Hodge ideals

A. Introduction

Let $X$ be a smooth complex variety of dimension $n$. To a reduced effective divisor $D$ on $X$ one associates the left $\mathscr{D}_{X}$-module of functions with poles along $D$,

i.e. the localization of $\mathscr{O}_{X}$ along $D$. In Saito’s theory [Saito-MHM], this $\mathscr{D}$-module underlies the mixed Hodge module $j_{*}{\mathbf{Q}}_{U}^{H}[n]$, where $U=X\smallsetminus D$ and $j:U\hookrightarrow X$ is the inclusion map. It therefore comes with an attached Hodge filtration $F_{k}\mathscr{O}_{X}(*D)$. Saito [Saito-B] shows that this filtration is contained in the pole order filtration, namely

The problem of how far these filtrations are from being equal is of great interest in the study of the singularities of $D$, and also in that of Deligne’s Hodge filtration on the singular cohomology $H^{\bullet}(U,{\mathbf{C}})$. The inclusion above leads to defining for each $k\geq 0$ a coherent sheaf of ideals $I_{k}(D)$ by the formula

Our main goal in this paper is to approach the definition and study of these ideal sheaves using methods from birational geometry, and to put them to use in a number of applications regarding singularities and Hodge theory. In sequels to this article we will present a framework for Hodge ideals associated to ${\mathbf{Q}}$-divisors and ideal sheaves, leading to further applications.

Given a log resolution $f\colon Y\rightarrow X$ of the pair $(X,D)$ which is an isomorphism over $X\smallsetminus D$, we define $E:=(f^{*}D)_{\rm red}$. We will see in §3 that there is a filtered complex of right $f^{-1}\mathscr{D}_{X}$-modules

which is exact except at the rightmost term, where the cohomology is $\omega_{Y}(*E)\otimes_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X}$; here $\mathscr{D}_{Y\to X}=f^{*}\mathscr{D}_{X}$ is the transfer module of $f$. Denoting it by $A^{\bullet}$, its filtration is provided by the subcomplexes $F_{k-n}A^{\bullet}$, for every $k\geq 0$, given by

We define the $k$-th Hodge ideal $I_{k}(D)$ associated to $D$ by the formula

after proving that this image is contained in \omega_{X}\big{(}(k+1)D\big{)}. We show that this definition is independent of the choice of log resolution, and that it indeed coincides with the ideals defined by Saito’s Hodge filtration.

The $0^{\rm th}$ Hodge ideal belongs to a class of ideal sheaves that is quite well understood, and has proved to be extremely useful; it is not hard to show that

the multiplier ideal associated to the ${\mathbf{Q}}$-divisor $(1-\epsilon)D$ with $0<\epsilon\ll 1$. In particular, $I_{0}(D)=\mathscr{O}_{X}$ if and only if the pair $(X,D)$ is log-canonical. Thus the sequence of ideals $I_{k}(D)$ can be seen as a refinement of this type of multiplier ideal.

Hodge ideals can be computed concretely when $D$ is a simple normal crossing divisor; see Proposition 8.2. In particular, if $D$ is smooth, then $I_{k}(D)=\mathscr{O}_{X}$ for all $k\geq 0$, which corresponds to equality between the Hodge filtration and the pole order filtration in (0.1). One of the main applications of the results below is an effective converse to this statement.

Let $X$ be a smooth complex variety of dimension $n$, and $D$ a reduced effective divisor on $X$. Then the following are equivalent:

the Hodge filtration and pole order filtration on $\mathscr{O}_{X}(*D)$ coincide.

$I_{k}(D)=\mathscr{O}_{X}$ for all $k\geq 0$.

$I_{k}(D)=\mathscr{O}_{X}$ for some $k\geq\frac{n-1}{2}$.

Saito introduced in [Saito-HF] a measure of the complexity of the Hodge filtration, and proved several results in the case of $\mathscr{O}_{X}(*D)$ (see e.g. Remark 20.11). Concretely, one says that the filtration $F_{\bullet}\mathscr{O}_{X}(*D)$ is generated at level $k$ if

If $D$ has simple normal crossings, the filtration is generated at level . It turns out that the same is true when $X$ is a surface. This is a special case of the following general result, which is a consequence of our main criterion for detecting the generation level, Theorem 17.1 below; there exist simple examples in which one cannot do better.

If $X$ has dimension $n\geq 2$, the Hodge filtration on $\mathscr{O}_{X}(*D)$ is generated at level $n-2$. More generally, for every $k\geq 0$ there exists an open subset $U_{k}$ in $X$ whose complement has codimension $\geq k+3$, such that the induced filtration on $\mathscr{O}_{X}(*D)|_{U_{k}}$ is generated at level $k$.

Going back to the study of the singularities of the pair $(X,D)$, the notion of log-canonical singularity is refined by the following:

If $D$ is a reduced effective divisor on the smooth variety $X$, we say that the pair $(X,D)$ is $k$-log-canonical if

We show in Proposition 13.1 that there is in fact a chain of inclusions

(Note that the definition gives automatically only an inclusion in the opposite direction, namely $I_{k-1}(D)\otimes\mathscr{O}_{X}(-D)\subseteq I_{k}(D)$.) Thus being $k$-log-canonical is equivalent to $I_{k}(D)=\mathscr{O}_{X}$.

Being log-canonical is of course equivalent to being -log-canonical in the above sense, while Theorem A says that $(n-1)/2$-log-canonical or higher is equivalent to $D$ being smooth. It turns out that any intermediate level of log-canonicity refines another basic notion, namely that of rational singularities. Recall that to $D$ one can also associate the adjoint ideal ${\rm adj}(D)$, see [Lazarsfeld, §9.3.E], which is a concrete measure of the failure of $D$ to have normal rational singularities.

Hence if $I_{k}(D)=\mathscr{O}_{X}$ for some $k\geq 1$, then $D$ is normal with rational singularities.

The condition of being $k$-log-canonical has Hodge-theoretic consequences for the cohomology $H^{\bullet}(U,{\mathbf{C}})$, where $U=X\smallsetminus D$. Using the definition of Hodge ideals and Lemma 7.4 below, if $X$ is smooth projective of dimension $n$ we have that

for all $i$, where $F_{\bullet}$ is the Hodge filtration and $P_{\bullet}$ is the pole order filtration on $H^{i}(U,{\mathbf{C}})$; see §7 for the definitions. One (difficult) calculation that we perform in §20 is the following; for the purpose of this paper, an ordinary singular point is a point whose projectivized tangent cone is smooth.

