Weighted Hodge ideals of reduced divisors

A. Introduction

In this paper, we continue the study of weighted Hodge ideals that started in [olano22a], where the focus was the -th weighted Hodge ideals, also called weighted multiplier ideals. We show that several results satisfied by the weighted multiplier ideals can be generalized under suitable conditions.

for all $p\geq 0$. Consequently, we can define the Hodge ideal $I_{p}(D)$ by

The $\mathscr{D}_{X}$-module $\mathscr{O}_{X}(*D)$ is also endowed with a weight filtration $W_{\bullet}\mathscr{O}_{X}(*D)$ by $\mathscr{D}_{X}$-submodules. The Hodge filtration of these submodules satisfies

and similarly we can define the weighted Hodge ideals by

The weighted Hodge ideals form a chain of inclusions

We can always understand the two extreme ideals in this chain. The first element in the list admits an easy description:

On the other end, the last ideal in this chain is the usual $p$-th Hodge ideal, that is,

Unlike $I_{p}^{W_{0}}(D)$, for all the other degrees, the support of the scheme defined by $I_{p}^{W_{l}}(D)$ is contained in the singular locus of $D$.

Birational definition We give an alternative description of the weighted Hodge ideals in terms of a resolution of singularities. Let $f:Y\to X$ be a resolution of singularities of the pair $(X,D)$ which is an isomorphism over $X\setminus D$, and let $E:=(f^{*}D)_{\rm red}$. This description stems from the birational definition of Hodge ideals in [hodgeideals]*§9, and uses right $\mathscr{D}$-modules. The $\mathscr{D}_{Y}$-module $\omega_{Y}(*E)$ admits a filtered resolution by $\mathscr{D}_{Y}$-modules given by

Similarly, using the weight filtration on the sheaves of logarithmic $p$-forms (see (1.4)), we show that the complex

is filtered quasi-isomorphic to the $\mathscr{D}_{X}$-module $W_{n+l}\omega(*E)$ (see Proposition 4.1).

The $\mathscr{D}_{X}$-module $\omega_{X}(*D)$ can be described using the filtered resolution of $\omega_{Y}(*E)$ described above. More precisely, we can define the complex $A^{\bullet}$ by

placed in degrees $-n,\ldots,0$, and we have that,

(see [hodgeideals]*§9). To give the alternative description of the weighted Hodge ideals, we introduce the complex $C^{\bullet}_{l,p-n}$ defined as

is precisely $F_{p-n}W_{n+l}\omega_{X}(*D)=I^{W_{l}}_{p}(D)\otimes\omega_{X}((p+1)D)$ (see Proposition 4.3).

Let $X$ be a smooth complex variety and $D$ a reduced divisor defined by a regular function $f\in\mathscr{O}_{X}(X)$. Then,

The proof is based on two ideas. First, we can relate the Hodge filtration of $V^{1}i_{+}\mathscr{O}_{X}$ with that of $\mathscr{O}_{X}(*D)$ (see 5.2). Second, the weight filtration on the nearby cycles sheaf can be related to that of the local cohomology sheaf (Proposition 5.3). This is enough to understand all the weighted Hodge ideals in the case when $D$ only has isolated weighted-homogeneous singularities (see Remark 5.7).

The description in Theorem A is useful to relate the weighted Hodge ideals with some invariants of the singularities, like the minimal exponent. Recall that to the variety $D\subseteq X$ we can associate the Bernstein-Sato polynomial $b_{D}(s)$. The polynomial $(s+1)$ divides $b_{D}(s)$, and we denote $\widetilde{b_{D}}(s)=b_{D}(s)/(s+1)$. The negative of the largest root of $\widetilde{b_{D}}(s)$ is called the minimal exponent of a $D$ and is denoted $\widetilde{\alpha_{D}}$. This invariant encodes important properties of the singularities of $D$. For instance, it is a refined version of the log-canonical threshold, since $lct(X,D)=\min\{\widetilde{\alpha_{D}},1\}$. In particular, this implies that $(X,D)$ is log-canonical if and only if $\widetilde{\alpha_{D}}\geq 1$. Moreover, it is a result of Saito that $D$ has rational singularities if and only if $\widetilde{\alpha_{D}}>1$.

The notions of log-canonicity and rationality can be described in terms of weighted Hodge ideals. Recall that 0-th weighted Hodge ideals, or weighted multiplier ideals, form a sequence of ideals interpolating between the adjoint ideal and a multiplier ideal. This is the case, as $I_{0}^{W_{1}}(D)=\operatorname{adj}(D)$ (see for instance [olano22a]*Theorem A) and $I_{0}(D)=\mathcal{J}((1-\varepsilon)D)$ for $0<\epsilon\ll 1$ [budursaito05]. These two ideals identify if a singularity is respectively rational or log-canonical. We give an analogous description for the higher weighted Hodge ideals. The Hodge ideal $I_{p}(D)$ is trivial if and only if $\widetilde{\alpha_{D}}\geq p+1$, in which case we say that $(X,D)$ is $p$-log-canonical. Also, the weighted Hodge ideal $I_{p}^{W_{1}}(D)$ is trivial if and only if $\widetilde{\alpha_{D}}>p+1$ (see Corollary 5.10), which some authors referred to as $D$ being $p$-rational. The rest of the $p$-weighted Hodge ideals filter and measure the “distance” between $(X,D)$ having $p$-log-canonical singularities and $D$ being $p$-rational.

