Hodge ideals for Q-divisors, V-filtration, and minimal exponent

A. Introduction

This paper establishes a connection between the Hodge ideals of a ${\mathbf{Q}}$-divisor, as defined in [MP3], and the $V$-filtration along an appropriately chosen hypersurface. It is inspired by Saito’s [Saito-MLCT], which explained such a connection expressed in terms of the microlocal $V$-filtration, in the case of the Hodge ideals of reduced divisors studied in [MP1]. In the ${\mathbf{Q}}$-divisor case, this relationship turns out to be crucial towards establishing some of the most basic properties of Hodge ideals, as well as of certain roots of Bernstein-Sato polynomials.

Let $X$ be a smooth complex variety, and $D$ an effective ${\mathbf{Q}}$-divisor on $X$. Such a divisor can be written locally as $D=\alpha H$, where $\alpha\in{\mathbf{Q}}$, and $H={\rm div}(f)$ is the divisor of a regular function, and it is this set-up that we focus on in what follows. To this data, by a standard construction one associates the left $\mathscr{D}_{X}$-module

a free $\mathscr{O}_{X}(*H)$-module of rank $1$ with generator the symbol $f^{\beta}$, where $\beta=1-\alpha$ (the $\mathscr{D}$-module action is recalled in §1). In [MP3] we observe that it carries a natural filtration $F_{p}\mathcal{M}(f^{\beta})$, with $p\geq 0$, which makes it a filtered direct summand in a $\mathscr{D}$-module underlying a mixed Hodge module. Moreover, we show that this can be written in the form

with $Z=H_{\rm red}$ the support of $H$, where $I_{p}(D)$ are coherent sheaves of ideals on $X$ called the Hodge ideals of $D$. Note that here and throughout the paper we make a slight abuse of notation, identifying the right-hand side with its image via the canonical injection into $\mathscr{O}_{X}(*H)f^{\beta}$.

The ideal $I_{0}(D)$ is identified in loc. cit. with the multiplier ideal \mathcal{I}\big{(}(1-\epsilon)D\big{)} associated to the ${\mathbf{Q}}$-divisor $(1-\epsilon)D$ with $0<\epsilon\ll 1$, which measures the failure of the pair $(X,D)$ to be log canonical. On the other hand, when $D$ is integral Budur and Saito [Budur-Saito] have shown the identification

where $V^{\bullet}\mathscr{O}_{X}$ denotes the $V$-filtration induced on $\mathscr{O}_{X}$. This is defined as $V^{\bullet}\mathscr{O}_{X}=V^{\bullet}\iota_{+}\mathscr{O}_{X}\cap\mathscr{O}_{X}$, where $V^{\bullet}\iota_{+}\mathscr{O}_{X}$ is the Kashiwara-Malgrange $V$-filtration on the graph embedding (via a local equation of $H$) of $\mathscr{O}_{X}$; see §2. Consequently we have $I_{0}(D)=V^{1}\mathscr{O}_{X}$.

When $D$ is a reduced divisor (corresponding in the notation above to $\beta=0$ and $D=H=Z$), Saito showed in [Saito-MLCT] that a relationship of this type continues to hold in a weaker sense even for $p\geq 1$, namely

or in other words $I_{p}(D)+(f)=\widetilde{V}^{p+1}\mathscr{O}_{X}+(f)$, where this time $\widetilde{V}^{\bullet}\mathscr{O}_{X}$ denotes the microlocal $V$-filtration induced on $\mathscr{O}_{X}$ by $f$, defined in [Saito_microlocal]. Examples show that in general this identification does not hold without modding out by $f$; even so however, it is significant for a number of reasons. Most importantly, it establishes a connection between Hodge ideals and the Bernstein-Sato polynomial of $f$. Moreover, to establish the triviality of the ideals on the two sides, it suffices to check it mod $f$.

In this paper we show that similar statements hold for arbitrary ${\mathbf{Q}}$-divisors. More precisely, we fully compute the Hodge ideals in terms of the (usual) $V$-filtration on $\iota_{+}\mathscr{O}_{X}$. This strengthens Saito’s result above even in the reduced case. In order to state our results, let us recall first that without loss of generality it suffices to focus on the case $\lceil D\rceil=Z$. Indeed, the ${\mathbf{Q}}$-divisor $B=D+Z-\lceil D\rceil$ satisfies this condition, while according to [MP3, Lemma 4.4], we have

It will also be convenient to express things equivalently in terms of a slightly different ideal, defined by the formula

The $V$-filtration will come into play via the following construction: for each $p\geq 0$, we consider the coherent sheaf of ideals in $\mathscr{O}_{X}$ given by

For $0<\alpha\leq 1$, this is just another way of expressing Saito’s microlocal $V$-filtration mentioned above; specifically, one has

It will also be convenient to make use of the polynomials

With these definitions and reductions, our main result can be phrased as follows:

In the set-up above, for every positive rational number $\alpha$ such that $D=\alpha H$ satisfies $\lceil D\rceil=Z$, and for every $p\geq 0$, we have

Note that in the case where $D=\alpha Z$, with $Z$ a reduced effective divisor, we have $I^{\prime\prime}_{p}(D)=I_{p}(D)$ for all $p\geq 0$. In this case we have the following variant of the theorem above, where we place no restrictions on the positive rational number $\alpha$.

If $Z$ is a reduced, effective divisor on $X$, defined by the global equation $f\in\mathscr{O}_{X}(X)$, then for every positive rational number $\alpha$, and every $p\geq 0$, if $D=\alpha Z$ we have

The proofs of these theorems, as well as various intermediate results, occupy §3 and §4. Some of the arguments follow [Saito-MLCT], and rely on the regular and quasi-unipotent property of filtered $\mathscr{D}$-modules underlying mixed Hodge modules. For the full calculation of Hodge ideals in terms of the $V$-filtration however, further techniques need to be developed as well. One technical point, of independent interest, is a calculation of the $V$-filtration on the (graph embedding of the) twisted $\mathscr{D}$-modules $\mathcal{M}(f^{\beta})$ in terms of the more tractable $V$-filtration on $\iota_{+}\mathscr{O}_{X}$; for the statement see Proposition 2.6.