Let $D$ be a reduced effective divisor on a smooth variety $X$ of dimension $n$, and let $x\in D$ be an ordinary singular point of multiplicity $m\geq 2$. Then

In particular, if $X$ is projective and $D$ has only such singularities, then

When all the singularities of $D$ are nodes, the equivalence in the theorem was established already in [DSW, §1.4], where all Hodge ideals were computed concretely; see Example 20.10.The paper [DSW] also obtains a range where the equality $F_{p}H^{\bullet}(U,{\mathbf{C}})=P_{p}H^{\bullet}(U,{\mathbf{C}})$ does not hold, for a general singular, hence nodal, hypersurface in ${\mathbf{P}}^{n}$. In the case of nodal surfaces in ${\mathbf{P}}^{3}$, a more precise result can be found in [DS2, Theorem 5.1].

It turns out that the nontriviality bound in Theorem D, i.e. the fact that $I_{k}(D)_{x}\neq\mathscr{O}_{X,x}$ for $k\geq\left[\frac{n}{m}\right]$, holds for any point $x\in D$ of multiplicity $m$; see Example 21.4. This, as well as Theorem A, follows from the following statement, proved in §21 using deformation to ordinary singularities. In most cases, examples given in §20 show its optimality.

Let $D$ be a reduced effective divisor on a smooth variety $X$, and let $W$ be an irreducible closed subset of codimension $r$, defined by the ideal sheaf $I_{W}$. If $m={\rm mult}_{W}(D)$, then for every $k$ we have

Here $I_{W}^{(q)}$ is the $q^{\rm th}$ symbolic power of $I_{W}$, and $I_{W}^{(q)}=\mathscr{O}_{X}$ if $q\leq 0$.

One ingredient in the proof of this result is the analogue of the Restriction Theorem for multiplier ideals, [Lazarsfeld, Theorem 9.5.1], which holds for all Hodge ideals as well: if $H$ is a smooth hypersurface with $H\not\subseteq{\rm Supp}(D)$, such that $D|_{H}$ is reduced, then

with equality for $H$ sufficiently general. Thus there is an inversion of adjunction for $k$-log-canonicity. The proof requires tools from the theory of mixed Hodge modules, and will be given in a separate paper [MP]. We do include however a proof using the methods of this paper in the generic case, see Theorem 16.1.

On the other hand, Theorem C is a consequence of another one of our main local results, Theorem 19.1, giving a lower bound for the order of vanishing of $I_{k}(D)$ along exceptional divisors over $X$ on carefully chosen log resolutions. We leave the slightly technical statement for the text, and note that it also leads to another nontriviality criterion, Corollary 19.4, that complements Theorem E. Just as in the theory of multiplier ideals, a precise measure of the nontriviality of Hodge ideals is crucial for applications, especially when combined with vanishing theorems; this is what we focus on next.

Multiplier ideals satisfy the celebrated Nadel vanishing theorem; see [Lazarsfeld, Theorem 9.4.8]. For the ideal $I_{0}(D)$, this says that given any ample line bundle $L$, one has

We obtain an analogous result for the entire sequence of Hodge ideals $I_{k}(D)$. Things however necessarily get more complicated; in brief, in order to have full vanishing, higher log-canonicity conditions and borderline Nakano vanishing type properties need to be satisfied.

Let $X$ be a smooth projective variety of dimension $n$, $D$ a reduced effective divisor, and $L$ a line bundle on $X$. Then, for each $k\geq 1$, assuming that the pair $(X,D)$ is $(k-1)$-log-canonical we have:

If $k\leq\frac{n}{2}$, and $L$ is a line bundle such that $L(pD)$ is ample for all $0\leq p\leq k$, then

holds if H^{j}\big{(}X,\Omega_{X}^{n-j}\otimes L((k-j+1)D)\big{)}=0 for all $1\leq j\leq k$.

If $k\geq\frac{n+1}{2}$, then $D$ is smooth by Theorem A, and so $I_{k}(D)=\mathscr{O}_{X}$. In this case, if $L$ is a line bundle such that $L(kD)$ is ample, then

If $D$ is ample, then (1) and (2) also hold with $L=\mathscr{O}_{X}$.

A slightly more precise statement for $I_{1}(D)$ is given in Theorem 23.2. The proof of Theorem F relies on the Kodaira-Saito vanishing theorem in the theory of mixed Hodge modules; at the moment we know how to give a more elementary proof only for $I_{1}(D)$. We explain in Corollary 24.1 how one can avoid the Nakano-type requirement in Theorem F by assuming that $D$ is sufficiently positive with respect to an ample divisor $A$ such that $T_{X}(A)$ is nef. It is worth noting that Hodge ideals also satisfy a local vanishing statement, Corollary 12.1, due to the strictness of the Hodge filtration.

Vanishing for Hodge ideals takes a particularly simple form on ${\mathbf{P}}^{n}$ (see Corollary 25.3) or more generally on smooth toric varieties (see Corollary 25.1), and on abelian varieties (see Theorem 28.2), as in these cases the hypotheses are automatically satisfied or can be relaxed. As mentioned above, in combination with Theorem E and related results, this leads to interesting applications. We present a few here, while further applications, as well as theoretical statements, will be treated elsewhere.

For instance, on ${\mathbf{P}}^{n}$ it is a consequence of Nadel vanishing that if an integral hypersurface $D$ of degree $d$ is not log-canonical, then its singular locus has dimension $\geq n-d+1$. Our vanishing theorem leads to a simultaneous extension of this fact and of a result of Deligne on the Hodge filtration on complements of hypersurfaces with isolated singularities. When the Hodge ideals are nontrivial, it imposes further restrictions on the corresponding subschemes in ${\mathbf{P}}^{n}$.

Let $D$ be a reduced hypersurface of degree $d$ in ${\mathbf{P}}^{n}$, and for each $k$ denote by $Z_{k}$ the subscheme associated to the ideal $I_{k}(D)$, and by $z_{k}$ its dimension. Then:

If $z_{k}<n-(k+1)d+1$, then in fact $Z_{k}=\emptyset$, i.e. $(X,D)$ is $k$-log-canonical. The converse is of course true if $n-(k+1)d+1\geq 0$.

with the convention that $\dim\emptyset$ and $\deg\emptyset$ are $-1$.

The dimension part of $Z_{k}$ imposes independent conditions on hypersurfaces of degree at least $(k+1)d-n-1$.

Part (1) says in particular that if $D$ has isolated singularities, then $I_{k}(D)=\mathscr{O}_{X}$ whenever $n-(k+1)d+1>0$; this is a result of Deligne, see [Saito-B, 4.6(iii)], that was originally phrased in terms of the equality $F_{k}H^{i}(U,{\mathbf{C}})=P_{k}H^{i}(U,{\mathbf{C}})$, where $U={\mathbf{P}}^{n}\smallsetminus D$. More generally, according to the theorem and (0.2), whenever $\dim Z_{k}+(k+1)d<n+1$ we have that

A related local result was proved by Saito [Saito-B, Theorem 0.11] in terms of the roots of the Bernstein-Sato polynomial of $D$; see Remark 26.1.