Isolated singularities. Recall that the weighted Hodge ideals satisfy

The difference between the two ideals can be described by the coherent sheaf $F_{p}\operatorname{gr}^{W}_{n+l}\mathscr{O}_{X}(*D)$ (see (6.1)). If $D$ has isolated singularities, we give a description of the dimension of this sheaf at the singular points in terms of a resolution of singularities. For this, possibly after restricting to an open set, assume $D$ has one isolated singularity $x\in D$. In this case, there exists a pure Hodge structure $H_{l}$ for $l\geq 2$, such that the dimension of their Hodge pieces describes the desired dimension. More concretely,

(see §6 for more details). For this reason, to find the difference between two consecutive weighted Hodge ideals, it is enough to compute the dimensions of the spaces $\operatorname{Gr}_{F}^{n-p}H_{l}$.

Let $g:\widetilde{D}\to D$ be a log-resolution of singularities that is an isomorphism outside of $x$. Let $G\subseteq\widetilde{D}$ be the exceptional divisor. Then

When $p=0$ the second summand in the description of $\dim(\operatorname{Gr}_{F}^{n-p}H_{2})$ is 0 because the dimension of $G$ is $n-2$, and therefore these dimensions are described as Hodge numbers of the middle cohomology of $G$. For $p\geq 1$ we cannot expect this term to be 0 in general, but this dimension admits a geometric interpretation (see Remark 6.8).

Vanishing results. Weighted Hodge ideals satisfy global results under suitable conditions. Let $X$ be a smooth projective variety and $D$ an ample divisor with at most isolated singularities. Under this assumptions, when $p=0$ we have that

for $i\geq 1$ and $l\geq 2$ [olano22a]*Theorem E. To generalize this result for all $p\geq 1$, we require the condition that $I_{p-1}^{W_{l}}(D)=\mathscr{O}_{X}$.

Let $X$ be a smooth projective variety of dimension $n$, and $D$ an ample reduced effective divisor with at most isolated singularities. Suppose that $I_{p-1}^{W_{1}}(D)$ is trivial. Then

If $H^{j}(X,\Omega_{X}^{n-j}((p-j+1)D))=0$ for all $1\leq j\leq p$, then

When $l=1$ and $i=1$ the vanishing does not hold in general. For an example see Remark 7.2. A Kodaira-type vanishing result is also satisfied for all $l\geq 1$, and the proof is based on a vanishing result by Saito [saito90]*Proposition 2.33 (see Proposition 8.1).

for $k\geq(p+1)d-n-1$ if $l\geq 2$, and $k\geq(p+1)d-n$ if $l\geq 1$.

This result gives a bound on a certain type of isolated singularities we describe next. For simplicity, suppose $D$ has at most one isolated singularity $x\in D$, and assume $\widetilde{\alpha_{D}}=p+1$. We describe first the case $p=0$. This case corresponds to a log-canonical and not rational singularity. In this case, according to (0.1), the length of the scheme described by $I_{0}^{W_{1}}(D)$ is determined by $\operatorname{Gr}_{F}^{0}(H^{n-2}(G))$, using the notation of Theorem B. Ishii proved that in this case, $\dim(\operatorname{Gr}_{F}^{0}(H^{n-2}(G)))=1$ [ishii85]*Proposition 3.7. This means that the ideal $I_{0}^{W_{1}}(D)$ is the maximal ideal of $x$ in $X$, and that there exists exactly one degree $l\geq 2$ such that

while the dimension for the other degrees is 0. A log-canonical singularity is of type $(0,n-l)$ in this case [ishii85]*Definition 4.1.

for $l\geq 2$ and $0\leq r\leq p-1$ by (0.1) and Theorem B. Moreover, by the same results, we know that there exists exactly one degree $l\geq 2$ such that

while the dimension for all the other degrees is 0. Related invariants in similar conditions have been studied by Friedman and Laza in [friedmanlaza22b]*Theorem 6.11 and Corollary 6.14.

Restriction theorem. Finally, we study the behavior of weighted Hodge ideals of a pair $(X,D)$ under the restriction of a hypersurface of $X$. Let $H\subseteq X$ be a smooth hypersurface, and $D_{H}$ the restriction of $D$ to $H$. If $D_{H}$ is reduced, then we can also consider the pair $(H,D_{H})$ and their respective weighted Hodge ideals.

Let $X$ be a smooth variety and $D$ an effective reduced divisor. Let $H\subseteq X$ be a smooth divisor such that $H\subsetneq\operatorname{Supp}(D)$ and D_{H}=D\big{|}_{H} is reduced. Then, for every $p\geq 0$ and $l\geq 0$ we have

Moreover, if $H$ is general, then we have an equality.

This is the analogue of the Restriction Theorem for Hodge ideals [mustatapopa18]*Theorem A, and for multiplier ideals [lazarsfeld2]*Theorem 9.5.1.

Acknowledgements. I would like to thank Mircea Mustaţă and Mihnea Popa for their constant support and many conversations during the project. I am also very grateful to the anonymous referees for their feedback on improving the presentation of the article and for suggesting a simpler proof of Lemma 5.4, explained in Remark 5.5.

B. Preliminaries

In this section, we recall some facts about mixed Hodge modules and set up the notation we use throughout this paper.

Let $X$ be a smooth variety of dimension $n$. Mixed Hodge modules introduced by Saito in [saito88] are the main object used throughout this article. For a graded-polarizable mixed Hodge module $M$, we denote the underlying left regular holonomic $\mathscr{D}_{X}$-module by $\mathcal{M}$. In some contexts, it is more useful to use right $\mathscr{D}_{X}$-modules. Recall that if $\mathcal{M}$ is a left $\mathscr{D}_{X}$-module, the corresponding right $\mathscr{D}_{X}$-module is $\mathcal{M}\otimes_{\mathscr{O}_{X}}\omega_{X}$, where $\omega_{X}$ is the canonical sheaf. We mostly use left $\mathscr{D}$-modules, and in case we are using right $\mathscr{D}$-modules instead, we will say it explicitly.