Remark. The microlocal $V$-filtration has been computed explicitly in various cases by Saito; for instance, it is computed for a large class of quasi-homogeneous isolated singularities in [Saito-MLCT, Proposition (2.2.4)]. In [Saito-HF] the Hodge filtration itself is computed combinatorially for all such singularities, and this is extended to the case of ${\mathbf{Q}}$-divisors in [Zhang]. This leads to an explicit calculation of Hodge ideals for isolated quasi-homogeneous singularities; see loc. cit. for examples.

In the ${\mathbf{Q}}$-divisor case, the results above allow us to deduce some basic properties of Hodge ideals that do not follow from the methods of [MP3]. We collect some of these, treated individually and discussed in detail in §5, in the following:

Let $D=\alpha Z$, where $Z$ is a reduced divisor and $\alpha\in{\mathbf{Q}}_{>0}$. Then the following hold:

$I_{p}(D)+\mathscr{O}_{X}(-Z)\subseteq I_{p-1}(D)+\mathscr{O}_{X}(-Z)$ for all $p$.

If $(X,D)$ is $(p-1)$-log canonical,This means that $I_{0}(D)=\cdots=I_{p-1}(D)=\mathscr{O}_{X}$. In particular it requires $\alpha\leq 1$. then $I_{p+1}(D)\subseteq I_{p}(D)=\widetilde{I}_{p}(D)$.

Fixing $p$, there exists a finite set of rational numbers $0=c_{0}<c_{1}<\cdots<c_{s}<c_{s+1}=1$ such that for each $0\leq i\leq s$ and each $\alpha\in(c_{i},c_{i+1}]$ we have

The last statement gives a picture analogous to that of jumping coefficients of multiplier ideals [Lazarsfeld, Lemma 9.3.21]. If $f$ is a local equation of $Z$, the set of $c_{i}$ is a subset of the set of jumping numbers for the $V$-filtration on $\iota_{+}\mathscr{O}_{X}$ associated to $f$. An example in §5 shows that the statement fails if we work directly with $I_{p}(\alpha Z)$ as opposed to $I_{p}(\alpha Z)\cdot\mathscr{O}_{Z}$.

There are interesting applications, obtained in §6 by combining the above results with the birational study of Hodge ideals in [MP3], concerning the Bernstein-Sato polynomial $b_{Z}(s)$ of $Z$. Assuming $Z\neq 0$, the polynomial $(s+1)$ divides $b_{Z}(s)$. Following [Saito-MLCT], we denote by $\widetilde{\alpha}_{Z}$ the negative of the largest root of $b_{Z}(s)/(s+1)$ (with the convention that this is $\infty$ if $b_{Z}(s)=s+1$). This invariant is called the minimal exponent of $Z$, and is a refined version of the log canonical threshold of the pair $(X,Z)$, which is equal to ${\rm min}\{\widetilde{\alpha}_{Z},1\}$; see §6 for a discussion. First, since by Theorem Theorem A′′ we have that $I_{p}(D)$ is trivial if and only if $\widetilde{I}_{p}(D)$ is so, results of Saito on the microlocal $V$-filtration will allow us to conclude:

If $Z\neq 0$ is a reduced effective divisor on the smooth variety $X$ and $\alpha\in(0,1]$ is a rational number, then

Now given a log resolution $\mu\colon Y\to X$ of the pair $(X,Z)$, assumed to be an isomorphism over $X\smallsetminus Z$, if $F_{1},\ldots,F_{m}$ are the irreducible components of its exceptional locus and $\widetilde{Z}$ is the strict transform of $Z$, assumed to be smooth, we write

where $K_{Y/X}$ is the relative canonical divisor. Denoting

it is well known that the log canonical threshold of $(X,Z)$ is also equal to $\min\{\gamma,1\}$. Such a precise interpretation in terms of log resolutions is however not known for other roots of the Bernstein-Sato polynomial, and Lichtin [Lichtin, Remark 2, p.303] posed the natural question whether $\widetilde{\alpha}_{Z}=\gamma$. As noted by Kollár [Kollar, Remark 10.8], in general the answer is negative, since $\gamma$ in fact depends on the choice of log resolution. Nevertheless:

With the notation above, we always have $\widetilde{\alpha}_{Z}\geq\gamma$.

The reason is that the triviality of $I_{p}(D)$ is related on one hand to $\widetilde{\alpha}_{Z}$ by Corollary C, and on the other hand to $\gamma$ by [MP3, Proposition 11.2]. It is worth noting that, although the statement is about the reduced divisor $Z$, the proof uses crucially the theory of Hodge ideals for ${\mathbf{Q}}$-divisors of the form $D=\alpha Z$.

Using further properties of Hodge ideals of ${\mathbf{Q}}$-divisors proved in [MP3], we deduce some general properties of the minimal exponent $\widetilde{\alpha}_{D}$ for any effective divisor $D$, extending important features of the log canonical threshold. In order to formulate the result, it is convenient to use a local version of this refined log canonical threshold, denoted $\widetilde{\alpha}_{D,x}$, for $x\in D$ (see §6 for the precise definition).

Let $X$ be a smooth $n$-dimensional complex variety, and $D$ an effective divisor on $X$.

If $Y$ is a smooth subvariety of $X$ such that $Y\not\subseteq D$, then for every $x\in D\cap Y$, we have

Consider a smooth morphism $\pi\colon X\to T$, together with a section $s\colon T\to X$ such that $s(T)\subseteq D$. If $D$ does not contain any fiber of $\pi$, so that for every $t\in T$ the divisor $D_{t}=D|_{\pi^{-1}(t)}$ is defined, then the function

For every $x\in X$, if $m={\rm mult}_{x}(D)\geq 2$, then

where $r$ is the dimension of the singular locus of the projectivized tangent cone ${\mathbf{P}}(C_{x}D)$ of $D$ at $x$ (with the convention that $r=-1$ if ${\mathbf{P}}(C_{x}D)$ is smooth).