When combined with nontriviality criteria like Theorem E or Theorem D, part (3) in Theorem G has a number of basic consequences describing the behavior of isolated singular points on hypersurfaces in ${\mathbf{P}}^{n}$, with $n\geq 3$. These are collected in §27; here is the most easily stated:

Let $D$ be a reduced hypersurface of degree $d$ in ${\mathbf{P}}^{n}$, with $n\geq 3$, and denote by $S_{m}$ the set of isolated singular points on $D$ of multiplicity $m\geq 2$. Then $S_{m}$ imposes independent conditions on hypersurfaces of degree at least $([\frac{n}{m}]+1)d-n-1$.

To put this in perspective, recall that a classical theorem of Severi [Severi] states that if a surface $S\subset{\mathbf{P}}^{3}$ of degree $d$ has only nodes as singularities, then the set of nodes imposes independent conditions on hypersurfaces of degree $2d-5$. The bound above is one worse in this case, but it becomes better than what is known for most other $n$ and $m$. For further discussion see §27. Results of a similar flavor hold on abelian varieties; see §30.

On principally polarized abelian varieties (ppav’s) we obtain an upper bound on the multiplicity of points on theta divisors whose singularities are isolated.

Let $(X,\Theta)$ be ppav of dimension $g$ such that $\Theta$ has isolated singularities. Then:

For every $x\in\Theta$ we have ${\rm mult}_{x}(\Theta)\leq\frac{g+1}{2}$, and also ${\rm mult}_{x}(\Theta)\leq\epsilon(\Theta)+2$, where $\epsilon(\Theta)$ is the Seshadri constant of the principal polarization.

Moreover, there is at most one point $x\in\Theta$ with ${\rm mult}_{x}(\Theta)=\frac{g+1}{2}$.

See §29 for a detailed discussion, including the conjectural context in which this result is placed, and for further numerical bounds. Suffice it to say here that, in the case of isolated singularities, this improves a well-known bound of Kollár saying that the multiplicity of each point is at most $g$. See also Remark 29.6 (1) for related work of Codogni-Grushevsky-Sernesi and communications from Lazarsfeld.

We conclude by noting that some of the statements in the text rely on local vanishing theorems of Akizuki-Nakano type for higher direct images of sheaves of differentials with log poles; some of these can already be found in [EV] and [Saito-LOG], while some are new and hopefully be of interest beyond the applications in this paper. We provide a uniform approach in the Appendix, using rather elementary arguments.

Finally, a word about the use of results from the theory of mixed Hodge modules; in the treatment given here many of the definitions, including that of Hodge ideals, as well as the proofs of most theorems, do not a priori depend of it. However, this is not always the case. For instance, the proof of Theorem F on vanishing, and that of Proposition 13.1, rely essentially on results regarding Hodge modules. One topic that does not appear in this paper though, is the connection with the theory of the $V$-filtration. This is used in [MP] in order to prove Theorem 16.9 and further results. It is treated systematically in the more recent preprint of Saito [Saito-MLCT], where in particular it leads to a different approach to many of the results in Theorems A, C and D; see also Remark 20.11. Both points of view will continue to play an important role in the sequels mentioned above.

Acknowledgements. This paper owes a great deal of inspiration to Morihiko Saito’s work on the Hodge filtration on localizations along hypersufaces [Saito-B], [Saito-LOG], [Saito-HF]. We are grateful to Christian Schnell for making us aware of these papers, which is one of the reasons our project got started. He also asked whether the equivalence between (1) and (2) in Theorem A might hold. We thank him, Rob Lazarsfeld, and Morihiko Saito for many other useful comments, Lawrence Ein for discussions regarding the results in the Appendix, and Hélène Esnault, Sam Grushevsky, Sam Payne, and Claire Voisin for comments and references. The second author would like to thank the math departments at the University of Michigan and Stony Brook University for hospitality during the preparation of the paper, and the Simons Foundation for fellowship support.

B. Preliminaries

Let $X$ be a smooth complex algebraic variety. We denote by $\mathscr{D}_{X}$ the sheaf of differential operators on $X$. A left or right $\mathscr{D}$-module on $X$ is simply a left, respectively right, $\mathscr{D}_{X}$-module.

We briefly recall some standard definitions from the theory of $\mathscr{D}$-modules that will be used in this paper. A very good source for further details is [HTT].

Left-right correspondence. We will work with both left and right $\mathscr{D}$-modules; while most definitions and results are best stated for left $\mathscr{D}$-modules, push-forwards are most natural in the context of right $\mathscr{D}$-modules. The standard one-to-one correspondence between left and right $\mathscr{D}_{X}$-modules is given by

where $\omega_{X}$ is endowed with its natural right $\mathscr{D}_{X}$-module structure; see [HTT, §1.2].

Filtrations and the de Rham complex. A filtered left $\mathscr{D}$-module on $X$ is a left $\mathscr{D}_{X}$-module $\mathcal{M}$ with an increasing filtration $F=F_{\bullet}\mathcal{M}$ by coherent $\mathscr{O}_{X}$-modules, bounded from below and satisfying

is finitely generated over $\operatorname{gr}_{\bullet}^{F}\mathscr{D}_{X}\simeq{\rm Sym}~{}T_{X}$. With the analogous definitions for a right $\mathscr{D}_{X}$-module $\mathcal{N}$, in case it corresponds to $\mathcal{M}$ via the left-right operation, the corresponding rule on filtrations is

Given a left $\mathscr{D}_{X}$-module $\mathcal{M}$, the associated de Rham complex is:

It is a ${\mathbf{C}}$-linear complex with differentials induced by the corresponding integrable connection $\nabla\colon\mathcal{M}\rightarrow\Omega_{X}^{1}\otimes\mathcal{M}$. We consider it to be placed in degrees $-n,\ldots,0$. (Strictly speaking, as such it is the de Rham complex associated to the corresponding right $\mathscr{D}$-module.) A filtration $F_{\bullet}\mathcal{M}$ on $\mathcal{M}$ induces a filtration on the de Rham complex of $\mathcal{M}$ by the formula

For any integer $k$, the associated graded complex for this filtration is

This is now a complex of coherent $\mathscr{O}_{X}$-modules in degrees $-n,\dotsc,0$, providing an object in ${\bf D}^{b}(X)$, the bounded derived category of coherent sheaves on $X$.

Transfer modules and pushforward. If $f\colon Y\rightarrow X$ is a morphism of smooth complex varieties, we consider the associated transfer module

It has the structure of a $\mathscr{D}_{Y}-f^{-1}\mathscr{D}_{X}$-bimodule; moreover, it is filtered by $f^{*}F_{k}\mathscr{D}_{X}$. It is simply $f^{*}\mathscr{D}_{X}$ as an $\mathscr{O}_{Y}$-module, and we will use this notation rather than $\mathscr{D}_{Y\to X}$ when thinking of it as such. For a right $\mathscr{D}_{Y}$-module $\mathcal{M}$, due to the left exactness of $f_{*}$ versus the right exactness of $\otimes$, the appropriate pushforward for $\mathscr{D}$-modules is at the level of derived categories, namely

See [HTT, §1.5] for more details, where this last functor is denoted by $\int_{f}$.