A mixed Hodge module $M$ is endowed with a weight filtration, which we denote by $W_{\bullet}M$, and

is the quotient, which is a polarizable Hodge module of weight $l$. We denote by $F_{\bullet}\mathcal{M}$ the Hodge filtration. The de Rham complex is defined as:

and the Hodge filtration of $\mathcal{M}$ induces a filtration on this complex:

The $p$-th subquotient of this filtration is the complex

(see for example [hodgeideals]*Lemma 2.2). Since $E$ is a simple normal crossings divisor, the weight filtration of the $\mathscr{D}_{Y}$-module $\mathscr{O}_{Y}(*E)$ can be described in terms of the intersections of its irreducible components. The lowest degree of the weight filtration is $n=\dim{Y}$, that is:

The lowest piece corresponds to the canonical Hodge module of $Y$:

To describe the rest of the subquotients, we introduce the following very useful notation. Let

with $E_{J}=\displaystyle\bigcap_{j\in J}E_{j}$, is a smooth and possibly disconnected variety. We denote $i_{l}:E(l)\to Y$ the map such that on each component is the inclusion. We have that

In order to describe the weight filtration of a pushforward of a projective morphism, a useful tool is to use the spectral sequence associated to the weight filtration:

which degenerates at $E_{2}$, and there is an isomorphism:

Finally, recall that the sheaf of $p$-forms with logarithmic poles along $E$ denoted by $\Omega^{p}_{Y}(\log{E})$ are endowed with a weight filtration. This increasing filtration consists of subsheaves

such that if $z_{1},\ldots,z_{n}$ are local coordinates on an open set $V$, and $E$ is given by the equation

then in $V$, $W_{l}\Omega^{p}(\log{E})$ is a $\mathscr{O}_{V}$ module generated by elements of the form

with $i_{l}\leq r$ and $s\leq k$ (see [elzeinetal]*3.4.1.2 for more details). For $I=\{i_{1},\ldots,i_{s}\}$ and $J=\{j_{1},\ldots,j_{p-s}\}$ we use the notation

C. Characterizations

In this section, we introduce weighted Hodge ideals using the theory of mixed Hodge modules.

A fundamental result by Saito about the Hodge filtration on $\mathscr{O}_{X}(*D)$ states that

(see [saito93]*Proposition 0.9). The definition of Hodge ideals follows from this result. These ideals are denoted by $I_{p}(D)$, and are defined using the formula

(see [hodgeideals]*Definition 9.4). In this article, we study weighted Hodge ideals which are defined similarly using the weight filtration with which $\mathscr{O}_{X}(*D)$ is endowed. The Hodge filtration of the sub-$\mathscr{D}_{X}$ modules $W_{n+l}\mathscr{O}_{X}(*D)$ satisfies

Let $X$ be a smooth complex variety and $D$ a reduced divisor. For $l\geq 0$ and $p\geq 0$, we define the ideal sheaf $I_{p}^{W_{l}}(D)$ on $X$ by the formula

We call $I_{p}^{W_{l}}(D)$ the $l$-th weighted $p$-th Hodge ideal of $D$.

for all $p\geq 0$. Indeed, the weight filtration of $\mathscr{O}_{X}(*D)$ is an increasing filtration, hence

Simple normal crossings divisor

Weighted Hodge ideals can be described completely when the reduced divisor $D$ has simple normal crossings. In this case, the Hodge filtration of $\mathscr{O}_{X}(*D)$ is fully understood, and from this information we can deduce the Hodge filtration of $W_{n+l}\mathscr{O}_{X}(*D)$.

Let $D$ be a simple normal crossings divisor. In this case, the Hodge filtration of $\mathscr{O}_{X}(*D)$ admits a simple description:

if $p\geq 0$, and 0 otherwise. Using this, one obtains a local description of the Hodge ideals. Let $x_{1},\ldots,x_{n}$ be coordinates around $z\in X$, such that $D$ is defined by $(x_{1}\cdots x_{r}=0)$. For every $p\geq 0$, the ideal $I_{p}(D)$ is generated around $z$ by

[hodgeideals]*Proposition 8.2. Weighted Hodge ideals of $D$ admit a similar local description.

Let $x_{1},\ldots,x_{n}$ be coordinates around $z\in X$, such that $D$ is defined by $(x_{1}\cdots x_{r}=0)$. Then, for every $p\geq 0$ and $l\leq r$, $I_{p}^{W_{l}}(D)$ is generated around $z$ by

where $I=\{1,\ldots,r\}$. For $l\geq r$, $I_{p}^{W_{l}}(D)=I_{p}(D)$ around $z$.

The Hodge filtration of $W_{n+l}\mathscr{O}_{X}(*D)$ also admits a simple description:

Indeed, this follows from the fact that $\operatorname{gr}^{W}_{n+l}\mathscr{O}_{X}(*D)\cong i_{l+}\mathscr{O}_{E(l)}$ with a Tate Twist, so that the analogous statement of $(\ref{wsncgen0})$ is true for $i_{l+}\mathscr{O}_{E(l)}$ (see e.g. [saito09]*Remark 1.1 iii.)