When $D$ has an isolated singularity at $x$ and $Y$ is a general hyperplane section through $x$, the inequality in (1) was proved in [Loeser, Théorème 1] (in fact, in this case the inequality is strict). The semicontinuity property in (2) was proved when every $D_{t}$ has an isolated singularity at $s(t)$ in [Steenbrink, Theorem 2.11], where it was deduced from more general semicontinuity properties of the spectrum. We stress that in (2) we do not assume that the restriction of the support of $D$ to the fibers of $\pi$ is reduced, as in the semicontinuity theorem [MP3, Theorem 14.1] (which we do use).

Yet more properties analogous to those of log canonical thresholds follow by combining Theorem E with a Thom-Sebastiani-type theorem due to Saito; see Proposition 6.6 for the concrete statement. We ask in Question 6.9 whether the analogue of the ACC property for log canonical thresholds holds for minimal exponents as well.

Finally, going back to the general relationship between Hodge ideals and the Bernstein-Sato polynomial, in Proposition 6.14 we give an extension of the fact that the negatives of the jumping coefficients of multiplier ideals in the interval $(0,1]$ are roots of the Bernstein-Sato polynomial, see [ELSV, Theorem B]. Namely, under a suitable log-canonicity hypothesis, the jumping coefficients of higher Hodge ideals in the same interval, in the sense of Corollary B (3), lead to further such roots. This follows quickly from results proved in the final two sections of the paper.

Acknowledgements. We are grateful to Morihiko Saito for comments and suggestions that helped improve a previous version of this paper. We would also like to thank Nero Budur and Mingyi Zhang for a few useful discussions, and a referee for several corrections.

B. Main results

Let $X$ be a smooth complex algebraic variety and $H$ an effective divisor on $X$. We assume that $H$ is defined by a global regular function $f\in\mathscr{O}_{X}(X)$. We denote by $\mathscr{O}_{X}(*H)$ the sheaf of rational functions on $X$ with poles along $H$, that is,

Given a rational number $\gamma$, we consider the $\mathscr{D}_{X}$-module

This is a free $\mathscr{O}_{X}(*H)$-module of rank $1$, with generator the symbol $f^{\gamma}$, on which a derivation $D$ of $\mathscr{O}_{X}$ acts by

We will keep the notation $\mathscr{O}_{X}(*H)$ for $\mathcal{M}(f^{0})$. The $\mathscr{D}_{X}$-modules $\mathcal{M}(f^{\gamma})$ are regular holonomic, with quasi-unipotent monodromy. In fact, they are filtered direct summands of $\mathscr{D}$-modules underlying mixed Hodge modules, see [MP3, §2].

Note that if $\gamma_{1}-\gamma_{2}=d$ is an integer, then we have a canonical isomorphism of $\mathscr{D}_{X}$-modules

We will be particularly interested in the $\mathscr{D}$-modules $\mathcal{M}(f^{\beta})$ as in the Introduction, which via the isomorphism above can also be identified with $\mathcal{M}(f^{-\alpha})$ with $\alpha=1-\beta$.

We begin by reviewing some basic facts about $V$-filtrations. Let

be the closed embedding given by the graph of $f$. For a $\mathscr{D}_{X}$-module $\mathcal{M}$, we consider the $\mathscr{D}$-module theoretic direct image

see for instance [HTT, Example 1.3.5]. This is a $\mathscr{D}_{X\times{\mathbf{C}}}$-module that can be described as follows. First, if $\mathcal{M}=\mathscr{O}_{X}$, then

with the obvious $\mathscr{D}_{X}$-module structure. If $\delta$ denotes the class of $\frac{1}{f-t}$ in $\iota_{+}\mathscr{O}_{X}$, it is straightforward to see that every element in $\iota_{+}\mathscr{O}_{X}$ can be written uniquely as

with $h_{j}\in\mathscr{O}_{X}$, only finitely many of these being nonzero. Note that by definition we have $t\delta=f\delta$.

Given an arbitrary $\mathscr{D}_{X}$-module $\mathcal{M}$, we have

which in particular shows the connection with the original definition above. With this description, multiplication by $t$ is given by

and the action of a derivation $D\in{\rm Der}_{{\mathbf{C}}}(\mathscr{O}_{X})$ is given by

In particular, every element in $\iota_{+}\mathcal{M}(f^{\beta})$ can be written uniquely as a finite sum

Each $V^{\gamma}$ is a coherent module over $\mathscr{D}_{X}[t,\partial_{t}t]$.

For every $\gamma\in{\mathbf{Q}}$, we have an inclusion

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

For every $\gamma\in{\mathbf{Q}}$, if we put $V^{>\gamma}=\bigcup_{\gamma^{\prime}>\gamma}V^{\gamma^{\prime}}$, then $\partial_{t}t-\gamma$ acts nilpotently on

It is easy to see that there exists at most one $V$-filtration (see for example [Saito-MHP, Lemme 3.1.2]). The existence of the $V$-filtration for $\mathcal{M}=\mathscr{O}_{X}$ and $\mathcal{M}=\mathscr{O}_{X}(*H)$ was proved by Malgrange [Malgrange]; the case of an arbitrary holonomic $\mathcal{M}$ is due to Kashiwara [Kashiwara3]. We note that this original $V$-filtration was indexed by integers; the indexing by ${\mathbf{Q}}$ (in the case of a regular holonomic $\mathcal{M}$, with quasi-unipotent monodromy) was introduced by Saito [Saito-GM].

Every element in the cokernel of the inclusion $\mathscr{O}_{X}\hookrightarrow\mathscr{O}_{X}(*H)$ is annihilated by some power of $f$. This implies that the canonical inclusion $\iota_{+}\mathscr{O}_{X}\hookrightarrow\iota_{+}\mathscr{O}_{X}(*H)$ induces equalities

In what follows, it will be convenient to also have a different description of $\iota_{+}\mathcal{M}$ under a minor extra assumption on $\mathcal{M}$. From now on we assume that multiplication by $f$ is bijective on $\mathcal{M}$ (in other words, $\mathcal{M}$ has a natural structure of $\mathscr{O}_{X}(*H)$-module). Note that this applies, in particular, if $\mathcal{M}=\mathcal{M}(f^{\beta})$.