Localization along a divisor

In this section, $X$ will always be a smooth complex variety of dimension $n$, and $D$ a reduced effective divisor on $X$. We denote the localization of $\mathscr{O}_{X}$ along $D$, as a left $\mathscr{D}_{X}$-module, by $\mathscr{O}_{X}(*D)$. If $h$ is a local equation of $D$, then this is $\mathscr{O}_{X}[\frac{1}{h}]$, with the obvious action of differential operators. The associated right $\mathscr{D}_{X}$-module is denoted $\omega_{X}(*D)$.

We will use the following well-known observation; see [HTT, Lemma 5.2.7].

If $\mathscr{F}$ is a coherent $\mathscr{O}_{X}(*D)$-module supported on $D$, then $\mathscr{F}=0$.

which is an isomorphism over $U:=X\smallsetminus D$. We assume that $Y$ is smooth, and let $E=(f^{*}D)_{\rm red}$ and $V=Y\smallsetminus E=f^{-1}(U)$. We denote by $j_{U}$ and $j_{V}$ the inclusions of $U$ and $V$ into $X$ and $Y$ respectively. Note that ${j_{U}}_{+}\omega_{U}\simeq\omega_{X}(*D)$ and ${j_{V}}_{+}\omega_{V}\simeq\omega_{Y}(*E)$. Since $j_{U}=f\circ j_{V}$, we have:

Given the morphism $f\colon Y\rightarrow X$, since $\mathscr{D}_{Y\to X}$ is a left $\mathscr{D}_{Y}$-module, we have a canonical morphism of left $\mathscr{D}_{Y}$-modules

that maps $1$ to $1$. This is in fact a morphism of $\mathscr{D}_{Y}-f^{-1}\mathscr{O}_{X}$ bimodules. It is clearly an isomorphism over $Y\smallsetminus E$. Since $\mathscr{D}_{Y}$ is torsion-free, we conclude that $\varphi$ is injective, with cokernel supported on $E$. For each $k$, we have induced inclusions $F_{k}\mathscr{D}_{Y}\hookrightarrow f^{*}F_{k}\mathscr{D}_{X}$ of $\mathscr{O}_{Y}-f^{-1}\mathscr{O}_{X}$ bimodules.

The canonical map of right $f^{-1}\mathscr{O}_{X}$-modules

induced by $\varphi$ is a split injection.

Denoting for simplicity $j=j_{V}$, we have a commutative diagram

where $\varphi$ and $\psi$ are the canonical maps. Since $\mathscr{D}_{Y\to X}|_{V}=\mathscr{D}_{V}$, it is clear that $\beta=j_{*}j^{*}(\alpha)$ is an isomorphism. On the other hand, as $\mathscr{O}_{Y}$-modules we have $\omega_{Y}(*E)\simeq j_{*}\omega_{V}$, hence $\varphi$ is an isomorphism. It follows from the above diagram that the composition $\psi\circ\alpha$ is an isomorphism, hence $\alpha$ is a split injection. ∎

The main result we are aiming for here is the following enhancement:

The canonical morphism induced by $\varphi$ in the derived category of right $f^{-1}\mathscr{O}_{X}$-modules

The proof we give below is inspired in part by arguments in [HTT, §5.2].

It suffices to show that the induced mappings

are all isomorphisms for $k\geq 0$. But this follows immediately from Lemma 2.1 (note that since $\mathscr{O}_{Y}(*E)$ is flat over $\mathscr{O}_{Y}$, these maps are injective). ∎

This is a sheaf of rings, and one can identify it with the subalgebra of {\rm End}_{{\mathbf{C}}}\big{(}\mathscr{O}_{Y}(*E)\big{)} generated by $\mathscr{D}_{Y}$ and $\mathscr{O}_{Y}(*E)$. Note that since $\mathscr{O}_{Y}(*E)$ is a flat $\mathscr{O}_{Y}$-module, we have $\mathscr{D}_{Y}(*E)\simeq\mathscr{O}_{Y}(*E)\overset{\mathbf{L}}{\otimes}_{\mathscr{O}_{Y}}\mathscr{D}_{Y}$. A basic fact is the following:

Via the isomorphism $\mathscr{D}_{Y}(*E)\simeq\mathscr{O}_{Y}(*E)\overset{\mathbf{L}}{\otimes}_{\mathscr{O}_{Y}}\mathscr{D}_{Y}$, the morphism in the statement gets identified to the morphism

induced by $\varphi$. Moreover, since $\mathscr{O}_{Y}(*E)$ is a flat $\mathscr{O}_{Y}$-module, the morphism (2.7) gets identified with the isomorphism in Lemma 2.5. ∎

Via the right $\mathscr{D}$-module structure on $\omega_{Y}$, we have that $\omega_{Y}(*E)$ has a natural right $\mathscr{D}_{Y}(*E)$-module structure. The morphism in the proposition gets identified with the morphism

induced by $\varphi$. In turn, this is obtained by applying $\omega_{Y}(*E)\overset{\mathbf{L}}{\otimes}_{\mathscr{D}_{Y}(*E)}-$ to the isomorphism in Lemma 2.6, hence it is an isomorphism. ∎

Filtrations on localizations and tensor products

We next include filtrations in the discussion. We fix again a smooth variety $X$ of dimension $n$, and a reduced effective divisor $D$ on $X$. Most obviously, on $\mathscr{O}_{X}(*D)$ there is a pole order filtration, whose nonzero terms are

Less obvious is the Hodge filtration $F_{k}\mathscr{O}_{X}(*D)$, again with nonzero terms for $k\geq 0$. This is our main topic of study in this paper. Its existence is guaranteed by general results on Hodge modules (see §4). However, we will take a hands-on approach and describe it explicitly now in the simple normal crossings case, and later in the general case via log resolutions.

For simple normal crossing divisors we fix different notation, in view of later use on log resolutions (note also the shift from left to right $\mathscr{D}$-modules). Let $E$ be a reduced simple normal crossing (SNC) divisor on a smooth $n$-dimensional variety $Y$. We define the Hodge filtration on the right $\mathscr{D}_{Y}$-module $\omega_{Y}(*E)$ to be given by

For instance, the first two nonzero terms are

where ${\rm Jac}(E)$ is the Jacobian ideal of $E$, i.e. $F_{1}\mathscr{D}_{Y}\cdot\mathscr{O}_{Y}(-E)$. See also Proposition 8.2 for a general local description.

Recall that the right $\mathscr{D}$-module $\omega_{Y}$ has a standard resolution

by induced $\mathscr{D}_{Y}$-modules; see [HTT, Lemma 1.2.57]. The following generalization will be a crucial technical point later on; cf. also [Saito-MHM, Proposition 3.11(ii)], where this is part of a more general picture.

