Hodge ideals for Q-divisors: birational approach

A. Introduction

In this paper we continue the study of Hodge ideals initiated in [MP1], [MP2], by considering an analogous theory for arbitrary ${\mathbf{Q}}$-divisors. The emphasis here is on a birational definition and study of Hodge ideals, while the companion paper [MP3] is devoted to a study based on their connection with the $V$-filtration, inspired by [Saito-MLCT]. Both approaches turn out to provide crucial information towards a complete understanding of these objects.

Let $X$ be a smooth complex variety. If $D$ is reduced divisor on $X$, the Hodge ideals $I_{k}(D)$, with $k\geq 0$, are defined in terms of the Hodge filtration on the $\mathscr{D}_{X}$-module $\mathscr{O}_{X}(*D)$ of functions with poles of arbitrary order along $D$. Indeed, this $\mathscr{D}_{X}$-module underlies a mixed Hodge module on $X$, and therefore comes with a Hodge filtration $F_{\bullet}\mathscr{O}_{X}(*D)$, which satisfies

See [MP1] for details, and for an extensive study of the ideals $I_{k}(D)$.

Our goal here is to provide a similar construction and study in the general case. A natural device for dealing with the fact that fractional divisors are not directly related to Hodge theory is to use new objects derived from covering constructions. Let $D$ be an arbitrary effective ${\mathbf{Q}}$-divisor on $X$. Locally, we can write $D=\alpha H$, for some $\alpha\in{\mathbf{Q}}_{>0}$ and $H={\rm div}(h)$, the divisor of a nonzero regular function; we also denote by $Z$ the support of $D$. A well-known construction associates to this data a twisted version of the localization $\mathscr{D}$-module above, namely

that is the rank $1$ free $\mathscr{O}_{X}(*Z)$-module with generator the symbol $h^{-\alpha}$, on which a derivation $D$ of $\mathscr{O}_{X}$ acts by

It turns out that this $\mathscr{D}_{X}$-module can be endowed with a natural filtration $F_{k}\mathcal{M}(h^{-\alpha})$, with $k\geq 0$, which makes it a filtered direct summand of a $\mathscr{D}$-module underlying a mixed Hodge module on $X$; see §2. This plays a role analogous to the Hodge filtration, and just as in the reduced case one can show that $F_{k}\mathcal{M}(h^{-\alpha})\subseteq\mathscr{O}_{X}(kZ)h^{-\alpha}$. This is done in §3 and §4, by first analyzing the case when $Z$ is a smooth divisor (in this case, if $\lceil D\rceil=Z$, then the inclusion is in fact an equality). It is therefore natural to define the $k$-th Hodge ideal of $D$ by the formula

Similarly to [MP1], one of our main goals here is to study Hodge ideals of ${\mathbf{Q}}$-divisors by means of log resolutions. To this end, let $f\colon Y\to X$ be a log resolution of the pair $(X,D)$ that is an isomorphism over $U=X\smallsetminus Z$, and denote $g=h\circ f$. There is a filtered isomorphism

Denoting $G=f^{*}D$ and $E={\rm Supp}(G)$, so that $E$ is a simple normal crossing divisor, it turns out that there exists a complex on $Y$:

which is placed in degrees $-n,\ldots,0$, whose differential is described in §6. This complex has a natural filtration given, for $k\geq 0$, by subcomplexes

Extending [MP1, Proposition 3.1], we show in Proposition 6.1 and Proposition 7.1 that there is a filtered quasi-isomorphism

where $\mathcal{M}_{r}(g^{-\alpha})$ is the filtered right $\mathscr{D}_{Y}$-module associated to $\mathcal{M}(g^{-\alpha})$. Thus one can use \big{(}C^{\bullet}_{g^{-\alpha}}(-\lceil G\rceil),F\big{)} as a concrete representative for computing the filtered $\mathscr{D}$-module pushforward of \big{(}\mathcal{M}_{r}(g^{-\alpha}),F\big{)}, hence for computing the ideals $I_{k}(D)$. More precisely, we have

See Theorem 8.1 for a complete picture regarding this push-forward operation.

This fact, together with special properties of the filtration on $\mathscr{D}$-modules underlying mixed Hodge modules, leads to our main results on Hodge ideals, which are collected in the following:

In the set-up above, the Hodge ideals $I_{k}(D)$ satisfy:

(i) $I_{0}(D)$ is the multiplier ideal \mathcal{I}\big{(}(1-\epsilon)D\big{)}, so in particular $I_{0}(D)=\mathscr{O}_{X}$ if and only if the pair $(X,D)$ is log canonical; see §9.

(ii) If $Z$ has simple normal crossings, then

while $I_{k}(Z)$ can be computed explicitly as in [MP1, Proposition 8.2]; see §7. In particular, if $Z$ is smooth, then $I_{k}(D)=\mathscr{O}_{X}(Z-\lceil D\rceil)$ for all $k$; cf. also Corollary 11.12.

(iii) The Hodge filtration is generated at level $n-1$, where $n=\dim X$, i.e.

(iv) There are non-triviality criteria for $I_{k}(D)$ at a point $x\in D$ in terms of the multiplicity of $D$ at $x$; see §11.

(v) If $X$ is projective, $I_{k}(D)$ satisfy a vanishing theorem analogous to Nadel Vanishing for multiplier ideals; see §12.

(vi) If $Y$ is a smooth divisor in $X$ such that $Z|_{Y}$ is reduced, then $I_{k}(D)$ satisfy

with equality when $Y$ is general; see §13 for a more general statement.

(vii) If $X\to T$ is a smooth family with a section $s\colon T\to X$, and $D$ is a relative divisor on $X$ that satisfies a suitable condition (see §14 for the precise statement) then

is an open subset of $T$, for each $q\geq 1$.

(viii) If $D_{1}$ and $D_{2}$ are ${\mathbf{Q}}$-divisors with supports $Z_{1}$ and $Z_{2}$, such that $Z_{1}+Z_{2}$ is also reduced, then the subadditivity property

holds; see §15 for a more general statement.