Our hypothesis on $\mathcal{M}$ implies that multiplication by $t$ is bijective on $\iota_{+}\mathcal{M}$. Indeed, if we consider on $\iota_{+}\mathcal{M}$ the filtration given by

then multiplication by $t$ preserves the filtration; moreover, it follows from (2.1) that for every $p\geq 0$, via the obvious isomorphism $G_{p}/G_{p-1}\simeq\mathcal{M}$, multiplication by $t$ gets identified with multiplication by $f$. We thus obtain by induction on $p$ the fact that multiplication by $t$ on $G_{p}$ is an isomorphism.

If we assume that $\mathcal{M}$ is as above and there is a $V$-filtration on $\iota_{+}\mathcal{M}$, then

where for simplicity we denote $V^{\alpha}=V^{\alpha}\iota_{+}\mathcal{M}$. Indeed, the inclusion “$\subseteq$”, as well as the reverse inclusion for $\alpha>0$, follow from general properties of the $V$-filtration. Moreover, the induced map

Suppose now that $\alpha\leq 0$ and $u=tw\in V^{\alpha+1}$. Let $\delta\ll 0$ be such that $w\in V^{\delta}$. If $\delta\geq\alpha$, then we are done. On the other hand, if $\delta<\alpha$, then $\delta\neq 0$ since $\alpha\leq 0$; since $tw\in V^{>\delta+1}$, we conclude that $w\in V^{>\delta}$. After repeating this argument finitely many times, we obtain $w\in V^{\alpha}$.

Let $\mathscr{D}\langle t,s\rangle$ be the subsheaf of $\mathscr{D}_{X\times{\mathbf{C}}}$ generated by $\mathscr{D}_{X}$, $t$, and $s=-\partial_{t}t$. Note that $t$ and $s$ satisfy $st=t(s-1)$ and more generally

We also consider the localization $\mathscr{D}_{X}\langle t,t^{-1},s\rangle=\mathscr{D}_{X}\langle t,t^{-1},\partial_{t}\rangle$ of $\mathscr{D}\langle t,s\rangle$. (This is the push-forward of the sheaf of differential operators from $X\times{\mathbf{C}}^{*}$ to $X\times{\mathbf{C}}$.) Note that in this ring we have $\partial_{t}=-st^{-1}$ and from (2.2) we obtain

A $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$-module is simply a $\mathscr{D}_{X\times{\mathbf{C}}}$-module on which $t$ acts bijectively.

We consider the $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$-module $\mathcal{M}[s]f^{s}$ defined as follows. As an $\mathscr{O}_{X}$-module, we have an isomorphism

The symbol $f^{s}$ motivates the $\mathscr{D}_{X}$-action: a derivation $D$ in ${\rm Der}_{{\mathbf{C}}}(\mathscr{O}_{X})$ acts by

The action of $s$ on $\mathcal{M}[s]f^{s}$ is the obvious one, while the action of $t$ is given by the automorphism “$s\to s+1$”, that is

In light of Remarks 2.2 and 2.3, if $\mathcal{M}$ is a $\mathscr{D}_{X}$-module on which multiplication by $f$ is bijective, a $V$-filtration on $\iota_{+}\mathcal{M}$ can be characterized as an exhaustive, decreasing, discrete, left continuous, rational filtration $(V^{\gamma}=V^{\gamma}\iota_{+}\mathcal{M})_{\gamma\in{\mathbf{Q}}}$, that satisfies the following conditions:

Each $V^{\gamma}$ is a coherent module over $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$.

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

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

For every $\gamma\in{\mathbf{Q}}$, the operator $s+\gamma$ acts nilpotently on ${\rm Gr}_{V}^{\gamma}$.

The next proposition contains the promised description of $\iota_{+}\mathcal{M}$. While the result is well known (in fact, in the case $\mathcal{M}=\mathscr{O}_{X}(*H)$, this has already been noticed in [Malgrange]), we sketch the proof since we will need the explicit description of the isomorphism. For every $i\geq 0$, we put

If $\mathcal{M}$ is a $\mathscr{D}_{X}$-module on which $f$ acts bijectively, then we have an isomorphism of $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$-modules

It is straightforward to check that the map in (2.4) is $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$-linear. In order to see that it is an isomorphism, consider on $\mathcal{M}[s]f^{s}$ and $\iota_{+}\mathcal{M}$ the filtrations given by

Note that the map (2.4) preserves the filtrations. Moreover, we have canonical isomorphisms

such that the map induced by (2.4) is given by multiplication with $(-1)^{p}f^{p}$. Since this is an isomorphism, we conclude by induction on $p$ that each induced map $G_{p}\mathcal{M}[s]f^{s}\to G_{p}\iota_{+}\mathcal{M}$ is an isomorphism.

The formula for the inverse isomorphism follows if we show that in $\iota_{+}\mathscr{O}_{X}(*H)$ we have

We argue by induction on $j$, the case $j=0$ being obvious. Assuming the formula for some $j$, we apply (2.3) and the fact that $\partial_{t}=-st^{-1}$ to write

This completes the proof of the proposition. ∎

We now come to the main result of this section, relating the $V$-filtrations on $\iota_{+}\mathcal{M}(f^{\beta})$ and $\iota_{+}\mathscr{O}_{X}(*H)$ as follows.

For every $\beta\in{\mathbf{Q}}$, we have an isomorphism of $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$-modules

where on the right-hand side $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$ acts via the automorphism $\mathscr{D}_{X}\langle t,t^{-1},s\rangle\to\mathscr{D}_{X}\langle t,t^{-1},s\rangle$ that maps $s$ to $s-\beta$ and is the identity on $\mathscr{D}_{X}$ and on $t$. The isomorphism $\Phi$ is given by

The isomorphism $\Phi$ translates the $V$-filtration by $-\beta$, in the sense that

The isomorphism is more transparently described via the identifications provided by Proposition 2.5, which gives isomorphisms

It is then straightforward to check that if we define

this is an isomorphism of $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$-modules, where the action on the right-hand side is via the automorphism of $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$ described in the statement of the proposition. If we take $\Phi=\varphi_{2}^{-1}\circ\Phi^{\prime}\circ\varphi_{1}$, we obtain an isomorphism that satisfies (2.6). The fact that $\Phi$ translates the $V$-filtration by $-\beta$ is a consequence of the uniqueness of the $V$-filtration and of the fact that our automorphism of $\mathscr{D}_{X}\langle t,t^{-1},s\rangle$ is the identity on $\mathscr{D}_{X}$ and $t$ and maps $s$ to $s-\beta$. ∎

In a first version of this paper, we showed that the isomorphism $\Phi$ translates the $V$-filtration by $-\beta$ using the description of this filtration in terms of Bernstein-Sato polynomials due to Sabbah [Sabbah] (see Proposition 6.11 below). The argument above, based on the uniqueness of the $V$-filtration, was pointed out to us by M. Saito.