The right $\mathscr{D}_{Y}$-module $\omega_{Y}(*E)$ has a filtered resolution with induced $\mathscr{D}_{Y}$-modules given by

is given by $\frac{\omega}{f}\otimes P\to\frac{\omega}{f}\cdot P$, and for each $p$ the morphism

is given by $\omega\otimes P\to d\omega\otimes P+\sum_{i=1}^{n}(dz_{i}\wedge\omega)\otimes\partial_{i}P$, in local coordinates $z_{1},\ldots,z_{n}$.

It is not hard to check that the expression in the statement is indeed a complex, which we call $A^{\bullet}$. We consider on $\Omega_{Y}^{p}(\log E)$ the filtration

and on $\Omega_{Y}^{p}(\log E)\otimes_{\mathscr{O}_{Y}}\mathscr{D}_{Y}$ the tensor product filtration. This filters $A^{\bullet}$ by subcomplexes $F_{k-n}A^{\bullet}$ given by

for each $k\geq 0$. Note that they can be rewritten as

where $T_{Y}(-\log E)$ is the dual of $\Omega_{Y}^{1}(\log E)$, and we use the isomorphisms $\omega_{Y}(E)\otimes\wedge^{i}T_{Y}(-\log E)\simeq\Omega_{Y}^{n-i}(\log E)$.

It is clear directly from the definition that every such complex is exact at the term $F_{k}\omega_{Y}(*E)$. We now check that they are exact at the term $\omega_{Y}(E)\otimes_{\mathscr{O}_{Y}}F_{k}\mathscr{D}_{Y}$. Let us assume that, in the local coordinates $z_{1},\ldots,z_{n}$, the divisor $E$ is given by $z_{1}\cdots z_{r}=0$. Using the notation $\omega=dz_{1}\wedge\cdots\wedge dz_{n}$, we consider an element

mapping to in $F_{k}\omega_{Y}(*E)=\omega_{Y}(E)\cdot F_{k}\mathscr{D}_{Y}$. This means that

We show that $u$ is in the image of the morphism $\beta_{k}$ by using a descending induction on $|\alpha|$. What we need to prove is the following claim: for each $\alpha$ in the sum above, with $|\alpha|=k$, there exists some $i$ with $\alpha_{i}>0$ such that $z_{i}$ divides $g_{\alpha}$. If so, an easy calculation shows that the term $u_{\alpha}=\frac{\omega}{z_{1}\cdots z_{r}}\otimes g_{\alpha}\partial^{\alpha}$ is in the image of $\beta_{k}$, and hence it is enough to prove the statement for $u-u_{\alpha}$. Repeating this a finite number of times, we can reduce to the case when all $|\alpha|\leq k-1$. But the claim is clear: if $z_{i}$ did not divide $g_{\alpha}$ for all $i$ with $\alpha_{i}>0$, then the Laurent monomial $z_{1}^{-\alpha_{1}}\cdots z_{r}^{-\alpha_{r}}$ would appear in the term $g_{\alpha}\cdot z_{1}^{-\alpha_{1}}\cdots z_{r}^{-\alpha_{r}}$ of the sum above, but in none of the other terms.

To check the rest of the statement, note that after discarding the term on the right, the associated graded complexes

are acyclic. Indeed, each such complex is, up to a twist, an Eagon-Northcott complex associated to the inclusion of vector bundles of the same rank

Concretely, in the notation on [Lazarsfeld, p.323], the complex above is $(EN_{k})$ tensored by $\omega_{Y}(E)$. According to [Lazarsfeld, Theorem B.2.2(iii)], $(EN_{k})$ is acyclic provided that

are the deneracy loci of $\varphi$. But locally $\varphi$ is given by the diagonal matrix

so this condition is verified by a simple calculation. ∎

Let now $X$ and $D$ be as at the beginning of the section, and let $f\colon Y\to X$ be a log resolution of the pair $(X,D)$ which is an isomorphism over $X\smallsetminus D$. Under the latter assumption, the log resolution condition simply means that $f$ is a projective morphism, $Y$ is smooth, and $E:=(f^{*}D)_{\rm red}$ has simple normal crossings. On the tensor product $\omega_{Y}(*E)\otimes_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X}$ we consider the tensor product filtration, that is,

where the map in the parenthesis is the natural map between the tensor product over $\mathscr{O}_{Y}$ and that over $\mathscr{D}_{Y}$.

Fix $i\geq-n$ and recall that $F_{i}\omega_{Y}(*E)=\omega_{Y}(E)\cdot F_{i+n}\mathscr{D}_{Y}$. The factor $F_{i+n}\mathscr{D}_{Y}$ can be moved over the tensor product once we pass to the image in the tensor product over $\mathscr{D}_{Y}$, and moreover we have an inclusion

Therefore inside $\omega_{Y}(*E)\otimes_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X}$, the image of $F_{i}\omega_{Y}(*E)\otimes_{\mathscr{O}_{Y}}f^{*}F_{k-i}\mathscr{D}_{X}$ is contained in the image of $\omega_{Y}(E)\otimes_{\mathscr{O}_{Y}}f^{*}F_{k+n}\mathscr{D}_{X}$. ∎

Propositions 3.1 and 2.4 have the following immediate consequence:

On $Y$ there is a filtered complex of right $f^{-1}\mathscr{D}_{X}$-modules

which is exact (though not necessarily filtered exact).

It follows from Proposition 3.1 that the complex

represents the object $\omega_{Y}(*E)\overset{\mathbf{L}}{\otimes}_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X}$ in the derived category, hence Proposition 2.4 implies the exactness of the entire complex in the statement. ∎

We record here the following lemma for later use.

in which $\gamma$ is the canonical inclusion. Since $\beta$ is injective by Proposition 3.1, it follows that $\alpha$ is injective, too. ∎

C. Saito’s Hodge filtration and Hodge modules

This section, unlike the previous one, only contains review material. The reader familiar with the topics it covers can skip to Section D, and use it as a reference.

The study of Hodge ideals relies in part on the fact that the filtered $\mathscr{D}_{X}$-module $\omega_{X}(*D)$ underlies a mixed Hodge module. For simplicity, we will call such objects Hodge $\mathscr{D}$-modules. It is not the place here to give a detailed account of the theory of Hodge modules; for details we refer to the original [Saito-MHP], [Saito-MHM], for summaries of the results needed here to the surveys [Saito-YPG] and [Schnell-MHM], and for a review of how they have been recently used for geometric applications to [Popa2]. We will however review properties of Hodge $\mathscr{D}$-modules that make them special among all filtered $\mathscr{D}$-modules, and that will be used here, as well as some results specific to $\omega_{X}(*D)$.

