we assume that $Z$ is a local
complete intersection closed subscheme of $X$, of pure codimension $r\geq 1$.
The above result motivates the following

######  Definition 6.2.

We denote by $\widetilde{\gamma}(Z)$ the negative of the largest root of
$b_{Z}(s)/(s+r)$ (with the convention that $\widetilde{\gamma}(Z)=\infty$ if
this polynomial is $1$).

######  Remark 6.3.

Note that since all roots of $b_{Z}(s)$ are rational numbers, the same is true
for $\widetilde{\gamma}(Z)$. Recall also that by formula (10), the largest
root of $b_{Z}(s)$ is $-\operatorname{lct}(X,Z)$, hence

(46) $\operatorname{lct}(X,Z)=\min\big{\\{}\widetilde{\gamma}(Z),r\\}.$

######  Remark 6.4.

Since $Z$ is locally a complete intersection in $X$, of pure codimension $r$,
it follows from [BMS, Theorem 4] that $\widetilde{\gamma}(Z)>r$ if and only if
$Z$ has rational singularities (the result in _loc_. _cit_. requires $Z$ to
also be reduced, but the hypothesis $\widetilde{\gamma}(Z)>r$ implies
$\operatorname{lct}(X,Z)=r$, and thus $Z$ is reduced by Remark 4.2).

We will need the following easy result:

######  Lemma 6.5.

If $X={\rm Spec}(R)$ is a smooth affine variety and $f_{1},\ldots,f_{r}\in R$
form a regular sequence, then for every $k\geq 0$, the sequence
$t_{1},\ldots,t_{r}$ is regular on the finitely generated
$R[t_{1},\ldots,t_{r}]$-module $F_{k}B_{{\bf f}}$.

###### Proof.

For every $k\geq 0$, the quotient $F_{k}B_{{\bf f}}/F_{k-1}B_{{\bf f}}$ is a
free $R$-module. Moreover, each $t_{i}$ acts on this quotient as
multiplication by $f_{i}$. The assertion in the lemma thus follows by
induction on $k$, using the fact that if

$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$

is a short exact sequence of $R[t_{1},\ldots,t_{r}]$-modules such that
$t_{1},\ldots,t_{r}$ is a regular sequence on both $M^{\prime}$ and
$M^{\prime\prime}$, then it is a regular sequence also on $M$. ∎

We can now prove our result relating $\widetilde{\gamma}(Z)$ and
$\widetilde{\alpha}(Z)$.

###### Proof of Theorem 1.6.

After possibly replacing $X$ by suitable affine open subsets, we may and will
assume that $X$ is affine and the ideal defining $Z$ is generated by a regular
sequence $f_{1},\ldots,f_{r}$. It follows from (34) and (46) that we have

$\min\big{\\{}\widetilde{\gamma}(Z),r\\}=\operatorname{lct}(X,Z)=\min\big{\\{}\widetilde{\alpha}(Z),r\\}.$

Therefore both assertions in the theorem hold if $\operatorname{lct}(X,Z)<r$.
Hence from now on we may and will assume that $\operatorname{lct}(X,Z)=r$,
which in light of (7), is equivalent to $\delta_{{\bf f}}\in V^{r}B_{{\bf
f}}$. In order to prove the two assertions in the theorem, it is enough to
show the following:

  1. (i)

If $q$ is a nonnegative integer and $\gamma\in(0,1]$ is a rational number such
that $\widetilde{\gamma}(Z)\geq r-1+q+\gamma$, then $\widetilde{\alpha}(Z)\geq
r-1+q+\gamma$; by definition of the minimal exponent, this is equivalent to
$\partial_{t}^{\beta}\delta_{{\bf f}}\in V^{r-1+\gamma}B_{{\bf f}}$ for all
$\beta\in{\mathbf{Z}}_{\geq 0}^{r}$ with $|\beta|\leq q$.

  2. (ii)

If $\gamma^{\prime}\in(0,1]$ is a rational number such that
$\widetilde{\alpha}(Z)\geq r+\gamma^{\prime}$, then $\widetilde{\gamma}(Z)\geq
r+\gamma^{\prime}$.

We first prove (i), arguing by induction on $q\geq 0$. If $q=0$, then we are
done, since we are assuming $\delta_{{\bf f}}\in V^{r}B_{{\bf f}}$. Suppose
now that $q\geq 1$. By the induction hypothesis, it is enough to show that for
every $\beta\in{\mathbf{Z}}_{\geq 0}^{r}$, with $|\beta|=q-1$, if
$u=\partial_{t}^{\beta}\delta_{{\bf f}}$, then $\partial_{t_{i}}u\in
V^{r-1+\gamma}B_{{\bf f}}$ for $1\leq i\leq r$. Furthermore, the induction
hypothesis gives $u\in V^{r}B_{{\bf f}}$. Let’s write
$b(s):=b_{Z}(s)=(s+r)p(s)$.