We next give a more explicit description of the transformation $\Phi$.

If $\Phi\colon\iota_{+}\mathcal{M}(f^{\beta})\to\iota_{+}\mathscr{O}_{X}(*H)$ is the map in Proposition 2.6, and if $\Phi(u)=v$, where

Letting $x=-s$ and $y=-\beta$ in Lemma 7.1 in the Appendix, and using the definition of $\Phi$, we get

Since the polynomials $Q_{i}(-s+\beta)$, with $0\leq i\leq p$, are linearly independent over ${\mathbf{Q}}$, the equality between the first and the last expressions above gives (2.8). ∎

Proposition 2.8 is used below in the proof of Theorem A. It was noted more recently in [JKSY, §2.4] that there is a proof of the theorem along the same lines as here, which however avoids the use of this precise formula.

Recall from the introduction that for every nonnegative integer $p$, there is an ideal sheaf $I^{\prime\prime}_{p}(D)$ such that

This ideal is related to the $p$-th Hodge ideal of $D$ by the formula

In particular, we see that $I^{\prime\prime}_{0}(D)=I_{0}(D)$.

For every $p\geq 0$, we define the subsheaf $\widetilde{I}_{p}(D)$ of $\mathscr{O}_{X}$ by

Since $V^{\alpha}\iota_{+}\mathscr{O}_{X}$ is an $\mathscr{O}_{X}$-module, it follows that $\widetilde{I}_{p}(D)$ is a (coherent) ideal in $\mathscr{O}_{X}$. As mentioned in the Introduction, when $0<\alpha\leq 1$ this is another way of expressing Saito’s microlocal $V$-filtration [Saito_microlocal] induced on $\mathscr{O}_{X}$.

We note that we have made an abuse of notation here: both ideals $I^{\prime\prime}(D)$ and $\widetilde{I}_{p}(D)$ depend on the choice of $H$, and not just on the ${\mathbf{Q}}$-divisor $D=\alpha H$. However, in what follows $H$ will be fixed, and we hope that this will not lead to any confusion.

For every nonnegative integer $p$ and every $g\in I^{\prime\prime}_{p}(D)$, there is

Before giving the proof, recall that the Hodge filtration on $\mathcal{M}(f^{\beta})$ induces a Hodge filtration on $\iota_{+}\mathcal{M}(f^{\beta})$, given by

This, just as with all the filtered $\mathscr{D}$-modules we consider here, satisfies the following special property.

Let $t$ be a nonzero function on the smooth variety $Y$, defining a smooth divisor $H$. If $(M,F)$ is a filtered $\mathscr{D}_{Y}$-module with no $t$-torsion, and which carries a $V$-filtration with respect to $t$ that is compatible with the $F$-filtration in the sense of [Saito-MHP, 3.2], such that the induced morphism

where $j\colon Y\smallsetminus H\hookrightarrow Y$ is the inclusion.

The conclusion of the lemma applies in particular when $(M,F)$ is a direct summand of a filtered $\mathscr{D}_{Y}$-module $(N,F)$ that underlies a mixed Hodge module (and hence is regular and quasi-unipotent, so it satisfies [Saito-MHP, 3.2]), and such that $N$ has no $t$-torsion and

is a filtered isomorphism. We may therefore apply it to the filtered $\mathscr{D}_{X\times{\mathbf{C}}}$-module $\iota_{+}\mathcal{M}(f^{\beta})$. Indeed, according to [MP3, Lemma 2.11], $\mathcal{M}(f^{\beta})$ is a filtered direct summand in a $\mathscr{D}$-module on $X$ of the form $j_{+}(Q,F)$, where $j\colon U\hookrightarrow X$ is the natural inclusion of $U=X\smallsetminus Z$, and $(Q,F)$ is the filtered $\mathscr{D}$-module underlying a mixed Hodge module on $U$; hence \big{(}\iota_{+}\mathcal{M}(f^{\beta}),F\big{)} is a summand in $(j\times{\rm id_{\mathbf{C}}})_{+}(\iota_{U})_{+}(Q,F)$, where $\iota_{U}$ is the graph embedding corresponding to $f|_{U}$. But filtered $\mathscr{D}$-modules such as the latter satisfy the properties above, by the general construction of direct images of Hodge modules via open embeddings in [Saito-MHM, Proposition 2.8] (cf. also [Saito-B, Proposition 4.2]).

We can now prove the main result of the section.

The argument is similar to that in [Saito-MLCT], which treats the case when $D$ is reduced (i.e. $H$ is reduced and $\alpha=1$). In what follows we may, and will assume, that $X$ is affine.