For comparison, the list of properties of Hodge ideals in the case when $D$ is reduced is summarized in [Popa, §4]. While much of the story carries over to the setting of ${\mathbf{Q}}$-divisors – besides of course the connection with the classical Hodge theory of the complement $U=X\smallsetminus D$, which only makes sense in the reduced case – there are a few significant points where the picture becomes more intricate. For instance, the bounds for the generation level of the Hodge filtration can become worse. Moreover, we do not know whether the inclusions $I_{k}(D)\subseteq I_{k-1}(D)$ continue to hold for arbitrary ${\mathbf{Q}}$-divisors. New phenomena appear as well: unlike in the case of multiplier ideals, for rational numbers $\alpha_{1}<\alpha_{2}$, usually the ideals $I_{k}(\alpha_{1}Z)$ and $I_{k}(\alpha_{2}Z)$ cannot be compared for $k\geq 1$; see for instance Example 10.5.

It turns out however that most of these issues disappear if one works modulo the ideal of the hypersurface, at least for rational multiples of a reduced divisor. This, as well as other basic facts, is addressed in the sequel [MP3], which studies Hodge ideals from a somewhat different point of view, namely by comparing them to the (microlocal) $V$-filtration induced on $\mathscr{O}_{X}$ by $h$. This is inspired by the work of Saito [Saito-MLCT] in the reduced case. In the statement below we summarize some of these properties, which complement the results in Theorem A, but which we do not know how to obtain with the methods of this paper.

[MP3] Let $D=\alpha Z$, where $Z$ is a reduced divisor and $\alpha\in{\mathbf{Q}}_{>0}$. Then the following hold:

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

If $\alpha\in(0,1]$, then $I_{k}(D)=\mathscr{O}_{X}\iff k\leq\widetilde{\alpha}_{Z}-\alpha$, where $\widetilde{\alpha}_{Z}$ is the negative of the largest root of the reduced Bernstein-Sato polynomial of $Z$.

If $I_{k-1}(D)=\mathscr{O}_{X}$ (we say that $(X,D)$ is $(k-1)$-log canonical), then $I_{k+1}(D)\subseteq I_{k}(D)$.

Fixing $k$, 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

Going back to the description of Hodge ideals by means of log resolutions, the strictness of the Hodge filtration for the push-forwards of (summands of) mixed Hodge modules leads to the following local Nakano-type vanishing result for ${\mathbf{Q}}$-divisors:

Let $D$ be an effective ${\mathbf{Q}}$-divisor on a smooth variety $X$ of dimension $n$, and let $f\colon Y\to X$ be a log resolution of $(X,D)$ that is an isomorphism over $X\smallsetminus{\rm Supp}(D)$. If $E=(f^{*}D)_{{\rm red}}$, then

Note that for $p=n$ this is the local vanishing for multiplier ideals [Lazarsfeld, Theorem 9.4.1], since $E-\lceil f^{*}D\rceil=-[(1-\epsilon)f^{*}D]$ for $0<\epsilon\ll 1$. In general, the statement extends the case of reduced $D$ in [Saito-LOG, Corollary 3] (cf. also [Saito-MLCT, §A.5]). Unlike [MP1, Theorem 32.1] regarding that case, at the moment we are unable to prove this corollary via more elementary methods.

A different series of applications, given in [MP3], uses the results proved in this paper together with the relationship between Hodge ideals of ${\mathbf{Q}}$-divisors and the $V$-filtration, in order to describe the behavior of the invariant $\widetilde{\alpha}_{Z}$ described in Theorem B (called the minimal exponent of $Z$). For instance, the triviality criterion proved here as Proposition 11.2 leads to a lower bound [MP3, Corollary D] for $\widetilde{\alpha}_{Z}$ in terms of invariants on a log resolution, addressing a question of Lichtin and Kollár. Moreover, the results in Theorem A (vi) and (vii), and Corollary 11.11, lead to effective bounds and to restriction and semicontinuity statements for $\widetilde{\alpha}_{Z}$, in analogy with well-known properties of log canonical thresholds; for details see [MP3, §6].

B. Hodge ideals via log resolutions, and first properties

Let $X$ be a smooth complex algebraic variety. Given an effective ${\mathbf{Q}}$-divisor $D$ on $X$, our goal is to attach to $D$ ideal sheaves $I_{k}(D)$ for $k\geq 0$; when $D$ is a reduced divisor, these will coincide with the Hodge ideals in [MP1].

A key ingredient for the definition of our invariants is Saito’s theory of mixed Hodge modules. In what follows, we give a brief presentation of the relevant objects, and recall a few facts that we will need. For details, we refer to [Saito-MHM].

Given a smooth $n$-dimensional complex algebraic variety $X$, we denote by $\mathscr{D}_{X}$ the sheaf of differential operators on $X$. This carries the increasing filtration $F_{\bullet}\mathscr{D}_{X}$ by order of differential operators. A left or right $\mathscr{D}$-module is a left, respectively right, $\mathscr{D}_{X}$-module, which is quasi-coherent as an $\mathscr{O}_{X}$-module. There is an equivalence between the categories of left and right $\mathscr{D}$-modules, which at the level of $\mathscr{O}_{X}$-modules is given by

For example, this equivalence maps the left $\mathscr{D}$-module $\mathscr{O}_{X}$ to the right $\mathscr{D}$-module $\omega_{X}$. For a thorough introduction to the theory of $\mathscr{D}$-modules, we refer to [HTT].

A filtered left (or right) $\mathscr{D}$-module is a $\mathscr{D}$-module $\mathcal{M}$, together with an increasing filtration $F=F_{\bullet}\mathcal{M}$ that is compatible with the order filtration on $\mathscr{D}_{X}$ and which is good, in a sense to be defined momentarily. A morphism of filtered $\mathscr{D}$-modules is required to be compatible with the filtrations. The equivalence between left and right $\mathscr{D}$-modules extends to the categories of filtered modules, with the convention that

A filtration $F_{\bullet}\mathcal{M}$ on a coherent $\mathscr{D}$-module $\mathcal{M}$ is good if the corresponding graded object ${\rm gr}_{\bullet}^{F}\mathcal{M}:=\bigoplus_{k}F_{k}\mathcal{M}/F_{k-1}\mathcal{M}$ is locally finitely generated over ${\rm gr}_{\bullet}^{F}\mathscr{D}_{X}$. We note that every coherent $\mathscr{D}$-module admits a good filtration, but this is far from being unique.