For the rest of the proof we use right $\mathscr{D}$-modules. By [olano22a]*Proposition 4.1,

Around $z$, $W_{l}\omega_{X}(D)$ is generated by

where $\omega$ is the standard generator of $\omega_{X}$. It is clear that $W_{l}\omega_{X}(D)\cdot F_{p}\mathscr{D}_{X}$ is generated by

The result follows from the equation $\frac{\omega}{x_{j_{1}}^{1+b_{1}}\cdots x_{j_{l}}^{1+b_{l}}}=\frac{\omega}{x_{I}^{p+1}}(x_{j_{1}}^{p-b_{1}}\cdots x_{j_{l}}^{p-b_{l}}x_{I\setminus J}^{p+1})$. The last statement follows from the fact that, if $l>r$, around $z$, $W_{l}\omega_{X}(D)=\omega_{X}(D)$. ∎

Birational definition

Let $X$ be a smooth variety and $D$ a reduced divisor. Consider a log-resolution $f:Y\to X$ of the pair $(X,D)$, which is an isomorphism over $X\setminus D$, and denote $E=(f^{*}D)_{\rm red}$. A birational definition is given for Hodge ideals in [hodgeideals]*§9. In this section, we give a similar equivalent definition for weighted Hodge ideals. For the rest of this section we use right $\mathscr{D}$-modules, as it is more convenient for the construction. Recall that the right $\mathscr{D}_{X}$-module corresponding to $\mathscr{O}_{X}(*D)$ is $\omega_{X}(*D)$, and

Consider the following complex which we denote by $A^{\bullet}$:

placed in degrees $-n,\ldots,0$. The results in [hodgeideals]*§3 say that the complex $A^{\bullet}$ 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. Moreover, $R^{0}f_{*}A^{\bullet}\cong\omega_{X}(*D)$.

For $p\geq 0$ define the subcomplex $C^{\bullet}_{p-n}=F_{p-n}A^{\bullet}$ of $A^{\bullet}$ by

The pushforward of this complex admits the following interpretation:

by [hodgeideals]*Remark 9.3, Corollary 12.1.

We prove similar results in order to obtain a birational definition. Consider the complex $B^{\bullet}$:

is given by $\omega\otimes P\to d\omega\otimes P+\sum{(dz_{i}\wedge\omega)\otimes\partial_{i}P}$. The complex $B^{\bullet}$ is filtered quasi-isomorphic to the object $\omega_{Y}(*E)$ in degree 0 [hodgeideals]*Proposition 3.1.

in degrees $-n,\ldots,0$ is quasi-isomorphic to $W_{n+l}\omega_{Y}(*E)$.

We see first that the complex $W_{l}B^{\bullet}$ is exact in degrees $-n,\ldots,-1$. Fix a degree $-p$. We need to see that

so that no $z_{i}$ that appears in $z_{I}$ divides $C^{\beta}_{I,J,\alpha}z^{\beta}$. From this description, it follows that for each summand $C^{\beta}_{I,J,\alpha}z^{\beta}\frac{dz_{I}}{z_{I}}\wedge dz_{J}\otimes\partial^{\alpha}$, $|I|$ determines the weight where the form $C^{\beta}_{I,J,\alpha}z^{\beta}\frac{dz_{I}}{z_{I}}\wedge dz_{J}$ lies.

Next, we write, $\omega=\omega_{\leq l}+\omega_{>l}$, where the first term consists of the summands with $|I|\leq l$, and the latter of the terms with $|I|>l$. Using the description of $d^{\prime}$, we see that $d^{\prime}\omega_{\leq l}$ is in the completion of $W_{l}\Omega_{Y}^{n-p}(\log{E})\otimes\mathscr{D}_{Y}$, and each summand of $d^{\prime}\omega_{>l}$ is not. Indeed,

Since $\eta\in\ker{\hat{b}}$, $d^{\prime}\omega_{>l}=0$, and $d^{\prime}\omega_{\leq l}=\eta$, with $\omega_{\leq l}$ in the completion of $W_{l}\Omega_{Y}^{n-p-1}(\log{E})\otimes\mathscr{D}_{Y}$.

given by $\frac{\omega}{f}\otimes P\to\frac{\omega}{f}\cdot P$. Fixing a degree of the Hodge filtration, and using the description of the Hodge filtration of $W_{n+l}\omega_{Y}(*E)$ (see for example Proposition 3.3), we see that this map is surjective. That the kernel is the image of $W_{l}\Omega^{n-1}_{Y}(\log{E})\otimes_{\mathscr{O}_{Y}}\mathscr{D}_{Y}$ follows from [hodgeideals]*Proposition 3.1 and an argument similar to the one above.

We have that $W_{l}A^{\bullet}=W_{l}B^{\bullet}\otimes_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X}$, where $\mathscr{D}_{Y\to X}=\mathscr{O}_{Y}\otimes_{f^{-1}\mathscr{O}_{X}}f^{-1}\mathscr{D}_{X}$ is the transfer module. Note that when we see it as an $\mathscr{O}_{Y}$ module, we simply write $f^{*}\mathscr{D}_{X}$.

The complex $W_{l}A^{\bullet}$ represents $W_{n+l}\omega_{Y}(*E)\overset{\mathbf{L}}{\otimes}_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X}$ in the derived category of filtered right $f^{-1}\mathscr{D}_{X}$-modules.

It is enough to show that the elements $W_{l}B^{k}$ are acyclic with respect to $-\otimes_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X}$. For any $k$ consider the following spectral sequence:

[weibel]*Theorem 5.6.6. As $\mathscr{D}_{Y}$ is a locally free $\mathscr{O}_{Y}$-module, then $E_{2}^{p,q}=0$ for $q\neq 0$. Therefore,

for $p\neq 0$, where the last equality follows from the fact that $f^{*}\mathscr{D}_{X}$ is locally free.

whose image is $W_{n+l}\omega_{X}(*D)$. Moreover, the complex $C_{p-n}^{\bullet}$ described above corresponds to the $F_{p-n}(\omega_{Y}(*E)\overset{\mathbf{L}}{\otimes}_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X})$ using the identification

whose image is $F_{p-n}\omega_{X}(*D)=I_{p}(D)\otimes\omega_{X}((p+1)D)$ (see [hodgeideals]*Sections 4, 9, and 12).