Let $g\in I^{\prime\prime}_{p}(D)$. It follows from the definition of $I^{\prime\prime}_{p}(D)$ that we have

Using Remark 3.4, we may apply Lemma 3.3 for the $\mathscr{D}_{X\times{\mathbf{C}}}$-module $\iota_{+}\mathcal{M}(f^{\beta})$, hence we can write

with $u^{(i)}\in V^{0}\iota_{+}\mathcal{M}(f^{\beta})\cap j_{*}j^{*}F_{p-i}\iota_{+}\mathcal{M}(f^{\beta})$ for all $i$. If we write

Note now that since $u^{(0)}\in V^{0}\iota_{+}\mathcal{M}(f^{\beta})$, we have

and by the definition of the action of $t$, we can write

We now use the transformation $\Phi$ in Proposition 2.6 to deduce that

where we use the fact that $V^{\gamma}\iota_{+}\mathscr{O}_{X}(*H)=V^{\gamma}\iota_{+}\mathscr{O}_{X}$ for every $\gamma>0$, by Remark 2.1. For every $j\geq i$, let

It follows from Proposition 2.8 that there are $v_{0},\ldots,v_{p}\in\mathscr{O}_{X}(X)$ such that

with the convention that $u^{(0)}_{p+1}=0$. Therefore we have

where the last equality follows from Lemma 7.2. We thus have (3.1). The last assertion in the statement is clear, since $v_{p}\in\widetilde{I}_{p}(\alpha H)$ and $Q_{p}(\alpha)\neq 0$. ∎

For every nonnegative integer $p$, if $v=\sum_{j=0}^{p}v_{j}\partial_{t}^{j}\delta\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$, then

We will prove the proposition by induction on $p$. Note that if we know it for all $q<p$, then we know the statements in (4.1) for $1\leq i\leq p$. Indeed, since $v\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$, we also have

Iterating this, we conclude that for every $1\leq i\leq p$, we have

Applying the inductive hypothesis for $(f-t)^{i}v$, we conclude that we have

Let us explain the significance of the sums on the left-hand side of (4.1). Suppose that $v=\sum_{i=0}^{p}v_{i}\partial_{t}^{i}\delta\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$ and

is such that $\Phi(u)=v$. Note that multiplication by $t$ is bijective on $\iota_{+}\mathcal{M}(f^{\beta})$, and let $w$ be such that $u=tw$. If we write $w=\sum_{i=0}^{p}w_{i}f^{\beta}\otimes\partial_{t}^{i}\delta$, then

are precisely the sums on the left hand side of (4.1).

To check (4.2), if we denote by $w^{\prime}_{i}$ the right hand side of the formula, it is enough to show that

and $fw^{\prime}_{p}=u_{p}$. Since $w^{\prime}_{p}=\frac{1}{f}v_{p}=\frac{1}{f}u_{p}$, the last equality is clear. Note now that

It follows that if $0\leq i\leq p-1$, then

where the last equality follows from the fact that

Using Proposition 2.8, we thus conclude that

It is shown in [MP3, Proposition 9.1] that, if ${\mathcal{I}}(\gamma H)$ denotes the multiplier ideal of the ${\mathbf{Q}}$-divisor $\gamma H$, we have

for $0<\epsilon\ll 1$. We refer to [Lazarsfeld, Chapter 9] for the definition and basic properties of multiplier ideals. In particular, for every $\alpha\leq 1$, we have

The following property of the Hodge filtration on $\iota_{+}\mathcal{M}(f^{\beta})$ is probably well known to the experts, but we include a proof for the benefit of the reader.

For every $p\in{\mathbf{Z}}$ and every $\gamma\geq 0$, we have

In order to simplify the notation, we write $V^{\gamma}$ for $V^{\gamma}\iota_{+}\mathcal{M}(f^{\beta})$ and $F_{p}$ for $F_{p}\iota_{+}\mathcal{M}(f^{\beta})$. Note first that since multiplication by $f$ is bijective on $\mathcal{M}(f^{\beta})$, it follows that multiplication by $t$ on $\iota_{+}\mathcal{M}(f^{\beta})$ is bijective and

The inclusion “$\subseteq$” is clear, and when $\gamma>0$ the equality follows from the compatibility of the $F$ and $V$ filtrations, see [Saito-MHP, §3.2]. (We use again the fact that $\iota_{+}\mathcal{M}(f^{\beta})$ is a filtered direct summand of a mixed Hodge module.) Suppose now that $\gamma=0$. We also know that

is a filtered isomorphism (see Remark 3.4). The statement follows then from the Five Lemma applied to the filtered commutative diagram

We can now prove the main result of this section.

We argue by induction on $p$. The case $p=0$ is known: if $v_{0}\otimes\delta\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$, then it follows from [Budur-Saito] that v_{0}\in{\mathcal{I}}\big{(}(\alpha-\epsilon)H\big{)}. On the other hand, we have

by Remark 4.4. We thus obtain the statement of the proposition for $p=0$.

Suppose now that the statement holds for all $q\leq p$, and let us prove it for $p+1$. Let

such that $\Phi(u)=v$. We also consider the unique $w=\sum_{i=0}^{p+1}w_{i}f^{\beta}\otimes\partial_{t}^{i}\delta$ such that $tw=u$. Note that $w\in V^{0}\iota_{+}\mathcal{M}(f^{\beta})$ by Lemma 4.5. We need to show that

We have seen in Remark 4.2 that for $1\leq i\leq p+1$ the assertion follows from the induction hypothesis, hence we only need to prove it for $i=0$.

On the other hand, it follows from Remark 4.3 that

The statement in (4.3) is thus equivalent to

Therefore, equivalently, we know the statement in (4.4) for $1\leq i\leq p+1$, and we need to show it for $i=0$. We also record the fact that, due to the way the $F$-filtration is defined on $\iota_{+}\mathcal{M}(f^{\beta})$, the conditions in (4.4) are equivalent to the statement $w\in F_{p+1}\iota_{+}\mathcal{M}(f^{\beta})$.

(with the convention $w_{p+2}=0$). Therefore

for $1\leq i\leq p+1$. Furthermore, since

we conclude that $u_{0}f^{\beta}\in F_{p+1}\mathcal{M}(f^{\beta})$ (which, given what we already know inductively, is equivalent to $u\in F_{p+1}\iota_{+}\mathcal{M}(f^{\beta})$) if and only if $fw_{0}f^{\beta}\in F_{p+1}\mathcal{M}(f^{\beta})$ (which again, given what we already know, is equivalent to $fw\in F_{p+1}\iota_{+}\mathcal{M}(f^{\beta})$). Using Lemma 4.5 we thus conclude that

We can now apply the same argument with $v$ replaced by $fv,\ldots,f^{p+1}v$ to conclude that

is a section of $\mathscr{O}_{X}$. Now using Remark 4.4 we see that

and putting everything together we conclude that $w_{0}f^{\beta}\in F_{p+1}\mathcal{M}(f^{\beta})$. As we have seen, this completes the proof of (4.4), and thus of the proposition. ∎

Theorem A now follows by combining Propositions 3.2 and 4.1. Let us explain how we can remove the condition on $\alpha$ when $H=Z$.