We now come to the key objects in Saito’s theory, the mixed Hodge modules from [Saito-MHM]. Such an object is given by the data $M=(\mathcal{M},F,\mathcal{P},\varphi,W)$, where:

$(\mathcal{M},F)$ is a filtered $\mathscr{D}$-module, with $\mathcal{M}$ a holonomic left (or right) $\mathscr{D}$-module, with regular singularities; $F$ is the Hodge filtration of $\mathcal{M}$.

$\mathcal{P}$ is a perverse sheaf of ${\mathbf{Q}}$-vector spaces on $X$.

$\varphi$ is an isomorphism between $\mathcal{P}_{{\mathbf{C}}}=\mathcal{P}\otimes_{{\mathbf{Q}}}{\mathbf{C}}$ and ${\rm DR}(\mathcal{M})$, i.e. the perverse sheaf corresponding to $\mathcal{M}$ via the Riemann-Hilbert correspondence.

$W$ is a finite, increasing filtration on $(\mathcal{M},F,\mathcal{P},\varphi)$, the weight filtration of the mixed Hodge module.

For a such an object to be a mixed Hodge module, it has to satisfy a complicated set of conditions of an inductive nature, which we do not discuss here. The main reference for the basic definitions and results of this theory is [Saito-MHM]; see also [Saito-YPG] for an introduction.

Given a mixed Hodge module $(\mathcal{M},F,\mathcal{P},\varphi,W)$, we say that the filtered $\mathscr{D}$-module $(\mathcal{M},F)$ is a Hodge $\mathscr{D}$-module (or that it underlies a mixed Hodge module). In fact, this is the only piece of information that we will be concerned with in this article. The basic example of a mixed Hodge module is ${\mathbf{Q}}_{X}^{H}[n]$, the trivial one. In this case, the filtered $\mathscr{D}$-module is the structure sheaf $\mathscr{O}_{X}$, with the filtration such that ${\rm gr}_{p}^{F}\mathscr{O}_{X}=0$ for all $p\neq 0$. The corresponding perverse sheaf is ${\mathbf{Q}}_{X}[n]$ and the weight filtration is such that ${\rm gr}_{p}^{W}\mathscr{O}_{X}=0$ for $p\neq n$.

The mixed Hodge modules on $X$ form an Abelian category, denoted ${\rm MHM}(X)$. Morphisms in this category are strict with respect to both the Hodge and the weight filtration. The corresponding bounded derived category is denoted {\bf D}^{b}\big{(}{\rm MHM}(X)\big{)}.

Mixed Hodge modules satisfy Grothendieck’s 6 operations formalism. The relevant fact for us is that to every morphism $f\colon X\to Y$ of smooth complex algebraic varieties we have a corresponding push-forward functor f_{+}\colon{\bf D}^{b}\big{(}{\rm MHM}(X)\big{)}\to{\bf D}^{b}\big{(}{\rm MHM}(Y)\big{)} (this is denoted by $f_{*}$ in [Saito-MHM]). Moreover, if $g\colon Y\to Z$ is another such morphism, we have a functorial isomorphism $(g\circ f)_{+}\simeq g_{+}\circ f_{+}$.

Regarding the push-forward functor for mixed Hodge modules, we note that on the level of $\mathscr{D}$-modules, it coincides with the usual $\mathscr{D}$-module push-forward. Moreover, if $f\colon X\to Y$ is proper and if we denote by ${\rm FM}(\mathscr{D}_{X})$ the category of filtered $\mathscr{D}$-modules on $X$ (here it is convenient to work with right $\mathscr{D}$-modules), then Saito defined in [Saito-MHP] a functor

This is compatible with the usual direct image functor for right $\mathscr{D}$-modules and it is used to define the push-forward between the derived categories of mixed Hodge modules at the level of filtered complexes. With a slight abuse of notation, if $(\mathcal{M},F)$ underlies a mixed Hodge module $M$ on $X$ and if $f\colon X\to Y$ is an arbitrary morphism, then we write $f_{+}(\mathcal{M},F)$ for the object in {\bf D}^{b}\big{(}{\rm FM}(\mathscr{D}_{Y})\big{)} underlying $f_{+}M$.

An important feature of the push-forward of Hodge $\mathscr{D}$-modules with respect to proper morphisms is strictness. This says that if $f\colon X\to Y$ is proper and $(\mathcal{M},F)$ underlies a mixed Hodge module on $X$, then $f_{+}(\mathcal{M},F)$ is strict as an object in {\bf D}^{b}\big{(}{\rm FM}(\mathscr{D}_{Y})\big{)} (and moreover, each $H^{i}f_{+}(\mathcal{M},F)$ underlies a Hodge $\mathscr{D}_{Y}$-module). This means that the natural mapping

is injective for every $i,k\in{\mathbf{Z}}$. Taking $F_{k}H^{i}f_{+}(\mathcal{M},F)$ to be the image of this map, we get the filtration on $H^{i}f_{+}(\mathcal{M},F)$.

The push-forward with respect to open embeddings is more subtle. For example, suppose that $Z$ is an effective divisor on the smooth variety $X$ and $j\colon U=X\smallsetminus Z\hookrightarrow X$ is the corresponding open immersion. Recall that $\mathscr{O}_{X}(*Z)$ is the push-forward $j_{*}\mathscr{O}_{U}$; on a suitable affine open neighborhood $V$ of a given point in $X$, this is given by localizing $\mathscr{O}_{X}(V)$ at an equation defining $Z\cap V$ in $V$. $\mathscr{O}_{X}(*Z)$ has a natural left $\mathscr{D}$-module structure induced by the canonical $\mathscr{D}$-module structure on $\mathscr{O}_{X}$. In fact, as such we have $\mathscr{O}_{X}(*Z)\simeq j_{+}\mathscr{O}_{U}$ (in general, for a $\mathscr{D}_{U}$-module $\mathcal{M}$, the $\mathscr{D}$-module push-forward $j_{+}\mathcal{M}$ agrees with $j_{*}\mathcal{M}$, with the induced $\mathscr{D}_{X}$-module structure). We thus see that $\mathscr{O}_{X}(*Z)$ carries a canonical filtration such that the corresponding filtered $\mathscr{D}$-module underlies $j_{+}{\mathbf{Q}}_{U}^{H}[n]$. This filtration is the one that leads to the Hodge ideals studied in [MP1].