If we denote by ${\rm FM}(\mathscr{D}_{X})$ the category of filtered $\mathscr{D}_{X}$-modules, one can construct an associated bounded derived category {\bf D}^{b}\big{(}{\rm FM}(\mathscr{D}_{X})\big{)}. Assuming that we are given a projective morphism of smooth varieties $f\colon Y\rightarrow X$, and that we are working with right $\mathscr{D}$-modules, Saito constructs in [Saito-MHP, §2.3] a filtered direct image functor

compatible with the usual direct image functor for right $\mathscr{D}$-modules.

A fundamental result about Hodge $\mathscr{D}$-modules is Saito’s Stability Theorem for direct images under projective morphisms, [Saito-MHP, Théorème 5.3.1]. This says that, in the above setting, if $(\mathcal{M},F)$ is a Hodge $\mathscr{D}_{Y}$-module, then $f_{+}(\mathcal{M},F)$ is strict as an object in {\bf D}^{b}\big{(}{\rm FM}(\mathscr{D}_{X})\big{)} (and moreover, each $H^{i}f_{+}(\mathcal{M},F)$ is a Hodge $\mathscr{D}_{X}$-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 $H^{i}f_{+}(\mathcal{M},F)$.

Strictness in this context can be seen as a generalization of the degeneration at $E_{1}$ of the classical Hodge-to-de Rham spectral sequence. Concretely, let $X$ be a smooth projective variety, and $(\mathcal{M},F)$ a Hodge $\mathscr{D}$-module on $X$. The natural inclusion of complexes $F_{k}\operatorname{DR}(\mathcal{M})\hookrightarrow\operatorname{DR}(\mathcal{M})$ induces, after passing to cohomology, a morphism

Now for the constant map $f\colon X\rightarrow{\rm pt}$, the definition of pushforward gives

and by the discussion above, the image of $\varphi_{k,i}$ is F_{k}H^{i}\big{(}X,\operatorname{DR}(\mathcal{M})\big{)}. Saito’s result on the strictness of $f_{+}(\mathcal{M},F)$ then implies that $\varphi_{k,i}$ is injective for all $k$ and $i$, which is in turn equivalent to

This is the same as the $E_{1}$-degeneration of the Hodge-to-de Rham spectral sequence

Vanishing theorem

Recall from §1 that given a $\mathscr{D}_{X}$-module with good filtration $(\mathcal{M},F)$ on a smooth variety $X$, for any integer $k$ the associated graded complex for the induced filtration on the de Rham complex is

seen as a complex of coherent $\mathscr{O}_{X}$-modules in degrees $-n,\dotsc,0$. When $X$ is projective and $(\mathcal{M},F)$ underlies a mixed Hodge module, these complexes satisfy the following Kodaira-type vanishing theorem [Saito-MHM, §2.g] (see also [Popa] and [Schnell]). A similar result can be formulated more generally, on singular projective varieties, but we will not make use of this here.

Let $(\mathcal{M},F)$ be a Hodge $\mathscr{D}$-module on a smooth projective variety $X$, and let $L$ be any ample line bundle. Then:

\mathbf{H}^{i}\bigl{(}X,\operatorname{gr}_{k}^{F}\operatorname{DR}(\mathcal{M})\otimes L\bigr{)}=0 for all $i>0$.

\mathbf{H}^{i}\bigl{(}X,\operatorname{gr}_{k}^{F}\operatorname{DR}(\mathcal{M})\otimes L^{-1}\bigr{)}=0 for all $i<0$.

Localization as a Hodge 𝒟𝒟\mathscr{D}-module

If $X$ is a smooth variety, and $D$ is a reduced effective divisor on $X$, then the right $\mathscr{D}_{X}$-module $\omega_{X}(*D)$ is a Hodge $\mathscr{D}$-module. Indeed, it underlies the mixed Hodge module $j_{*}{\mathbf{Q}}_{U}^{H}[n]$, where $j\colon U=X\smallsetminus D\hookrightarrow X$ is the inclusion, and ${\mathbf{Q}}_{U}^{H}[n]$ is the trivial Hodge module on $U$; see e.g. [Schnell, Example 5.4].

Let $f\colon Y\rightarrow X$ be a log resolution of the pair $(X,D)$ which is an isomorphism over $X\smallsetminus D$, and let $E=(f^{*}D)_{\rm red}$. If $V=Y\smallsetminus E$, the proof of Lemma 2.2 shows more generally that we have an isomorphism of mixed Hodge modules

in {\bf D}^{b}\big{(}{\rm FM}(\mathscr{D}_{X})\big{)}. According to §4, the filtration on the right hand side is given by

and the mapping in the parenthesis is in fact injective.

We note also that, although not used in the sequel, the following result of Saito is suggestive for some of the constructions below.

in the derived category of filtered complexes of ${\mathbf{C}}$-vector spaces (and more generally of filtered differential complexes), where the filtration on the log-de Rham complex on the left is the “stupid” filtration.

The log-de Rham complex on the left is of course also filtered quasi-isomorphic to \operatorname{DR}\big{(}\mathscr{O}_{Y}(*E),F\big{)}, by a well-known result of Deligne [Deligne].

In particular, Saito deduces from Theorem 6.1 that the object $\mathbf{R}f_{*}\Omega_{Y}^{\bullet}(\log E)$ is independent of the log resolution, and that $R^{q}f_{*}\Omega_{Y}^{p}(\log E)$ is for $p+q>n$. A different proof, together with other results on forms with log-poles, is given in the Appendix.

Hodge filtration on the complement

In this section we assume that $X$ is a smooth projective variety. In Hodge theory it is of interest to understand Deligne’s Hodge filtration on $H^{\bullet}(U,{\mathbf{C}})$; see for instance [DD] and [Saito-B]. Saito showed that there is a close relationship between this Hodge filtration and the ideals $I_{k}(D)$, or equivalently $F_{k}\mathscr{O}_{X}(*D)$.

For every integer $i$ there is a natural morphism

which is a filtered isomorphism. Here the right hand side is endowed with Deligne’s Hodge filtration, and the left hand side with Saito’s filtration given by the image of H^{i}\big{(}X,F_{\bullet}\operatorname{DR}(\mathscr{O}_{X}(*D))\big{)}.

Now according to Saito’s theory, the left-hand side in the isomorphism above is the cohomology

of the filtered direct image, where $a_{X}\colon X\rightarrow{\rm pt}$. This filtration is strict, and we saw in Example 4.2 that this is equivalent to the degeneration at $E_{1}$ of the Hodge-to-de Rham spectral sequence

For every integer $k$ there is a decomposition

The spaces $H^{\bullet}(U,{\mathbf{C}})$ also have a pole order filtration. Indeed, using the pole order filtration on $\mathscr{O}_{X}(*D)$, i.e.

we obtain a filtration on the de Rham complex

For each $i\in{\mathbf{Z}}$, we define the pole order filtration on $H^{i+n}(U,{\mathbf{C}})$ by

where the image is considered via the isomorphism $\varphi_{i}$. This is defined in a slightly different, but equivalent fashion by Deligne-Dimca [DD], and the main result of that paper is the inclusion

for all $j$ and $k$. Using Lemma 7.1, this also follows from the stronger statement $F_{k}\mathscr{O}_{X}(*D)\subseteq P_{k}\mathscr{O}_{X}(*D)$, which is proved in [Saito-B, Proposition 0.9]; for a different proof see also Lemma 9.2 below.