It is of course enough to prove only the first assertion of the theorem. For $\alpha\in(0,1]$, this follows from Theorem A. Therefore it suffices to show that if we know (0.1) for $\alpha$, then we also know it for $\alpha+1$. Let us temporarily denote the right-hand side of (0.1) by $\sigma(\alpha Z)$.

Note that I_{p}\big{(}(\alpha+1)Z\big{)}=f\cdot I_{p}(\alpha Z) by [MP3, Lemma 4.4]), hence it is enough to show that we also have \sigma\big{(}(\alpha+1)Z\big{)}=f\cdot\sigma(\alpha Z). By the definition of the $V$-filtration, since $\alpha>0$ we have

It follows that, given $v=\sum_{j=0}^{p}v_{j}\partial_{t}^{j}\delta\in V^{\alpha+1}\iota_{+}\mathscr{O}_{X}$, we can find $w=\sum_{j=0}^{p}w_{j}\partial_{t}^{j}\delta\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$ such that $v=tw$. This means that

with the convention that $w_{p+1}=0$. Let us denote by $h$ and $g$ the elements of $\sigma_{p}(\alpha Z)$ and \sigma_{p}\big{(}(\alpha+1)Z\big{)} corresponding to $w$ and $v$, respectively. We thus have

where the last equality follows from the fact that

The equality $g=fh$ implies that \sigma\big{(}(\alpha+1)Z)=f\cdot\sigma(\alpha Z), and thus completes the proof of the theorem. ∎

We conclude with a few remarks regarding the statements of the main theorems.

If we write $f=f_{1}^{m_{1}}\cdots f_{r}^{m_{r}}$, where $f_{i}$ correspond to the irreducible components of $H$ and $m_{i}\geq 1$, then $Z$ is given by the equation $g=f_{1}\cdots f_{r}$, and so

Thus when $p\geq 2$ and $m_{i}\geq 2$ for all $i$, we have $I^{\prime\prime}_{p}(D)\subseteq(f)$, and so the only content of the last statement in Theorem A is that $\widetilde{I}_{p}(D)\subseteq(f)$.

The last assertion in Theorem Theorem A′′ is only interesting for $\alpha\leq 1$, since for $\alpha>1$ both sides are equal to $(f)$.

Saito introduced and studied in [Saito_microlocal] a microlocal $V$-filtration. This induces a filtration on $\mathscr{O}_{X}$ denoted $(\widetilde{V}^{\gamma}\mathscr{O}_{X})_{\gamma\in{\mathbf{Q}}}$. Using the definition of this filtration, when $H$ is reduced one can reformulate the last assertion in Theorem A as saying that

for all $p\geq 0$. As mentioned in the Introduction, when $\alpha=1$, this was proved in [Saito-MLCT].

In the setting of Theorem A, the fact that the F_{p}\mathcal{M}(f^{\beta})=I^{\prime\prime}_{p}(D)\otimes\mathscr{O}_{X}\big{(}(p+1)H\big{)}f^{\beta} give a filtration on $\mathcal{M}(f^{\beta})$ compatible with the order filtration on $\mathscr{D}_{X}$ is equivalent to the following properties:

Each $I^{\prime\prime}_{p}(D)$ is an $\mathscr{O}_{X}$-module.

We have $f\cdot I^{\prime\prime}_{p}(D)\subseteq I^{\prime\prime}_{p+1}(D)$ for every $p\geq 0$.

For every $D\in{\rm Der}_{{\mathbf{C}}}(\mathscr{O}_{X})$ and every $h\in I^{\prime\prime}_{p}(D)$, we have

One can easily check that these properties can also be deduced from the formula in Theorem A and the general properties of the $V$-filtration.

C. Consequences

From now on we consider the case $H=Z$, that is $D=\alpha Z$, with $Z$ a reduced divisor and $\alpha$ a positive rational number. We will see that Theorem Theorem A′′ implies a number of fundamental properties of Hodge ideals that cannot be easily deduced directly from the definition.

Note that in the statements below we do not require that $Z$ be defined by a global equation; however, the assertions immediately reduce to this case, hence in the proofs we will tacitly make this assumption, and denote by $f$ the equation defining $Z$.

If $v=\sum_{j=0}^{p}v_{j}\partial_{t}^{j}\delta\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$, then

We thus see that $\widetilde{I}_{p}(D)\subseteq\widetilde{I}_{p-1}(D)$. The assertion now follows from Theorem Theorem A′′. ∎

In the case $\alpha=1$ we have the stronger statement $I_{p}(D)\subseteq I_{p-1}(D)$, see [MP1, Proposition 13.1]. However, for $\alpha<1$ this seems likely to fail, though at the moment we do not have an example. It does hold when $Z$ has simple normal crossings [MP3, Proposition 7.1] and when $Z$ has isolated quasi-homogeneous singularities [Zhang].

In what follows we will use of the following triviality criterion for the ideals $\widetilde{I}_{p}(D)$:

It is clear by definition that if $\partial_{t}^{p}\delta\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$, then $1\in\widetilde{I}_{p}(D)$, giving one implication. On the other hand, the converse is clear for $p=0$, and in general we argue by induction. If $\widetilde{I}_{p}(D)=\mathscr{O}_{X}$, then there is an element

By considering $(f-t)^{i}v$, for $1\leq i\leq p$, we see that $\widetilde{I}_{p-i}(D)=\mathscr{O}_{X}$, hence $\partial_{t}^{p-i}\delta\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$ by induction. Therefore we have

Recall now from [MP1] and [MP3] the following notion which extends that of a log canonical pair.

Corollary 5.1 implies that this is equivalent to $I_{k}(D)=\mathscr{O}_{X}$. Note that for this to hold, we need $\alpha\leq 1$. We make the convention that $(X,D)$ is $(-1)$-log canonical if and only if $\alpha\leq 1$.

For the first nontrivial ideal we have a statement that is stronger than that of Corollary 5.1.