By definition of $b_{Z}(s)$, we have

(47) $b(s)V^{0}\mathcal{R}\cdot\delta_{{\bf f}}\subseteq
V^{1}\mathcal{R}\cdot\delta_{{\bf f}}.$

We first note that for every $i$ with $1\leq i\leq q-1$, if
$\beta^{\prime}\in{\mathbf{Z}}_{\geq 0}^{r}$ is such that
$|\beta^{\prime}|=q-i$, then there is
$\beta^{\prime\prime}\in{\mathbf{Z}}^{r}_{\geq 0}$ with
$|\beta^{\prime\prime}|=q-i-1$ such that

(48) $b(s-q+i)\partial_{t}^{\beta^{\prime}}V^{0}\mathcal{R}\cdot\delta_{{\bf
f}}\subseteq\partial_{t}^{\beta^{\prime\prime}}V^{0}\mathcal{R}\cdot\delta_{{\bf
f}}.$

Indeed, it follows from Lemma 7.3 that if we take any
$\beta^{\prime\prime}\in{\mathbf{Z}}_{\geq 0}^{r}$ with
$|\beta^{\prime\prime}|=q-i-1$ and
$\beta^{\prime}_{j}\geq\beta^{\prime\prime}_{j}\geq 0$ for all $j$, then

$b(s-q+i)\partial_{t}^{\beta^{\prime}}V^{0}\mathcal{R}\cdot\delta_{{\bf
f}}=\partial_{t}^{\beta^{\prime}}b(s)V^{0}\mathcal{R}\cdot\delta_{{\bf
f}}\subseteq\partial_{t}^{\beta^{\prime}}V^{1}\mathcal{R}\cdot\delta_{{\bf
f}}\subseteq\partial_{t}^{\beta^{\prime\prime}}V^{0}\mathcal{R}\cdot\delta_{{\bf
f}}.$

Applying (48) for all $i$ with $1\leq i\leq q-1$, as well as (47), we obtain

$b(s)b(s-1)\cdots b(s-q+1)\partial_{t}^{\beta}\delta_{{\bf f}}\subseteq
b(s)V^{0}\mathcal{R}\cdot\delta_{{\bf f}}\subseteq
V^{1}\mathcal{R}\cdot\delta_{{\bf f}}\subseteq V^{r+1}B_{{\bf f}}.$

Note now that we can write

$b(s)b(s-1)\cdots b(s-q+1)=(s+r)p_{1}(s),$

where

$p_{1}(s)=(s+r-1)\cdots(s+r-q+1)\cdot\prod_{i=0}^{q-1}p(s-i).$

The assumption that $\widetilde{\gamma}(Z)\geq r-1+q+\gamma$ implies that
$p_{1}(s)$ has no roots in the interval $(-r-\gamma,-r]$. Since $(s+r)u\in
V^{r}B_{{\bf f}}$ (recall that $u\in V^{r}B_{{\bf f}}$) and $p_{1}(s)(s+r)u\in
V^{r+1}B_{{\bf f}}$, we conclude that $(s+r)u\in V^{r+\gamma}B_{{\bf f}}$. By
assumption, we have $u\in F_{q-1}B_{{\bf f}}$, hence $(s+r)u\in
F_{q}V^{r+\gamma}B_{{\bf f}}$ (where for every $p\in{\mathbf{Z}}_{\geq 0}$ and
every $\alpha\in{\mathbf{Q}}$, we put $F_{p}V^{\alpha}B_{{\bf f}}=F_{p}B_{{\bf
f}}\cap V^{\alpha}B_{{\bf f}}$).

Consider now the map

$(F_{q}V^{r-1+\gamma}B_{{\bf f}})^{\oplus
r}\overset{(t_{1},\ldots,t_{r})}{\longrightarrow}F_{q}V^{r+\gamma}B_{{\bf
f}}.$

Since $\gamma>0$, it follows from [CD, Theorem 1.1] that the map is
surjective, hence we may write

$(s+r)u=\sum_{i=1}^{r}t_{i}u_{i},$

with $u_{i}\in F_{q}V^{r-1+\gamma}B_{{\bf f}}$ for $1\leq i\leq r$. Since
$s+r=-\sum_{i=1}^{r}t_{i}\partial_{t_{i}}$, we obtain

$\sum_{i=1}^{r}t_{i}(u_{i}+\partial_{t_{i}}u)=0.$

Using the fact that $t_{1},\ldots,t_{r}$ form a regular sequence on
$F_{q}B_{{\bf f}}$ by Lemma 6.5, we conclude that for every $i$, we have

$u_{i}+\partial_{t_{i}}u\in\sum_{i=1}^{r}t_{i}\cdot F_{q}B_{{\bf f}}\subseteq
V^{r}B_{{\bf f}},$

where the inclusion follows from the fact that we already know, by induction,
that $F_{q-1}B_{{\bf f}}\subseteq V^{r}B_{{\bf f}}$ and thus $F_{q}B_{{\bf
f}}\subseteq\sum_{i=1}^{r}\partial_{t_{i}}\cdot V^{r}B_{{\bf f}}\subseteq
V^{r-1}B_{{\bf f}}$. We thus conclude that

$\partial_{t_{i}}u=(u_{i}+\partial_{t_{i}}u)-u_{i}\in V^{r-1+\gamma}B_{{\bf
f}}.$

This completes the proof of (i).