Filtered 𝒟𝒟\mathscr{D}-modules associated to 𝐐𝐐{\mathbf{Q}}-divisors

Let $X$ be a smooth complex algebraic variety, with $\dim(X)=n$. The ideals we associate to effective ${\mathbf{Q}}$-divisors on $X$ arise from certain Hodge $\mathscr{D}$-modules. The $\mathscr{D}$-modules themselves have been extensively studied: these are the $\mathscr{D}$-modules attached to rational powers of functions on $X$. We proceed to recall their definition.

Consider a nonzero $h\in\mathscr{O}_{X}(X)$ and $\beta\in{\mathbf{Q}}$. We denote by $Z$ the reduced divisor on $X$ with the same support as $H={\rm div}(h)$ and let $j\colon U=X\smallsetminus{\rm Supp}(Z)\hookrightarrow X$ be the inclusion map. We consider the left $\mathscr{D}_{X}$-module $\mathcal{M}(h^{\beta})$, which is a rank 1 free $\mathscr{O}_{X}(*Z)$-module with generator the symbol $h^{\beta}$, on which a derivation $D$ of $\mathscr{O}_{X}$ acts by

We will denote the corresponding right $\mathscr{D}_{X}$-module by $\mathcal{M}_{r}(h^{\beta})$. This can be described as $h^{\beta}\omega_{X}(*Z)$, an $\mathscr{O}_{X}$-module isomorphic to $\omega_{X}(*Z)$, and such that if $x_{1},\ldots,x_{n}$ are local coordinates, then

When $\beta\in{\mathbf{Z}}$, we have a canonical isomorphism of left $\mathscr{D}_{X}$-modules

where on the localization $\mathscr{O}_{X}(*Z)$ we consider the natural $\mathscr{D}_{X}$-module structure induced from $\mathscr{O}_{X}$. Note that $\mathscr{O}_{X}(*Z)$ is also the $\mathscr{D}$-module push-forward $j_{+}\mathscr{O}_{U}$.

For every positive integer $m$, we have a canonical isomorphism of left $\mathscr{D}_{X}$-modules

If $r$ is an integer, then we have an isomorphism of left $\mathscr{D}_{X}$-modules

We have an isomorphism of left $\mathscr{D}_{X}$-modules

with the convention that the first summand is $\mathscr{O}_{X}(*Z)$.

Since $p$ is finite étale, it follows that we have a canonical isomorphism $\tau\colon p^{*}\mathscr{D}_{U}\simeq\mathscr{D}_{V}$, and for every $\mathscr{D}_{V}$-module $\mathcal{M}$ we have $p_{+}\mathcal{M}\simeq p_{*}\mathcal{M}$, with the action of $\mathscr{D}_{U}$ induced via the isomorphism $\tau$.

which via our map corresponds to $D(h^{-i\alpha})$. This implies the assertion. ∎

It follows from the lemma that the right-hand side of (2.4) is the $\mathscr{D}$-module corresponding to the mixed Hodge module push-forward $(j\circ p)_{+}{\mathbf{Q}}_{V}^{H}[n]$. In particular, it carries a canonical structure of filtered $\mathscr{D}$-module.

where $\widetilde{j}\colon Y\smallsetminus{\rm Supp}(E)\hookrightarrow Y$ is the inclusion map.

We can deduce the assertion in the lemma from an explicit computation of the filtration on $j_{+}p_{+}\mathscr{O}_{V}$ via the isomorphism (2.6), as follows. First, since we deal with $\mathscr{D}$-module push-forward, it is more convenient to work with right $\mathscr{D}$-modules. We will thus compute $g_{+}\omega_{Y}(*E)$, where $\omega_{Y}(*E)$ is the filtered right $\mathscr{D}$-module corresponding to $\mathscr{O}_{Y}(*E)$.

Since $E$ is a simple normal crossing divisor, $\omega_{Y}(*E)$ has a resolution by a complex $C^{\bullet}$ of filtered right $\mathscr{D}_{Y}$-modules

where $C^{i}=\Omega_{Y}^{i+n}(\log E)\otimes_{\mathscr{O}_{Y}}\mathscr{D}_{Y}$, with the filtration given by

For a description of the maps in this complex, see the beginning of §6 below; a proof of the fact that it resolves $\omega_{Y}(*E)$ is given in [MP1, Proposition 3.1]. We can thus compute $F_{k}g_{+}\omega_{Y}(*E)$ as the image of the injective map

We note that the filtration on $j_{+}p_{+}\mathscr{O}_{V}$ induces the canonical filtration on the first summand $\mathscr{O}_{X}(*Z)$. Indeed, on $U$ we have a morphism of mixed Hodge modules ${\mathbf{Q}}_{U}^{H}[n]\to p_{+}{\mathbf{Q}}_{V}^{H}[n]$. Applying $j_{+}$ and only considering the underlying filtered $\mathscr{D}$-modules, we obtain a morphism $j_{+}\mathscr{O}_{U}\to j_{+}p_{+}\mathscr{O}_{V}$, which is an isomorphism onto the first summand.

In this definition, a priori different covers have to be considered for each of the summands $\mathcal{M}(h^{-i\alpha})$. However, we have:

With the filtration defined above, the isomorphism (2.4) is an isomorphism of filtered $\mathscr{D}$-modules.