If $(X,D)$ is $(p-1)$-log canonical, then

In particular, we always have $I_{1}(D)\subseteq I_{0}(D)$ when $D=\alpha Z$ with $\alpha\leq 1$.

The inclusion $\widetilde{I}_{p}(D)\subseteq I_{p}(D)$ follows from the identity

combined with the fact that $(f)\subseteq I_{p}(D)$ due to $(p-1)$-log canonicity; see assertion ii) in Remark 4.9 (note that the inclusion also holds if $p=0$, by Remark 4.4). To prove the opposite inclusion, it suffices to show that we also have $(f)\subseteq\widetilde{I}_{p}(D)$. To this end, the triviality of $I_{p-1}(D)$ implies that we also have $\widetilde{I}_{p-1}(D)=\mathscr{O}_{X}$, which in turn is equivalent to

as well, which gives $f\in\widetilde{I}_{p}(D)$. This proves the first statement.

The second statement follows since by Corollary 5.1 we have

the last equality again being due to $(p-1)$-log canonicity. ∎

We also obtain information about the behavior of the Hodge ideals $I_{p}(\alpha Z)$ when $\alpha$ varies. In the case of $I_{0}$, via the connection with multiplier ideals (or directly from the description in terms of $V^{\alpha}\iota_{+}\mathscr{O}_{X}$), it is well known that they get smaller as $\alpha$ increases, and that there is a discrete set of values of $\alpha$ (called jumping coefficients) where the ideal actually changes; see [Lazarsfeld, Lemma 9.3.21]. This is not the case for higher $k$; for instance, for the cusp $Z=(x^{2}+y^{3}=0)$ and $\alpha\leq 1$ and close to $1$, we see in [MP3, Example 10.5] that

and thus we obtain incomparable ideals. However, Theorem Theorem A′′ implies that the picture does becomes similar to that for multiplier ideals if one considers the images in $\mathscr{O}_{Z}$.

Given any $p\geq 0$, there exists a finite set of rational numbers $0=c_{0}<c_{1}<\cdots<c_{s}<c_{s+1}=1$ such that for each $0\leq i\leq s$ and each $\alpha\in(c_{i},c_{i+1}]$ we have

In fact, if $Z$ is defined by a global equation $f$, the set of $c_{i}$ is a subset of the set of jumping numbers for the $V$-filtration on $\iota_{+}\mathscr{O}_{X}$ associated to $f$.

for every $\alpha\in(0,1]$, where $0<\epsilon\ll 1$. It follows that for $p=0$, the jumping coefficients in Corollary 5.6 coincide with those jumping coefficients for the multiplier ideals of $Z$, in the sense of [ELSV], that lie in $(0,1]$.

Note that Theorem Theorem A′′ implies further facts about elements in the $V$-filtration on $\iota_{+}\mathscr{O}_{X}$. For example, if $v=\sum_{j=0}^{p}v_{j}\partial_{t}^{j}\delta\in V^{\alpha}\iota_{+}\mathscr{O}_{X}$, with $p\geq 2$ and $\alpha<1$, then

For $p=2$, this says that $fv_{2}\in I_{2}(D)$, so that $f\cdot\widetilde{I}_{2}(D)\subseteq I_{2}(D)$.

Indeed, it follows from Theorem Theorem A′′ that

another application of Theorem Theorem A′′ gives

Therefore we have $fg\in I_{p}(D)$ (see assertion ii) in Remark 4.9). Note also that we always have $(f^{p+1})\subseteq I_{p}(D)$, by combining Remark 4.4 with the assertion ii) in Remark 4.9. We thus obtain

Dividing by $(1-\alpha)$, which is assumed to be nonzero, we obtain (5.1).

Bernstein-Sato polynomials and minimal exponent

In this section we relate the $p$-log canonicity of a pair $(X,D)$, with $D=\alpha Z$, to the Bernstein-Sato polynomial of $Z$. We begin by recalling the definition and some basic facts about Bernstein-Sato polynomials.

Suppose that $X$ is a smooth complex variety and $f\in\mathscr{O}_{X}(X)$ is a nonzero regular function on $X$. The Bernstein-Sato polynomial $b_{f}(s)\in{\mathbf{C}}[s]$ of $f$ is the (nonzero) monic polynomial of minimal degree such that

If $f$ is not invertible, by setting $s=-1$ in (6.1), we see that $(s+1)$ divides $b_{f}(s)$. We can thus write $b_{f}(s)=(s+1)\cdot\widetilde{b}_{f}(s)$, and $\widetilde{b}_{f}(s)$ is called the reduced Bernstein-Sato polynomial of $f$. This invariant was studied by Saito in [Saito_microlocal]. In particular, he showed that it is related to the microlocal $V$-filtration mentioned in Remark 4.8; consequently, $\widetilde{b}_{f}(s)$ was also called the microlocal $b$-function in loc.cit.

The existence of a nonzero polynomial $b_{f}(s)$ that satisfies (6.1) was proved by Bernstein [Bernstein] when $X={\mathbf{A}}^{n}$. For a proof in the case of arbitrary $X$ (or, more generally, when $f$ is a holomorphic function on a complex manifold), see [Kashiwara2] and [Bjork]. It follows from the definition that if $X=\bigcup_{i\in I}U_{i}$ is a finite open cover, then $b_{f}(s)$ is the least common multiple of the polynomials $(b_{f|_{U_{i}}})_{i\in I}$. Moreover, one can show that if $g$ is an invertible function, then $b_{f}(s)=b_{fg}(s)$. If $E$ is an effective divisor on $X$, we can thus define the Bernstein-Sato polynomial $b_{E}(s)$ such that if $X=\bigcup_{i\in I}U_{i}$ is a finite open cover and $f_{i}\in\mathscr{O}_{X}(U_{i})$ is an equation of $E|_{U_{i}}$, then $b_{E}(s)$ is the least common multiple of the polynomials \big{(}b_{f_{i}}(s)\big{)}_{i\in I}. If $E\neq 0$, then $b_{E}(s)=(s+1)\cdot\widetilde{b}_{E}(s)$, for a polynomial $\widetilde{b}_{E}(s)$.