The lowest $k$ for which $F_{k}H^{\bullet}(U,{\mathbf{C}})$ is not automatically zero is $k=-n$, and similarly for $P_{k}$. Note that

On the other hand, we will see later that

where I_{0}(D)=\mathcal{I}\big{(}X,(1-\epsilon)D\big{)}, the multiplier ideal of the ${\mathbf{Q}}$-divisor $(1-\epsilon)D$, with $0<\epsilon\ll 1$. By Corollary 7.2 this is a direct summand of $H^{i+n}(U,{\mathbf{C}})$; it may be different from $P_{-n}H^{i+n}(U,{\mathbf{C}})$ if the pair $(X,D)$ is not log-canonical.

In any case, it is clear from the descriptions above that any statement relating the two filtrations on $\mathscr{O}_{X}(*D)$ automatically leads to a similar statement for those on $H^{\bullet}(U,{\mathbf{C}})$. For instance:

D. Birational definition of Hodge ideals

Throughout this section $X$ is a smooth complex variety of dimension $n$ and $D$ is a reduced effective divisor on $X$. We define the Hodge ideals $I_{k}(D)$ associated to $D$, for $k\geq 0$, first explicitly in the simple normal crossing case, and then in general in terms of log resolutions. We show directly that they are independent of the choice of log resolution, and then note that they coincide with the ideals defined by Saito’s Hodge filtration. We establish a few first properties of these ideals.

When $D$ is a simple normal crossing divisor, we define the ideals $I_{k}(D)$ by the following expression:

where the left-hand side is considered via the natural injective image in $\mathscr{O}_{X}(*D)$. Note that this includes the statement $I_{0}(D)=\mathscr{O}_{X}$, and that a simple local calculation shows that F_{k}\mathscr{D}_{X}\cdot\mathscr{O}_{X}(D)\subseteq\mathscr{O}_{X}\big{(}(k+1)D\big{)}. It is clear from definition that if we use the filtration on $\omega_{X}(*D)$ introduced in §3, then

Suppose that around a point $p\in X$ we have coordinates $x_{1},\ldots,x_{n}$ such that $D$ is defined by $(x_{1}\cdots x_{r}=0)$. Then, for every $k\geq 0$, the ideal $I_{k}(D)$ is generated around $p$ by

In particular, if $r=1$ (that is, when $D$ is smooth), we have $I_{k}(D)=\mathscr{O}_{X}$ and if $r=2$, then $I_{k}(D)=(x_{1},x_{2})^{k}$.

It is clear that $F_{k}\mathscr{D}_{X}\cdot\mathscr{O}_{X}(D)$ is generated as an $\mathscr{O}_{X}$-module by

According to (8.1), the expression for $I_{k}(D)$ now follows by multiplying these generators by $(x_{1}\cdots x_{r})^{k+1}$. The assertions in the special cases $r=1$ and $r=2$ are clear. ∎

The general case

When $D$ is arbitrary, we consider a log resolution $f\colon Y\to X$ of the pair $(X,D)$ which is an isomorphism over $X\smallsetminus D$, and let $E=(f^{*}D)_{\rm red}$. Note that by assumption $E$ has simple normal crossings. Because we need to deal with pushforwards, we will work in the setting of right $\mathscr{D}$-modules.

placed in degrees $-n,\ldots,0$. We have seen in §3 that it comes with a filtration, and that as such it represents the object $\omega_{Y}(*E)\overset{\mathbf{L}}{\otimes}_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X}$ in the derived category of filtered right $f^{-1}\mathscr{D}_{X}$-modules. In particular, we have

Moreover, by Corollary 3.3, we know that if we ignore the filtration, $A^{\bullet}$ is exact everywhere except at the last term on the right, i.e. the natural mapping

is a quasi-isomorphism. For every $k\geq 0$, we also consider the subcomplex $C_{k-n}^{\bullet}=F_{k-n}A^{\bullet}$ of $A^{\bullet}$, given by

For every $k\geq 0$, the inclusion $C^{\bullet}_{k-n}\hookrightarrow A^{\bullet}$ induces a canonical morphism of (quasi-coherent) $\mathscr{O}_{X}$-modules

Let $F_{k-n}\omega_{X}(*D)\subseteq\omega_{X}(*D)$ be the image of this map. Since $C_{k-n}^{\bullet}$ is a complex of quasi-coherent $f^{-1}\mathscr{O}_{X}$-modules, it follows that $F_{k-n}\omega_{X}(*D)$ is a quasi-coherent $\mathscr{O}_{X}$-module.

For every $k\geq 0$, we have an inclusion

Since $D$ is reduced, we can find an open subset $U\subseteq X$ with the property that ${\rm codim}(X\smallsetminus U,U)\geq 2$, the induced morphism $f^{-1}(U)\to U$ is an isomorphism, and $D|_{U}$ is a smooth (possibly disconnected) divisor. Let $j\colon U\hookrightarrow X$ be the inclusion. By assumption, on $f^{-1}(U)$ we have $\mathscr{D}_{Y\to X}=\mathscr{D}_{Y}$. From Proposition 3.1, on $U$ we obtain

Now $F_{k-n}\omega_{X}(*D)$ is torsion-free, being a subsheaf of $\omega_{X}(*D)$, and so the following canonical map is injective:

The inclusion F_{k-n}\omega_{X}(*D)\subseteq\omega_{X}\big{(}(k+1)D\big{)} given by Lemma 9.2 is equivalent to the inclusion of the Hodge filtration in the pole order filtration, i.e.

proved in [Saito-B, Proposition 0.9] using the $V$-filtration; see §12.

We can now introduce the main objects we are concerned with in this paper.

Given the inclusion in Lemma 9.2, for each $k\geq 0$ we define the ideal sheaf $I_{k}(D)$ on $X$ by the formula

We call $I_{k}(D)$ the $k$-th Hodge ideal of $D$. We will show in Theorem 11.1 that the definition is independent of the choice of log resolution.

