Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension

2.1 $\mathscr {D}$ -modules and mixed Hodge modules

Given a smooth, irreducible, n-dimensional complex algebraic variety X, we denote by $\mathscr {D}_X$ the sheaf of differential operators on X. For basic facts in the theory of $\mathscr {D}_X$ -modules, we refer to [Reference Hotta, Takeuchi and Tanisaki18].

We only recall here a couple of things: first, an exhaustive filtration $F_{\bullet }$ on a left $\mathscr {D}_X$ -module $\mathcal {M}$ (always assumed to be compatible with the filtration $F_\bullet \mathscr {D}_X$ by the order of differential operators) is good if $F_p\mathcal {M}$ is coherent for all p and there is q, such that

$$ \begin{align*}F_{q+k}\mathcal{M}=F_k\mathscr{D}_X\cdot F_q\mathcal{M}\quad\text{for all}\quad k\geq 0.\end{align*} $$

In this case, we say that the filtration is generated at level q.

Next, there is an equivalence of categories between left and right $\mathscr {D}_X$ -modules, such that, if $\mathcal {M}^r$ is the right $\mathscr {D}_X$ -module corresponding to the left $\mathscr {D}_X$ -module $\mathcal {M}$ , we have an isomorphism of underlying $\mathscr {O}_X$ -modules $\mathcal {M}^r\simeq \omega _X\otimes _{\mathscr {O}_X}\mathcal {M}$ . We will mostly work with left $\mathscr {D}_X$ -modules, but it will sometimes be convenient to work with their right counterparts. We note that if they are endowed with a good filtration as above, then the left-right rule for the filtration is by convention

(2.1) $$ \begin{align} F_{p-n}\mathcal{M}^r=\omega_X\otimes_{\mathscr{O}_X}F_p\mathcal{M}\quad\text{for all}\quad p\in {\mathbf Z}. \end{align} $$

For the basic notions and results on mixed Hodge modules, we refer to [Reference Saito48] and [Reference Saito49]. In what follows, we refer to a mixed Hodge module on X simply as a Hodge module. Recall that such an object consists of a $\mathscr {D}_X$ -module on X (always assumed to be holonomic and with regular singularities), endowed with a good filtration called the Hodge filtration, and with several other pieces of data satisfying an involved set of conditions that we will not be directly concerned with in this paper. We will say that this filtered $\mathscr {D}_X$ -module underlies the respective Hodge module. An important fact is that every morphism of Hodge modules is strict with respect to the Hodge filtrations (see also [Reference Mustaţă and Popa36, Chapter C] and [Reference Mustaţă and Popa37, §1] for further review).

An important example of (pure) Hodge module is the trivial Hodge module ${\mathbf Q}_X^H[n]$ . The underlying $\mathscr {D}_X$ -module is $\mathscr {O}_X$ , while the Hodge filtration is defined by the condition $\mathrm {Gr}^F_p(\mathscr {O}_X)=0$ for all $p\neq 0$ .

Given a filtered $\mathscr {D}_X$ -module $(\mathcal {M},F)$ and $q\in {\mathbf Z}$ , the Tate twist $(\mathcal {M},F)(q)$ is the filtered $\mathscr {D}_X$ -module $\big (\mathcal {M}, F[q]\big )$ , where

$$ \begin{align*}F[q]_p\mathcal{M}=F_{p-q}\mathcal{M} \,\,\,\,\,\,\mathrm{for~ all} \,\,\,\, p\in {\mathbf Z}.\end{align*} $$

For every left $\mathscr {D}_X$ -module $\mathcal {M}$ , the de Rham complex $\mathrm {DR}_X(\mathcal {M})$ is the complex

$$ \begin{align*}0\to \mathcal{M}\to \Omega_X^1\otimes_{\mathscr{O}_X}\mathcal{M}\to\cdots\to\omega_X\otimes_{\mathscr{O}_X}\mathcal{M}\to 0,\end{align*} $$

placed in cohomological degrees $-n,\ldots ,0$ . If $\mathcal {M}$ carries a good filtration F, the de Rham complex has an induced filtration, such that, for every integer k, the graded piece $\mathrm {Gr}^F_k\mathrm {DR}_X(\mathcal {M},F)$ is given by

$$ \begin{align*}0\to \mathrm{Gr}^F_k \mathcal{M} \to \Omega_X^1\otimes_{\mathscr{O}_X}\mathrm{Gr}^F_{k+1} \mathcal{M} \to\cdots\to\omega_X\otimes_{\mathscr{O}_X}\mathrm{Gr}^F_{k+n} \mathcal{M} \to 0,\end{align*} $$

where $\mathrm {Gr}^F_i \mathcal {M} =F_i\mathcal {M}/F_{i-1}\mathcal {M}$ for all $i\in {\mathbf Z}$ . Note that this is a complex of coherent $\mathscr {O}_X$ -modules. The filtered de Rham complex of a filtered right $\mathscr {D}_X$ -module is the filtered de Rham complex of the corresponding left $\mathscr {D}_X$ -module. When $(\mathcal {M}, F)$ underlies a Hodge module M, we sometimes use, alternatively, the notation $\mathrm {DR}_X(M)$ and $\mathrm {Gr}^F_p\mathrm {DR}_X(M)$ .