Similarly, we define $C^{\bullet}_{l,p-n}$ by

which corresponds to $F_{p-n}(W_{n+l}\omega_{Y}(*E)\overset{\mathbf{L}}{\otimes}_{\mathscr{D}_{Y}}\mathscr{D}_{Y\to X})$ under the identification

whose image is $F_{p-n}H^{0}f_{+}W_{n+l}\omega_{X}(*D)$ (see for instance [hodgeideals]*§4). Taking the composition

and using strictness in the middle morphism (since it underlies a morphism of mixed Hodge modules), the image corresponds to $F_{p-n}W_{n+l}\omega_{X}(*D)=I^{W_{l}}_{p}(D)\otimes\omega_{X}((p+1)D).$ ∎

The description in Proposition 4.3 for $I_{0}^{W_{l}}(D)$ coincides with the description in [olano22a]*Proposition 3, since $f_{*}W_{l}\omega_{Y}(E)\to f_{*}\omega_{Y}(E)$ is an inclusion. The complex $C^{\bullet}_{l,1-n}$ also has a simple description. Recall that by definition

is injective [hodgeideals]*Lemma 3.4. Using the fact that $W_{l}\Omega^{n-1}_{Y}(\log{E})\hookrightarrow\Omega^{n-1}_{Y}(\log{E})$ and $W_{l}\omega_{Y}(E)\otimes f^{*}F_{1}\mathscr{D}_{X}\hookrightarrow\omega_{Y}(E)\otimes f^{*}F_{1}\mathscr{D}_{X}$ are injective (since $F_{1}\mathscr{D}_{X}$ is a locally free $\mathscr{O}_{X}$-module), we obtain that the differential in $C^{\bullet}_{l,1-n}$ is also an inclusion. Let $\mathcal{F}_{l,1}$ be the cokernel. This means that

This map can be interpreted by using the complex $C^{\bullet}_{1-n}$. Indeed, let $\mathcal{F}_{1}$ be the cokernel of the differential in $C^{\bullet}_{1-n}$. We have an induced map $\mathcal{F}_{l,1}\to\mathcal{F}_{1}$. Since $f_{*}\mathcal{F}_{1}=I_{1}(D)\otimes\omega_{X}(2D)$,

Note that since weighted Hodge ideals were defined in terms of the Hodge and weight filtrations of $\mathscr{O}_{X}(*D)$, the constructions presented in this section are independent of the resolution of singularities.

Weighted Hodge ideals and V𝑉V-filtration

Let $X$ be a smooth variety and $D$ be an effective reduced divisor defined by the global equation $f\in\mathscr{O}_{X}(X)$. The Hodge ideals $I_{p}(D)$ can be described using the $V$-filtration of $i_{+}\mathscr{O}_{X}$, where $i$ is the graph embedding defined by $f$. Namely,

where $Q_{j}(x)=\prod_{i=0}^{j-1}(x+i)$, [mustatapopa20b]*Theorem A’. An equivalent description is obtained using the following map:

The map $\tau$The map $\tau$ corresponds to $\tau_{1}$ in the notation of [mustatapopa20]. See §1 for the discussion about the reduced case. is a surjective morphism of $\mathscr{D}_{X}$-modules, and

see [mustatapopa20]*Proposition 5.4 and Lemma 5.1. Moreover, the map $\tau$ induces a map

Indeed, it is enough to see that $\tau(V^{>1}i_{+}\mathscr{O}_{X})\subseteq\mathscr{O}_{X}$. This follows from the fact that $V^{>1}i_{+}\mathscr{O}_{X}=V^{1+\alpha}i_{+}\mathscr{O}_{X}=t\cdot V^{\alpha}i_{+}\mathscr{O}_{X}$, with $\alpha>0$, and that if $j>0$, $tu\partial_{t}^{j}\delta=fu\partial_{t}^{j}\delta-ju\partial_{t}^{j-1}\delta$, and $tu\delta=fu\delta$. For $v=\sum_{j=0}^{p}v_{j}\partial_{t}^{j}\delta\in V^{>1}i_{+}\mathscr{O}_{X}$, there exists $u=\sum_{j=0}^{p}u_{j}\partial_{t}^{j}\delta\in V^{\alpha}i_{+}\mathscr{O}_{X}$ such that, $tu=v$. Hence,

The $\mathscr{D}_{X}$-module $\operatorname{gr}^{1}_{V}i_{+}\mathscr{O}_{X}$ underlies the mixed Hodge module $\psi_{f,1}\mathscr{O}_{X}$ and its weight filtration can be described in terms of the nilpotent operator $t\partial_{t}$. In order to complete the description in Theorem 5.6, we first need to show that the map $\bar{\tau}$ also preserves the weight filtration.