We end this section by mentioning another sequence of ideals that can be defined in this context. We discuss them only briefly, since they will not play an important role in what follows; it is however a somewhat more intuitive definition that helps with a first approximation understanding of the Hodge ideals. For every $k\geq 0$, define

The morphism $C^{\bullet}_{k-n}\to A^{\bullet}$ induces a morphism

whose image we denote by $\overline{\mathscr{F}}_{k-n}$. An argument similar to that in Lemma 9.2 shows that

hence there is a coherent ideal $I_{k}^{f}(D)$ of $\mathscr{O}_{X}$ such that

for every $k$. Indeed, we have a commutative diagram

in which $\varphi$ is a quasi-isomorphism. This induces a commutative diagram

hence via the isomorphism $\delta$ we have

We do not know whether $I_{k}^{f}(D)$ is independent of resolution. If $k=0$ or $k=1$, then $I_{k}(D)=I_{k}^{f}(D)$. This is a consequence of the fact that the canonical morphism $C^{\bullet}_{k-n}\to H^{0}C_{k-n}^{\bullet}$ is a quasi-isomorphism in these cases (this is trivial for $k=0$ and it is a consequence of Lemma 3.4 for $k=1$). At the moment we do not know however whether this equality also holds for higher $k$; this is an intriguing question.

The case k=0𝑘0k=0.

Before engaging in a detailed study, let’s note that the first ideal in the sequence can be identified with a multiplier ideal; for the general theory of multiplier ideals see [Lazarsfeld, Ch.9].

the multiplier ideal associated to the ${\mathbf{Q}}$-divisor $(1-\epsilon)D$ on $X$, for any $0<\epsilon\ll 1$.

Recall that $C^{\bullet}_{-n}=\omega_{Y}(E)$, hence

Therefore the statement to be proved is that

On the other hand, the right-hand side is by definition

An equivalent result for $F_{0}\mathscr{O}_{X}(*D)$ can be found in Saito [Saito-HF], stated and proved using the theory of the $V$-filtration.

The following is a direct consequence of the definition; see [Lazarsfeld, 9.3.9].

We have $I_{0}(D)=\mathscr{O}_{X}$ if and only if the pair $(X,D)$ is log-canonical.

Independence of resolution, and filtration property

We will remark in the next section that the ideals $I_{k}(D)$ are the same as those defined by the Hodge filtration on $\omega_{X}(*D)$; thus their independence of the choice of log resolution can be deduced from Saito’s results on the uniqueness of open direct images [Saito-MHM, Proposition 2.11]. We also include below an elementary proof that does not appeal to the theory of mixed Hodge modules.

The ideals $I_{k}(D)$ are independent of the choice of log resolution.

Since every two log resolutions can be dominated by a third one, it is enough to consider two morphisms $f\colon Y\to X$ and $g\colon Z\to Y$ such that both $f$ and $h=f\circ g$ are log resolutions of $(X,D)$ which are isomorphisms over $X\smallsetminus D$. We put $E=(f^{*}D)_{\rm red}$ and $G=(g^{*}f^{*}D)_{\rm red}$.

We denote by $C^{Y,\bullet}_{k-n}$ the complex

on $Y$, and by $C^{Z,\bullet}_{k-n}$ the complex

on $Z$, both of them placed in degrees $-n,\ldots,0$. Since each $f^{*}F_{j}\mathscr{D}_{X}$ is a locally free $\mathscr{O}_{Y}$-module, we may apply the projection formula and Theorem 31.1i) to deduce that

for all $q\geq 0$ and all $p>0$, and that we have canonical isomorphisms

for all $q\geq 0$. We thus obtain a canonical isomorphism

The assertion in the theorem is now a consequence of the fact that the induced isomorphism

commutes with the two morphisms to $\omega_{X}(*D)$. Since the whole picture is compatible with restriction to open subsets, this follows by restricting to $U=X\smallsetminus D$, where the assertion is straightforward. ∎

It is instructive to also give an elementary proof of the fact that $F_{\bullet}\omega_{X}(*D)$ gives a filtration for the $\mathscr{D}_{X}$-module $\omega_{X}(*D)$ (without appealing to push-forwards of filtered $\mathscr{D}_{X}$-modules).

We employ the usual log resolution notation. With the notation in §9, we see that using the multiplication maps $F_{i}\mathscr{D}_{X}\otimes_{\mathscr{O}_{X}}F_{k}\mathscr{D}_{X}\rightarrow F_{i+k}\mathscr{D}_{X}$ we get a morphism of complexes

Since $F_{k}\mathscr{D}_{X}$ is a locally free $\mathscr{O}_{X}$-module, applying $\mathbf{R}f_{*}$ and taking $H^{0}$, we obtain an induced morphism

compatible with the canonical multiplication map

Comparison with Hodge filtration, and strictness property

It follows from the discussion in §6 and the results from §3 used in the definition, that the filtration $F_{\bullet}\omega_{X}(*D)$ we introduced in §9 is the same as the Hodge filtration on $\omega_{X}(*D)\simeq H^{0}f_{+}\omega_{Y}(*E)$. Our definition of Hodge ideals is independent of this, but equating it with the Hodge filtration highlights the following important extra consequence that comes from strictness. For $k\geq 0$, recall that $C^{\bullet}_{k-n}$ are the complexes providing the filtration on $A^{\bullet}$.

With the notation in §9, the following hold for every $k\geq 0$:

${\rm(}$Local vanishing for $I_{k}(D).$${\rm)}$ We have

We have $R^{i}f_{*}A^{\bullet}=0$ for all $i\neq 0$ by Lemma 2.2 and Proposition 2.4. On the other hand, strictness implies that $R^{i}f_{*}C^{\bullet}_{k-n}$ injects in $R^{i}f_{*}A^{\bullet}$. ∎

The case $k=0$ in part ii) of the corollary is local vanishing for multiplier ideals; in this case $C^{\bullet}_{-n}=\omega_{Y}(E)$. We note that in fact all vanishings $R^{i}f_{*}C^{\bullet}_{k-n}=0$ are more elementary when $i>0$. (This completely takes care of the case $k=1$ for instance, since $C^{\bullet}_{1-n}$ is quasi-isomorphic to $H^{0}C^{\bullet}_{1-n}$, whose negative direct images trivially vanish.) Indeed, if $C^{\bullet}=C^{\bullet}_{k-n}$, we have

by Theorem 32.1. The first quadrant spectral sequence

then implies that $R^{i}f_{*}C^{\bullet}=0$ for $i>0$.

Chain of inclusions

It follows from the definition and Lemma 11.2 that

for each $k\geq 1$. However, the Hodge ideals also satisfy a more subtle sequence of inclusions.

For every reduced effective divisor $D$ on the smooth variety $X$, and for every $k\geq 1$, we have

We give an argument using the theory of mixed Hodge modules. Consider the canonical inclusion

of filtered left $\mathscr{D}_{X}$-modules that underlie mixed Hodge modules. Since the category ${\rm MHM}(X)$ of mixed Hodge modules on $X$ constructed in [Saito-MHM] is abelian, the cokernel $\mathcal{M}$ of $\iota$ underlies a mixed Hodge module on $X$ too, and it is clear that $\mathcal{M}$ has support $D$. Since morphisms between Hodge $\mathscr{D}$-modules preserve the filtrations and are strict, for each $k\geq 0$ we have a short exact sequence