Note that we have a finite morphism $\psi\colon V\to V^{\prime}$ of varieties over $U$, that pulls-back $y$ to $y^{i}$. We have a canonical morphism of mixed Hodge modules ${\mathbf{Q}}_{V^{\prime}}^{H}[n]\to\psi_{+}{\mathbf{Q}}_{V}^{H}[n]$. Applying $j_{+}p^{\prime}_{+}$ and passing to the underlying filtered $\mathscr{D}$-modules, we obtain a morphism of filtered $\mathscr{D}$-modules $j_{+}p^{\prime}_{+}\mathscr{O}_{V^{\prime}}\to j_{+}p_{+}\mathscr{O}_{V}$ that is the identity on the summand $\mathcal{M}(h^{-i\alpha})$. This proves our claim. ∎

It is clear from definition that for every $\alpha>0$ and every positive integer $m$, the isomorphism

is an isomorphism of filtered $\mathscr{D}$-modules.

We have an isomorphism $\varphi\colon V^{\prime}\to V$ of schemes over $U$, where $\varphi^{*}(y)=uy$. This induces an isomorphism of filtered $\mathscr{D}_{X}$-modules

which via the identifications given by Lemma 2.6 is the direct sum

of the isomorphisms (2.2). For $i=1$, we obtain our assertion.

A special case of the above remark implies that for every $\alpha>0$ the isomorphism

is an isomorphism of filtered $\mathscr{D}$-modules. We use this to put a structure of filtered $\mathscr{D}$-module on $\mathcal{M}(h^{\beta})$ for every $\beta\in{\mathbf{Q}}$, such that for every $r\in{\mathbf{Z}}$, we have an isomorphism of filtered $\mathscr{D}$-modules

For example, we have have an isomorphism of filtered $\mathscr{D}$-modules $\mathcal{M}(h^{0})\simeq\mathscr{O}_{X}(*Z)$.

Suppose that $h,\bar{h}\in\mathscr{O}_{X}(X)$ are nonzero, and $\alpha,\bar{\alpha}\in{\mathbf{Q}}_{>0}$ are such that we have the equality of ${\mathbf{Q}}$-divisors

Indeed, this follows from the definition of the filtrations and the isomorphism of schemes over $U$

It is clear that the filtration on $\mathcal{M}(h^{-\alpha})$ is compatible with restriction to open subsets. More generally, it is compatible with smooth pullback, as follows. Suppose that $h\in\mathscr{O}_{X}(X)$ is nonzero and $\alpha\in{\mathbf{Q}}$. If $\varphi\colon Y\to X$ is a smooth morphism and $g=h\circ\varphi$, then there is an isomorphism of $\mathscr{D}_{Y}$-modules

where $j$ and $p$ are as in Lemma 2.6 and $j_{Y}$ and $p_{Y}$ are the corresponding morphisms for $Y$ and $g$. Note that we have a base-change theorem that gives

The case of smooth divisors

Our goal now is to describe the filtrations introduced in the previous section when $Z$ is a smooth divisor. We will then use this to define Hodge ideals for arbitrary ${\mathbf{Q}}$-divisors. The key result in the smooth case is the following:

As usual, it is easier to do the computation for the filtered right $\mathscr{D}$-module $\omega_{Y}(*Z)$ corresponding to $\mathscr{O}_{Y}(*Z)$. Note that this is filtered quasi-isomorphic to the complex

placed in degrees $-1$ and , where $w(1)=\frac{dt}{t}\otimes t\partial_{t}$; see e.g. [MP1, Proposition 3.1]. Since $\psi$ is finite, the functor $\psi_{*}$ is exact on quasi-coherent $\mathscr{O}_{Y}$-modules, hence $\psi_{+}\omega_{Y}(*Z)$ is computed by the -th cohomology of the complex

where we use the fact that $t\partial_{t}t^{j}=t^{j}t\partial_{t}+jt^{j}$. In other words, we have have an eigenspace decomposition

where $B^{\bullet}_{j}$ is identified with the complex

where the filtration on the $j$-th component is such that

The assertion in the lemma now follows immediately from the explicit description of the equivalence between the categories of left and right $\mathscr{D}$-modules on $X={\mathbf{A}}^{1}$. Indeed, recall that if $\tau$ is the ${\mathbf{C}}$-linear endomorphism of the Weyl algebra $\Gamma({\mathbf{A}}^{1},\mathscr{D}_{{\mathbf{A}}^{1}})$ such that $\tau(PQ)=\tau(Q)\cdot\tau(P)$ for all $P$ and $Q$, and such that $\tau(t)=t$ and $\tau(\partial_{t})=-\partial_{t}$, then the left $\mathscr{D}$-module $N$ corresponding to a right $\mathscr{D}$-module $M$ is isomorphic to $M$ itself, with scalar multiplication given via the map $\tau$. Moreover, for filtered $\mathscr{D}$-modules, via this isomorphism $F_{k}N$ corresponds to $F_{k-1}M$. In particular, we see that if $M=\mathscr{D}_{X}/P\cdot\mathscr{D}_{X}$, then $N\simeq\mathscr{D}_{X}/\mathscr{D}_{X}\cdot\tau(P)$, and we obtain the statement. ∎

In what follows, we denote by $\lceil\alpha\rceil$ the smallest integer that is $\geq\alpha$. For a ${\mathbf{Q}}$-divisor $D=\sum_{i=1}^{r}a_{i}D_{i}$, we put $\lceil D\rceil=\sum_{i=1}^{r}\lceil a_{i}\rceil D_{i}$.

If $h\in\mathscr{O}_{X}(X)$ is nonzero and such that the support $Z$ of ${\rm div}(h)$ is smooth (possibly disconnected), then for every $\alpha\in{\mathbf{Q}}_{>0}$ the filtration on $\mathcal{M}(h^{-\alpha})$ is given by

where $D=\alpha\cdot{\rm div}(h)$, and $F_{k}\mathcal{M}(h^{-\alpha})=0$ if $k<0$.

We first reduce to the case when $Z={\rm div}(h)$. We can check the assertion in the proposition locally, hence we may assume that $Z={\rm div}(g)$, for some $g\in\mathscr{O}_{X}(X)$, and $h=ug^{m}$, for some $u\in\mathscr{O}_{X}^{*}(X)$. Furthermore, by Remark 2.15, it is enough to prove the assertion after passing to a surjective étale cover, hence we may assume that $u=v^{m}$ for some $v\in\mathscr{O}_{X}^{*}(X)$. After replacing $g$ by $vg$, we may thus assume that $h=g^{m}$. In this case we have an isomorphism of filtered $\mathscr{D}$-modules $\mathcal{M}(h^{-\alpha})\simeq\mathcal{M}(g^{-m\alpha})$, hence we may and will assume that ${\rm div}(h)=Z$.