The map $\bar{\tau}$ sends the weight and Hodge pieces to the same image as the map $\tau_{\mathscr{D}_{X}}$ that underlies a morphism of mixed Hodge modules

The map $\bar{\tau}$ is surjective and using its description, we observe that its kernel is the image of the map $\partial_{t}t-1$ on $\operatorname{gr}^{1}_{V}i_{+}\mathscr{O}_{X}$. The same is true for the map $\tau_{\mathscr{D}_{X}}$. Indeed, the map $\partial_{t}t-1$ underlies the composition $Var\circ can$ on $\psi_{f,1}\mathscr{O}_{X}$. As $can:\psi_{f,1}\mathscr{O}_{X}\to\phi_{f,1}\mathscr{O}_{X}$ is surjective because $i_{+}\mathscr{O}_{X}$ has strict support (see for instance [schnellsurvey]*§11), the cokernel of $Var\circ can$ coincides with the cokernel of

The cokernel of $Var$ is isomorphic to $i_{D*}\mathcal{H}^{1}i_{D}^{!}\mathscr{O}_{X}$, where $i_{D}:D\to X$ is the inclusion [saito90]*Corollary 2.24. Moreover, $i_{D*}\mathcal{H}^{1}i_{D}^{!}\mathscr{O}_{X}$ is isomorphic to $\mathcal{H}^{1}_{D}(\mathscr{O}_{X})$ [saito09]*§2.2. This means that $\bar{\tau}$ and $\tau_{\mathscr{D}_{X}}$ could only differ by a $\mathscr{D}_{X}$-automorphism of $\mathcal{H}^{1}_{D}(\mathscr{O}_{X})$ and the result is a consequence of Lemma 5.4. ∎

A $\mathscr{D}_{X}$-automorphism of $\mathcal{H}^{1}_{D}(\mathscr{O}_{X})$ preserves the Hodge and weight filtration.

We are very grateful to Mircea Mustaţă for suggesting the argument of Lemma 5.4.

A consequence of the result above, is that we can write a description of the weighted Hodge ideals in a similar way to (5.1). Let $W_{l}V^{1}i_{+}\mathscr{O}_{X}$ be the submodule of $V^{1}i_{+}\mathscr{O}_{X}$ which maps to $W_{n+l-2}\operatorname{gr}^{1}_{V}i_{+}\mathscr{O}_{X}$ via the canonical projection.

It follows from Proposition 5.3 that $\tau(F_{p+1}W_{l}V^{1}i_{+}\mathscr{O}_{X})=F_{p}W_{n+l}\mathscr{O}_{X}(*D)=I_{p}^{W_{l}}(D)\otimes\mathscr{O}_{X}((p+1)D).$ ∎

The result above can be simplified even more using the description of the weight filtration of $\psi_{f,1}\mathscr{O}_{X}$, and that is the statement of Theorem A.

First, we note that if $v\in V^{1}i_{+}\mathscr{O}_{X}$, then $\tau(t\partial_{t}v)=0$. Indeed, let $v=\sum_{j=0}^{p}v_{j}\partial_{t}^{j}\delta\in V^{1}i_{+}\mathscr{O}_{X}$, then

The weight filtration of $\psi_{f,1}\mathscr{O}_{X}$ admits the following description for $k\geq 0$:

(see [saito94]*2.7 and for the monodromy filtration see e.g. [variationsofmhsi]*Remark 2.3). The only piece that is not an image of $(t\partial_{t})$ is $\ker{(t\partial_{t})^{k+1}}$. That means that the subset $\ker{(t\partial_{t})^{l}}\subseteq W_{l}V^{1}i_{+}\mathscr{O}_{X}$ has the same image as $W_{l}V^{1}i_{+}\mathscr{O}_{X}$ via $\tau$. ∎

Let $(X,D)$ be a pair such that $D$ has at most isolated weighted homogeneous singularities. Theorem A gives a complete description of the weighted Hodge ideals using the description of the $V$-filtration in [saito09]. Using the notation above, in this case, $(t\partial_{t})^{2}u\in V^{>1}i_{+}\mathscr{O}_{X}$ for all $u\in V^{1}i_{+}\mathscr{O}_{X}$. For this reason, $I_{p}^{W_{2}}(D)=I_{p}(D)$, for all $p\geq 0$. An argument without the use of the $V$-filtration in the case of $p=0$ is described in [olano22a]*§10.

A direct application of Theorem A is that we can recover the following result proved in [hodgeideals]*Theorem C. The proof we give differs from the one in [hodgeideals] and is also much shorter.

Let $X$ be a smooth variety and $D$ an effective reduced divisor. Then

Recall that $\operatorname{adj}(D)=I_{0}^{W_{1}}(D)$ [olano22a]*Theorem A. Moreover, as $I_{p}(D)\subseteq I_{1}(D)$ [hodgeideals]*Proposition 13.1, it is enough to prove that $I_{1}(D)\subseteq I_{0}^{W_{1}}(D).$

Let $u\in I_{1}(D)$. By (5.1), $u=u_{0}f+u_{1}$, where $f$ is defining equation of $D$, and $u_{0}\delta+u_{1}\partial_{t}\delta\in V^{1}i_{+}\mathscr{O}_{X}$. We also have that

Finally, as $\delta\in V^{>0}i_{+}\mathscr{O}_{X}$, then $u_{0}f\delta=t(u_{0}\delta)\in V^{>1}i_{+}\mathscr{O}_{X}\subseteq W_{1}V^{1}i_{+}\mathscr{O}_{X}$. This means that $u_{0}f\delta+u_{1}\delta\in W_{1}V^{1}i_{+}\mathscr{O}_{X}$, hence $u_{0}f+u_{1}\in I_{0}^{W_{1}}(D).$ ∎

Using these ideas, we obtain the following result for the 1st weighted Hodge ideals.

Suppose first that $\widetilde{\alpha_{f}}>p+1$. Then, by Lemma 5.9, $\partial_{t}^{p}\delta\in V^{1}i_{+}\mathscr{O}_{X}$. Moreover, there exists $\alpha\in(0,1]$ such that $\widetilde{\alpha_{f}}\geq p+1+\alpha$. Again, by Lemma 5.9, $\partial_{t}^{p+1}\delta\in W_{1}V^{1}i_{+}\mathscr{O}_{X}$, and therefore, $I_{p}^{W_{1}}(D)=\mathscr{O}_{X}$.