We next prove (ii). We will make use of the results in Section 3. By
definition of $\widetilde{\alpha}(Z)$, we know that
$\partial_{t_{i}}\delta_{{\bf f}}\in V^{r-1+\gamma^{\prime}}B_{{\bf f}}$ for
$1\leq i\leq r$. Theorem 3.3 (see also equation (16)) thus gives
$y_{i}\delta_{g}\in V^{r+\gamma^{\prime}}\widetilde{B}_{g}$ for $1\leq i\leq
r$. By Lemma 3.1, we have
$v:=(s+r)\delta_{g}=\sum_{i=1}^{r}\partial_{y_{i}}y_{i}\delta_{g}\in
V^{r+\gamma^{\prime}}\widetilde{B}_{g}$. Using the description (13), we deduce
that all roots of $\widetilde{b}_{v}(s)$ are $\leq-(r+\gamma^{\prime})$. On
the other hand, the inclusions

$\widetilde{b}_{v}(s)(s+r)\delta_{g}\subseteq
V^{1}\widetilde{\mathcal{R}}\cdot(s+r)\delta_{g}\subseteq
V^{1}\widetilde{\mathcal{R}}\cdot\delta_{g},$

imply that $\widetilde{b}_{\delta_{g}}$ divides $\widetilde{b}_{v}(s)(s+r)$.
Since $b_{Z}(s)=\widetilde{b}_{\delta_{g}}$ (see Remark 3.5), we conclude that
$\widetilde{\gamma}(Z)\geq r+\gamma^{\prime}$. This completes the proof of the
theorem. ∎

We can now show that the equality $F_{1}=O_{1}$ on
$\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ implies that $Z$ has rational
singularities.

###### Proof of Corollary 1.7.

It follows from Theorem 1.6 that $\widetilde{\alpha}(Z)>r$ if and only if
$\widetilde{\gamma}(Z)>r$. By [BMS, Theorem 4], this holds if and only if $Z$
has rational singularities (see also Remark 6.4). The last assertion in the
corollary now follows from Theorem 1.3. ∎

##  7\. Appendix: some formulas involving differential operators

In this appendix we collect for ease of reference some easy computations
involving differential operators. We work in the ring ${\mathbf{C}}\langle
t,\partial_{t},\partial_{t}^{-1}\rangle$ and put $s=-\partial_{t}t$.

######  Lemma 7.1.

For every $m\geq 1$ we have $[\partial_{t},t^{m}]=mt^{m-1}$ and for every
$m\in{\mathbf{Z}}$ we have $[t,\partial_{t}^{m}]=-m\partial_{t}^{m-1}$.

###### Proof.

The first formula follows from the more general fact that for every derivation
$D$ and every regular function $f$, we have $[D,f]=D(f)$. The second formula
is clear if $m=1$ and the general case follows by induction on $|m|$, using
the fact that

$[t,\partial_{t}^{m+1}]=(t\partial_{t}^{m}-\partial_{t}^{m}t)\partial_{t}+\partial_{t}^{m}(t\partial_{t}-\partial_{t}t)=[t,\partial_{t}^{m}]\partial_{t}-\partial_{t}^{m},$

which immediately implies that the formula holds for $m$ if and only if it
holds for $m+1$. ∎

######  Lemma 7.2.

For every $m\geq 1$, we have

$\partial_{t}^{m}t^{m}=(-1)^{m}s(s-1)\cdots(s-m+1).$

###### Proof.

The assertion is clear when $m=1$ and the general case follows by induction on
$m$, writing

$\partial_{t}^{m+1}t^{m+1}=\partial_{t}t\partial_{t}^{m}t^{m}+\partial_{t}[\partial_{t}^{m},t]t^{m}=(\partial_{t}t+m)\partial_{t}^{m}t^{m},$

where the last equality follows using Lemma 7.1. ∎

######  Lemma 7.3.

For every $m\in{\mathbf{Z}}$ and every $P\in{\mathbf{C}}[s]$, we have
$P(s)\partial_{t}^{m}=\partial_{t}^{m}P(s+m)$.

###### Proof.

It is easy to see that it is enough to prove the equality when $P$ is a
monomial and then that it is enough to prove it when $P=s$. In this case we
have

$s\partial_{t}^{m}-\partial_{t}^{m}s=-\partial_{t}t\partial_{t}^{m}+\partial_{t}^{m+1}t=-\partial_{t}[t,\partial_{t}^{m}]=m\partial_{t}^{m},$

where the last equality follows from Lemma 7.1. ∎

The proof of the next lemma is similar and we leave it for the reader.

######  Lemma 7.4.

For every $m\in{\mathbf{Z}}_{\geq 0}$ and every $P\in{\mathbf{C}}[s]$, we have
$P(s)t^{m}=t^{m}P(s-m)$.

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