The morphism $h\colon X\to{\mathbf{A}}^{1}$ is smooth over some open neighborhood of . Using Remark 2.15, we see that in order to prove the corollary, we may assume that $X={\mathbf{A}}^{1}$ and $h=x$, the standard coordinate on ${\mathbf{A}}^{1}$. Consider the Cartesian diagram

Let $\varphi\colon\widetilde{W}=\operatorname{Spec}{\mathbf{C}}[t]\to W$ be the normalization, given by

that maps the class of $1$ to $x^{-\alpha}$. The formula for the filtration on $\mathcal{M}(h^{\alpha})$ now follows from Lemma 3.1. When $\alpha>1$, we put $m=\lceil\alpha\rceil-1$, and use the fact from Remark 2.4, namely that we have an isomorphism of filtered modules

to reduce the assertion to the case $\alpha\in(0,1)$. This completes the proof of the corollary. ∎

Definition of Hodge ideals for 𝐐𝐐{\mathbf{Q}}-divisors

In general, we obtain an upper bound for the terms in the filtration on $\mathcal{M}(h^{-\alpha})$ by restricting to the open subset where the support of ${\rm div}(h)$ is smooth, as follows.

Given a nonzero $h\in\mathscr{O}_{X}(X)$ and a positive rational number $\alpha$, for every $k\geq 0$ we have

where $D=\alpha\cdot{\rm div}(h)$ and $Z={\rm Supp}(D)$, while $F_{k}\mathcal{M}(h^{-\alpha})=0$ for $k<0$.

Let $\iota\colon X_{0}\to X$ be an open immersion such that the codimension of its image in $X$ is $\geq 2$ and $Z|_{X_{0}}$ is smooth (though possibly disconnected). Note that our constructions are compatible with restrictions to open subsets. Moreover, since $\mathcal{M}(h^{-\alpha})$ is clearly torsion-free, it follows that $F_{k}:=F_{k}\mathcal{M}(h^{-\alpha})$ is torsion free, hence the canonical map F_{k}\to\iota_{*}\big{(}F_{k}|_{X_{0}}\big{)} is injective. Therefore it is enough to prove the assertion on $X_{0}$, hence we may assume that $Z$ is smooth. However, in this case the assertion follows from Corollary 3.2. ∎

We can now define the Hodge ideals for ${\mathbf{Q}}$-divisors. Let $X$ be a smooth complex algebraic variety and $Z$ a reduced effective divisor on $X$. Given an effective ${\mathbf{Q}}$-divisor $D$ with ${\rm Supp}(D)=Z$, we define coherent ideals sheaves $I_{k}(D)$ in $\mathscr{O}_{X}$ as follows. Suppose first that there is a nonzero $h\in\mathscr{O}_{X}(X)$, with $H={\rm div}(h)$, and a positive rational number $\alpha$ such that $D=\alpha H$. It turns out to be more convenient to work with the $\mathscr{D}_{X}$-module $\mathcal{M}(h^{\beta})$, where $\beta=1-\alpha$. Recall that we have a filtered isomorphism

and therefore, if $k\geq 0$, it follows from Proposition 4.1 that there is a unique coherent ideal $I_{k}(D)$ such that

(note that we always have $\lceil D\rceil\geq Z$). The definition is independent of the choice of $\alpha$ and $h$: indeed, using Remark 2.15, it is enough to check this after the pullback by a suitable étale surjective map, hence we deduce the independence assertion using Remark 2.14. This implies that the general case of the definition follows by covering $X$ with suitable affine open subsets such that $D$ can be written as above in each of them. Note that when $D=Z$ we have $\beta=0$, and so the ideals $I_{k}(D)$ are the Hodge ideals studied in [MP1].

From the definition and the filtration property, it follows that we always have the inclusion

We note that for the reduced divisor $Z$, we have the more subtle inclusions

(see [MP1, Proposition 13.1]). We do not know however whether this holds for arbitrary ${\mathbf{Q}}$-divisors $D$, and in fact we suspect that this is not the case. (Note that it does hold when $D$ has simple normal crossings support by Proposition 7.1. It is also shown to hold when $D$ has an isolated weighted homogeneous singularity in the upcoming [Zhang].) However, when $D=\alpha Z$ these inclusions do hold modulo the ideal $\mathscr{O}_{X}(-Z)$, see [MP3, Corollary B]. More precisely, we have

This implies in particular that if $I_{k}(\alpha Z)=\mathscr{O}_{X}$ for some $k\geq 1$, then $I_{k-1}(\alpha Z)=\mathscr{O}_{X}$.

According to Proposition 4.1, we also have ideals $I_{k}^{\prime}(D)$ given by

which are related to $I_{k}(D)$ by the formula

The following periodicity property often allows us to reduce our study to the case $\lceil D\rceil=Z$.

If $D^{\prime}$ is an integral divisor with ${\rm Supp}(D^{\prime})\subseteq{\rm Supp}(D)$, then

with $B=D+Z-\lceil D\rceil$ satisfying $\lceil B\rceil=Z$.

Using the notation in Remark 4.3, the equivalent statement

follows from the definition and Remark 2.13. ∎

Note that $I_{k}(D)\subseteq\mathscr{O}_{X}(Z-\lceil D\rceil)$ for all $k$, and so if $\lceil D\rceil\neq Z$, then one can never have $I_{k}(D)=\mathscr{O}_{X}$. It is however still interesting to ask whether $I_{k}(B)=\mathscr{O}_{X}$.

A global setting for the study of Hodge ideals

We now consider a setting in which we can define global filtered $\mathscr{D}_{X}$-modules that are locally isomorphic to the \big{(}\mathcal{M}(h^{-\alpha}),F\big{)} discussed in the previous sections.

We denote by $U$ the complement of $Z={\rm Supp}(H)$ and by $j$ the inclusion $U\hookrightarrow X$.