Suppose now that $I_{p}^{W_{1}}(D)=\mathscr{O}_{X}$. Then, $I_{p}(D)=\mathscr{O}_{X}$, and in particular $\delta,\partial_{t}\delta,\ldots,\partial_{t}^{p}\delta\in V^{1}i_{+}\mathscr{O}_{X}$. Moreover, there exists $v=\sum_{j=0}^{p}v_{j}\partial_{t}^{j}\delta\in W_{1}V^{1}i_{+}\mathscr{O}_{X}$ such that $\sum_{j=0}^{p}Q_{j}(1)f^{p-j}v_{j}=1$. It is enough to show that $\partial_{t}^{p}\delta\in W_{1}V^{1}i_{+}\mathscr{O}_{X}$. Indeed, by Proposition 5.6 and the injectivity of $t:\operatorname{gr}_{V}^{0}i_{+}\mathscr{O}_{X}\to\operatorname{gr}_{V}^{1}i_{+}\mathscr{O}_{X}$ (see e.g. [schnellsurvey]*§11), this means that $\partial_{t}^{p+1}\delta\in V^{\alpha}i_{+}\mathscr{O}_{X}$ with $\alpha\in(0,1]$, and therefore, $\widetilde{\alpha_{f}}\geq p+1+\alpha>p+1$. We argue by induction. Suppose $p=0$. Then $v=v_{0}\delta$ and by the second condition, $v_{0}=1$. Hence, $\delta\in W_{1}V^{1}i_{+}\mathscr{O}_{X}$. By the induction hypothesis, we assume now that $\partial_{t}^{k}\delta\in W_{1}V^{1}i_{+}\mathscr{O}_{X}$ for $k=0,\ldots,p-1$. It follows from the description of $v$ that

The result follows if we show that $f\partial_{t}^{p}\delta\in W_{1}V^{1}i_{+}\mathscr{O}_{X}$, and this is a consequence of $f\partial_{t}^{p}\delta=t\partial_{t}(\partial_{t}^{p-1}\delta)+p\partial_{t}^{p-1}\delta\in W_{1}V^{1}i_{+}\mathscr{O}_{X}$.

In general, we cannot obtain more information about the other $p$-weighted Hodge ideals. In [olano22a]*§13, the case of isolated log-canonical singularities, that are not rational, is discussed. This case corresponds to $\widetilde{\alpha_{f}}=1$. By the discussion above, it is clear that $I_{0}(D)=\mathscr{O}_{X}$ and that $I_{0}^{W_{1}}(D)$ is not trivial. For $l=2,\ldots,n-1$, there are examples of $f$ where the weighted multiplier ideals $I_{0}^{W_{l}}(D)$ are trivial, and other examples where they are non-trivial [ishii85]*Theorem 5.2.

D. Local study

There is a short exact sequence that arises from the definition of the weight filtration on $\mathscr{O}_{X}(*D)$:

Applying $F_{p}$, we obtain the short exact sequence

When $D$ has at most isolated singularities and $l\geq 2$, $\operatorname{gr}^{W}_{n+l}\mathscr{O}_{X}(*D)$ is supported on the singular points. To simplify the notation, we use the following definition.

Suppose $D$ has at most one isolated singularity $x\in D$, and let $i_{x}:\{x\}\hookrightarrow X$. For $l\geq 2$, we denote by $H_{l}$ the complex pure Hodge structure of weight $n+l$ such that

In order to describe the dimension of $F_{p}(i_{x})_{+}H_{l}$, it is enough to describe the dimension of $\operatorname{Gr}_{F}^{n-k}H_{l}$ for $0\leq k\leq p$. This is a consequence of the local description of the Hodge filtration of $(i_{x})_{+}H_{l}$. Let $x_{1},\ldots,x_{n}$ be a set of coordinates around the point $x\in X$. We have the following description of the pushforward of $H_{l}$ as a $\mathscr{D}$-module:

where $\partial_{i}=\frac{\partial}{\partial_{x_{i}}}$, and

where $\partial^{\nu}=\partial_{1}^{\nu_{1}}\cdots\partial_{n}^{\nu_{n}}$, $|\nu|=\nu_{1}+\ldots+\nu_{n}$, and $F_{k}H_{l}=F^{-k}H_{l}$. Since the lowest degree of the Hodge filtration of $\mathscr{O}_{X}(*D)$ is 0, and $\operatorname{DR}((i_{x})_{+}H_{l})\cong(i_{x})_{*}H_{l}$, that is, the push-forward of the pure Hodge structure $H_{l}$ is a skyscraper sheaf, then the highest degree of the Hodge filtration of $H_{l}$ is $n$, in other words, $F^{n+1}H_{l}=0$. Using this, we obtain, for instance, that

Since $F_{p}(i_{x})_{+}H_{l}$ is a skyscraper sheaf, we denote by $\dim(F_{p}(i_{x})_{+}H_{l})$ the dimension of the complex vector space $J_{p}$ that satisfies $F_{p}(i_{x})_{+}H_{l}=(i_{x})_{*}J_{p}$. From the discussion above, we obtain that

The dimension of $\operatorname{Gr}_{F}^{n-k}H_{l}$ is described in Theorem B.

We can and will assume that $X$ is a projective variety. Indeed, there is an open set around $x$ which has a smooth projective compactification $\bar{X}$. Let $\bar{D}$ be the closure of $D$ in $\bar{X}$. Consider a log-resolution of $(\bar{X}\smallsetminus x,\bar{D}\smallsetminus x)$ given by a sequence of blow ups with centers over the singular locus of $\bar{D}\smallsetminus x$. By blowing up the same sequence of centers over $\bar{X}$, we obtain a map $X_{1}\to\bar{X}$. Let $D_{1}$ be the strict transform of $\bar{D}$. By construction, the map is an isomorphism over $(X,D)$, and $D_{1}$ has only one isolated singularity corresponding to $x\in D$. We replace $(X,D)$ with $(X_{1},D_{1})$.

First, we prove that these dimensions do not depend on the log-resolution of singularities that is an isomorphism outside of $\{x\}$. Since for a pair of resolution of singularities one can find a third one that dominates the two of them, it is enough to show that the dimensions are equal if we have two resolutions of singularities $g_{1}:D_{1}\to D$ and $g_{2}:D_{2}\to D$ such that there is a morphism $h:D_{1}\to D_{2}$ such that $g_{1}=g_{2}\circ h$. Let $G_{i}\subseteq D_{i}$ be the exceptional divisor of $g_{i}$. Consider the exact sequence of mixed Hodge structures

(see [PS]*Proof of Theorem 6.15). For $l\geq 3$, applying $H^{p,n-l-p}$, we obtain that

For $l=2$, applying $H^{p,n-p-2}$ and $H^{n-p-1,p+1}$, and noting that $h^{p,n-p-2}(D_{i})=h^{n-p-1,p+1}(D_{i})$, we obtain that

Let $f:Y\to X$ be a log-resolution that is an isomorphism outside of $x$, and $E:=f^{-1}(D)_{red}$. This resolution defines a log-resolution of singularities $g:\widetilde{D}\to D$ by restriction, that is an isomorphism outside of $x$. We use the spectral sequence (1.3) for the constant map from $X$ to a point. In this case, it says

Moreover, the degeneration of the Hodge-to-de-Rham spectral sequence says that

(see for example [hodgeideals]*Example 4.2).

Applying $\operatorname{gr}^{F}_{-n+p}$, using (6.7), and the $E_{2}$-degeneration of the spectral sequence, we obtain that

where the last isomoprhism follows from Poincaré duality (see [PS]*Theorem 6.23). Using the long exact sequence of the pair $(X,D)$, we obtain that

as $H^{n-1}(X)$ and $H^{n}(X)$ have pure Hodge structures. Finally, as $g$ has $\{x\}$ as discriminant, we have a long exact sequence,

As this is a sequence of mixed Hodge structures, we obtain

Consider now $l=2$. In this case, the maps

where $\beta=\operatorname{gr}^{F}_{-n+p}\widetilde{\beta}$ and $\gamma=\operatorname{gr}^{F}_{-n+p}\widetilde{\gamma}$, we obtain that

Indeed, this follows from the descriptions of $E_{2}^{n-2+s,n+2}$ for $s=0,1,2$ and Poincaré duality. More precisely, we have three short exact sequences

and also that $\operatorname{Gr}_{F}^{n-p}E_{2}^{n-1,n+2}\cong H^{p,n-p-2}(H^{n-1}_{c}(U))^{*}$. Using the long exact sequence associated to the pair $(X,D)$ to relate these three sequences, we obtain

Finally, using that the map $g$ has $\{x\}$ as discriminant, we obtain that

In general, the term $h^{n-p-1,p+1}(H^{n}(G))$ might not be 0. Consider for instance $n=4$ and $p=1$. In this case, $h^{2,2}(H^{4}(G))=k$ where $k$ is the number of irreducible components of $G$. Using similar computations as above, we also see that

that is, the failure of Poincaré duality. Still, in the case $p=0$, the term $h^{n-p-1,p+1}(H^{n}(G))$ is always 0, as $G$ is $(n-2)$-dimensional (see [olano22a]*Theorem B).

E. Vanishing Theorems

Let $X$ be a smooth projective variety of dimension $n$, and $D$ an ample divisor. Let $U=X\smallsetminus D$. As $U$ is smooth and affine, $H^{i+n}(U)=0$ for $i>0$ (see for instance [lazarsfeld2]*Theorem 3.1.1). In this setting we have the following result.

In [olano22a]*Proof of Proposition 12.1, using the spectral sequences

corresponding to the map $E^{-n-k-1,i+n+k}_{1}\to E_{1}^{-n-k,i+n+k}$. Then we have the following short exact sequence:

As $E_{2}^{-n-l,n+l+i}=0$, using the analysis above, we obtain a short exact sequence

When $p=0$, the result above is enough to obtain that

for $l\geq 2$ and $i\geq 1$. Indeed, as 0 is the lowest degree of the Hodge filtration on $\mathscr{O}_{X}(*D)$, we have

This is no longer the case when we consider $\operatorname{gr}^{F}_{-n+p}$ for $p\geq 1$ instead. Nonetheless, following the idea in [hodgeideals]*Proof of Theorem F, we give conditions in Theorem C to obtain an analogue vanishing theorem.

Since $I_{p-1}^{W_{l}}(D)=\mathscr{O}_{X}$, we have the following short exact sequence

Using the long exact sequence of cohomologies and Kodaira-vanishing, we note that it is enough to prove that

The complex $C^{\bullet}$ can be identified with the complex

concentrated in degrees $-p$ to , since $F_{0}W_{n+l}\mathscr{O}_{X}(*D)=\mathscr{O}_{X}(D)$ and $\operatorname{gr}^{F}_{k}W_{n+l}\mathscr{O}_{X}(*D)\cong\mathscr{O}_{D}((k+1)D)$ for $k\leq p-1$ (see §1 for the definition of $\operatorname{gr}^{F}_{p}\operatorname{DR})$.

Suppose now that $D$ has at most isolated singularities. By Lemma 7.1, we obtain that

for $i\geq 1$ and $l\geq 2$. In particular, this means that