We also see that the filtration on $\mathcal{M}$ is the direct sum filtration, since this holds locally. Moreover, we have isomorphisms of $\mathscr{O}_{X_{0}}$-modules

which glue to isomorphisms of $\mathscr{O}_{X}$-modules

Via these isomorphisms, it follows from the definition of Hodge ideals (see also Remark 4.3) that we have

A complex associated to simple normal crossing divisors

We now discuss a complex that, as we will see later, gives a filtered resolution of $\mathcal{M}_{r}(h^{-\alpha})$ by filtered induced $\mathscr{D}_{X}$-modules in the case when $h$ defines a simple normal crossing divisor.

Let $X$ be a smooth, $n$-dimensional, complex variety, $h\in\mathscr{O}_{X}(X)$ nonzero, and $\alpha$ a nonzero rational number (we allow $\alpha$ to be either positive or negative). Let $D=\alpha\cdot{\rm div}(h)$. We denote by $Z$ the support of $D$, and assume that it has simple normal crossings.

Associated to $Z$ we have the following complex of right $\mathscr{D}_{X}$-modules:

placed in degrees $-n,\ldots,0$. We denote by $D_{i}\colon C^{i}\to C^{i+1}$ its differentials. If $x_{1},\ldots,x_{n}$ are local coordinates on $X$, then

In fact $C^{\bullet}$ is a filtered complex, where

This filtered complex is quasi-isomorphic to the filtered right $\mathscr{D}_{X}$-module $\omega_{X}(*Z)$ corresponding to the filtered left $\mathscr{D}_{X}$-module $\mathscr{O}_{X}(*Z)$ (see [MP1, Proposition 3.1], and [Saito-MHM, Proposition 3.11(ii)] for a more general statement).

Given $h$ and $\alpha$ as above, we also consider the filtered complex $C^{\bullet}_{h^{-\alpha}}$ consisting of the same sheaves, but with differential $C^{i}_{h^{-\alpha}}\to C^{i+1}_{h^{-\alpha}}$ given by

It is easy to see that this is indeed a filtered complex.

Suppose now that we also have an effective divisor $T$ supported on $Z$. It is not hard to check that the formula for the map

This is due to the fact that if locally $T={\rm div}(u)$ and $\eta$ is a local section of $\Omega^{i+n}_{X}(\log Z)$, then we can write $d(u\eta)=ud(\eta)+u\cdot{\rm dlog}(u)\wedge\eta$. We thus obtain a filtered subcomplex $C^{\bullet}_{h^{-\alpha}}(-T)$ of $C^{\bullet}_{h^{-\alpha}}$. We emphasize that this is not obtained by tensoring $C^{\bullet}_{h^{-\alpha}}$ with $\mathscr{O}_{X}(-T)$.

If no coefficient of $D-T$ lies in ${\mathbf{Z}}_{<0}$, then the complex $C^{\bullet}_{h^{-\alpha}}(-T)$ is filtered quasi-isomorphic to \big{(}h^{-\alpha}\omega_{X}(*Z),G_{\bullet}\big{)}, where

It is immediate to check that the differential induced on ${\rm gr}^{F}_{p}C^{\bullet}_{h^{-\alpha}}(-T)$ does become equal to the differential $D_{i}$ twisted with the identity on $\mathscr{O}_{X}(-T)$, and therefore for every $p$ we have

by the result in [MP1] quoted above. Consider now the morphism of right $\mathscr{D}_{X}$-modules

We first check that this morphism is surjective. We do this locally, hence we may assume that we have a system of coordinates $x_{1},\ldots,x_{n}$ on $X$ such that $\mathscr{O}_{X}(-Z)$ is generated by $x_{1}\cdots x_{r}$ and $\mathscr{O}_{X}(-T)$ by $x_{1}^{\beta_{1}}\cdots x_{r}^{\beta_{r}}$. We also write $h=ux_{1}^{a_{1}}\cdots x_{r}^{a_{r}},$ where $u$ is an everywhere nonvanishing function, and define $\alpha_{i}=\alpha a_{i}$ and $\gamma_{i}=\alpha_{i}-\beta_{i}$ for all $i$. Note for later use that

The surjectivity of $\varphi$ follows from the fact that

and the second equality in (6.1) is a consequence of the fact that $-\gamma_{i}-1\not\in{\mathbf{Z}}_{\geq 0}$ for all $i$, by assumption.

In order to complete the proof of the proposition it is enough to show that, for every $k\geq 0$, the following sequence is exact:

where $\varphi_{k}$ is the restriction of $\varphi$ to the $(k-n)$-th level of the filtration and $\psi_{k}$ is the restriction of the differential of $C^{\bullet}_{h^{-\alpha}}(-T)$. The surjectivity of $\varphi_{k}$ is an immediate consequence of the surjectivity of $\varphi$ and the definition of the filtration on $h^{\alpha}\omega_{X}(*Z)$.

Keeping the above notation for the local coordinates on $X$, it follows from the definition of $\psi_{k}$ that

and it is straightforward to see that this is contained in ${\rm Ker}(\varphi_{k})$. We now prove by induction on $k$ that if $\varphi_{k}(x_{1}^{\beta_{1}}\cdots x_{r}^{\beta_{r}}\otimes\eta\otimes P)=0$ for some $P\in F_{k}\mathscr{D}_{X}$, then $x_{1}^{\beta_{1}}\cdots x_{r}^{\beta_{r}}\otimes\eta\otimes P\in{\rm Im}(\psi_{k})$. Note that the case $k=0$ is trivial. Let’s write $P=\sum_{u,v}c_{u,v}\partial^{u}x^{v}$, where $u$ and $v$ vary over ${\mathbf{Z}}_{\geq 0}^{n}$. After subtracting suitable terms from $P$, we may assume that whenever $c_{u,v}\neq 0$, we have $u_{i}=0$ for $i>r$. Furthermore, note that if $u_{i},v_{i}>0$ for some $i\leq r$, then we can write

with both $A$ and $B$ of order $\leq k-1$. Therefore we may also assume that whenever $c_{u,v}\neq 0$ and $|u|:=\sum_{i}u_{i}=k$, we have

and since (6.2) implies that for every $(u,v)$ and $(u^{\prime},v^{\prime})$ with $|u|=k$ and $c_{u,v},c_{u^{\prime},v^{\prime}}\neq 0$ we have $x^{v-u}\neq x^{v^{\prime}-u^{\prime}}$, we conclude that in fact $P\in F_{k-1}\mathscr{D}_{X}$, hence we are done by induction. ∎

The Hodge ideals of simple normal crossing divisors

In this section we show that the Hodge ideals of divisors with simple normal crossing support essentially depend only on the support of the divisor, and therefore can be computed as in [MP1, §8].

Let $X$ be a smooth variety, and $D$ an effective divisor on $X$ with simple normal crossing support $Z$. Then for all $k$ we have

with $j_{0}^{*}(z)=y^{-1}$. We will make use of some standard facts about cyclic covers with respect to simple normal crossing divisors, exploiting the toric variety structure on the normalization of $W$. For basic facts regarding toric varieties, we refer to [Fulton].

Let $N$ be the lattice ${\mathbf{Z}}^{n}$ and $M$ its dual. We also consider the lattice

We thus have an isomorphism $N^{\prime}_{{\mathbf{R}}}\simeq N_{{\mathbf{R}}}={\mathbf{R}}^{n}$. The strongly convex cone $\sigma={\mathbf{R}}_{\geq 0}^{n}$ in $N_{{\mathbf{R}}}={\mathbf{R}}^{n}$ gives the toric variety $X={\mathbf{C}}^{n}$. As a cone in $N^{\prime}_{{\mathbf{R}}}$, $\sigma$ gives an affine toric variety $\widetilde{W}$, and the lattice map $N^{\prime}\to N$ corresponds to a toric map $\psi\colon\widetilde{W}\to X$. Note that we have a morphism of $\mathscr{O}(X)$-algebras $\mathscr{O}(W)\to\mathscr{O}(\widetilde{W})$ that maps $x_{i}$ to the element of ${\mathbf{C}}[\sigma^{\vee}\cap M^{\prime}]$ corresponding to the class of the $i$-th element of the standard basis of ${\mathbf{Z}}^{n}$, and $z$ to the class of $(0,\ldots,0,1)$. It is easy to check that if we denote by $\lfloor\gamma\rfloor$ the largest integer $\leq\gamma$, then

and consequently to deduce that $\mathscr{O}(\widetilde{W})$ is integral over $\mathscr{O}(W)$. As the coordinate ring of a toric variety, $\mathscr{O}(\widetilde{W})$ is normal, hence it is the integral closure of $\mathscr{O}(W)$ in its field of fractions. Moreover, since $\widetilde{W}$ is a toric variety, we may choose a toric resolution of singularities $Y\to\widetilde{W}$, and let $f\colon Y\to X$ be the composition. Since the map $Y\to W$ is an isomorphism over the complement of $g^{-1}(\sum H_{i})$, it follows that there is an open embedding $\iota\colon V\hookrightarrow Y$ such that $f\circ\iota=j\circ p$. The support $E_{Y}$ of $Y\smallsetminus\iota(V)$ is the sum of all prime toric divisors on $Y$.

The equality $f\circ\iota=j\circ p$ implies that we have an isomorphism of filtered $\mathscr{D}_{X}$-modules

As usual, in order to compute the push-forward of $\mathscr{O}_{Y}(*E_{Y})$, it is more convenient to work with right $\mathscr{D}$-modules. Recall that there is a complex of right $\mathscr{D}_{Y}$-modules

located in degrees $-n,\ldots,0$, that is filtered quasi-isomorphic to $\omega_{Y}(*E_{Y})$; see the beginning of §6. Since $Y$ is a toric variety, we have a canonical isomorphism $\Omega_{Y}^{1}(\log E_{Y})\simeq M^{\prime}\otimes_{{\mathbf{Z}}}\mathscr{O}_{Y}$ (see [Fulton, Section 4.3]). We will also consider the corresponding complex on $X$:

It follows from the definition that, forgetting about the filtration, we have

Note that $\mathscr{D}_{Y\to X}=f^{*}\mathscr{D}_{X}$ as $\mathscr{O}_{Y}$-modules, hence the projection formula implies

for $i>0$, since $f$ is the composition of a finite map with a toric resolution. Therefore $f_{+}\omega_{Y}(*E_{Y})$ is represented by the complex $B^{\bullet}$, where

In order to describe the differential of this complex, it is convenient to use the isomorphism $M_{{\mathbf{Q}}}\simeq M^{\prime}_{{\mathbf{Q}}}$ and the decomposition (7.1). With a little care, it follows from the definitions that if we put

where $\delta_{A_{X}}$ is the differential on $A_{X}^{\bullet}$ and

It follows from Proposition 6.1 that we have a morphism

We now bring the filtrations into the picture. It follows from Saito’s strictness results (see the discussion in §1; cf. also [MP1, §4, §6]) that

In other words, $(f_{+}\omega_{Y}(*E_{Y}),F)$ is represented by the filtered complex $B^{\bullet}$, and using Proposition 6.1, we conclude that

where the filtration on $\mathcal{M}_{r}(x_{1}^{w_{j,1}}\cdots x_{n}^{w_{j,n}})$ is given by

It is now a straightforward computation to see that $I_{k}^{\prime}(D)$ is the ideal generated by the monomials $\prod_{i=1}^{n}x_{i}^{c_{i}}$, where $0\leq c_{i}\leq k$ for all $i$ and $\sum_{i}c_{i}=(n-1)k$. This coincides with $I_{k}(Z)$ according to [MP1, Proposition 8.2], completing the proof of the proposition. ∎

Computation in terms of a log resolution

We use the results of the previous two sections in order to describe Hodge ideals of ${\mathbf{Q}}$-divisors in terms of log resolutions. Let $X$ be a smooth variety, $h\in\mathscr{O}_{X}(X)$ a nonzero function, $H={\rm div}(h)$, and $\alpha\in{\mathbf{Q}}_{>0}$. We are interested in computing $I_{k}(D)$, where $D=\alpha H$. As always, let $Z={\rm Supp}(D)$ and $\beta=1-\alpha$.