Since morphisms of Hodge modules are strict with respect to the Hodge filtration, $\mathrm {Gr}^F_k\mathrm {DR}_X(-)$ gives an exact functor from the abelian category of Hodge modules on X to the abelian category of bounded complexes of coherent sheaves on X. This induces an exact functor, also denoted $\mathrm {Gr}^F_k\mathrm {DR}_X(-)$ , from the derived category $\mathbf {D}^b\big (\mathrm {MHM}(X)\big )$ of Hodge modules on X to the derived category $\mathbf {D}^b\big (\mathrm {Coh}(X)\big )$ . By considering the truncation functors associated to the standard t-structure on $\mathbf {D}^b\big (\mathrm {MHM}(X)\big )$ , one obtains for every $u\in \mathbf { D}^b\big (\mathrm {MHM}(X)\big )$ and every $k\in {\mathbf Z}$ a spectral sequence

(2.2) $$ \begin{align} E_2^{pp'}={\mathcal H}^p\mathrm{Gr}^F_k\mathrm{DR}_X\big({\mathcal H}^{p'}(u)\big)\Rightarrow {\mathcal H}^{p+p'}\mathrm{Gr}^F_k\mathrm{DR}_X(u). \end{align} $$

Given a morphism $f\colon Y\to X$ of smooth complex varieties, we use the notation $f_*$ for the pushforward of Hodge modules. Note that at the level of underlying $\mathscr {D}$ -modules, this corresponds to the usual pushforward $f_+$ , defined for right $\mathscr {D}$ -modules as

$$ \begin{align*}f_+ \mathcal{M}: = \mathbf{R} f_* \big(\mathcal{M} \overset{\mathbf{L}}{\otimes}_{\mathscr{D}_Y} \mathscr{D}_{Y\to X} \big),\end{align*} $$

where the object on the right is in the derived category of right $\mathscr {D}_X$ -modules. Here,

$$ \begin{align*}\mathscr{D}_{Y\to X} : = \mathscr{O}_Y \otimes_{f^{-1} \mathscr{O}_X} f^{-1} \mathscr{D}_X\end{align*} $$

is the associated transfer $(\mathscr {D}_Y, f^{-1} \mathscr {D}_X)$ -bimodule, which is isomorphic to $f^* \mathscr {D}_X$ as an $\mathscr {O}_Y$ -module, and is filtered by $f^* F_k \mathscr {D}_X$ (see [Reference Hotta, Takeuchi and Tanisaki18, §1.5] for more details). In general, $f_+$ is different from the pushforward $\mathbf {R} f_*$ on quasi-coherent $\mathscr {O}$ -modules, but the two definitions agree if f is an open immersion, and, in this case, we also use the $\mathbf {R} f_*$ notation.

Moreover, if f is proper and if we denote by $\mathrm {FM}(\mathscr {D}_X)$ the category of filtered $\mathscr {D}$ -modules on X, then there is a functor

$$ \begin{align*}f_+ \colon \mathbf{D}^b \big(\mathrm{FM}(\mathscr{D}_Y)\big) \rightarrow \mathbf{D}^b \big(\mathrm{FM}(\mathscr{D}_X)\big)\end{align*} $$

defined in [Reference Saito48], which is compatible with the functor $f_+$ above. If $(\mathcal {M},F)$ underlies a Hodge module M on X, we write $f_+(\mathcal {M},F)$ for the object in $\mathbf {D}^b\big (\mathrm {FM}(\mathscr {D}_Y)\big )$ underlying $f_* M$ .

An important feature of the pushforward of Hodge modules under projective morphisms is the strictness property of the Hodge filtration (see [Reference Saito49, Theorem 2.14]). This says that if $f\colon Y\to X$ is projective and $(\mathcal {M},F)$ underlies a Hodge module on Y, then $f_+ (\mathcal {M}, F)$ is strict as an object in $\mathbf {D}^b \big (\mathrm {FM}(\mathscr {D}_X)\big )$ (and moreover, each ${\mathcal H}^i f_+ (\mathcal {M}, F)$ underlies a Hodge module). Concretely, this means that the natural mapping

(2.3) $$ \begin{align} R^i f_* \big(F_k (\mathcal{M} \overset{\mathbf{L}}{\otimes}_{\mathscr{D}_Y} \mathscr{D}_{Y\to X}) \big) \longrightarrow R^i f_* (\mathcal{M} \overset{\mathbf{L}}{\otimes}_{\mathscr{D}_Y} \mathscr{D}_{Y\to X}) \end{align} $$

is injective for every $i, k\in {\mathbf Z}$ . The filtration on ${\mathcal H}^if_+(\mathcal {M},F)$ is obtained by taking $F_k{\mathcal H}^if_+(\mathcal {M},F)$ to be the image of this map. Note that this strictness property is a vast generalisation of the degeneration at $E_1$ of the Hodge-to-de Rham spectral sequence (cf., e.g. [Reference Mustaţă and Popa36, §4]).

2.2 Brief review of local cohomology

Let X be a smooth, irreducible n-dimensional complex algebraic variety and Z a proper closed subscheme of X defined by the coherent ideal sheaf $\mathcal {I}_Z$ . For a quasi-coherent $\mathscr {O}_X$ -module $\mathcal {M}$ and $j\geq 0$ , we denote by ${\mathcal H}^q_Z(\mathcal {M})$ the $q^{\mathrm {th}}$ local cohomology sheaf of $\mathcal {M}$ , with support in Z. This is the $q^{\mathrm {th}}$ derived functor of the functor $\underline {\Gamma _Z} (-)$ given by the subsheaf of local sections with support in Z. The sheaves ${\mathcal H}^q_Z(\mathcal {M})$ only depend on the support of Z, so in many situations, it is convenient to assume that Z is reduced. For the basic facts on local cohomology, see [Reference Hartshorne15].

The sheaf ${\mathcal H}^q_Z(\mathcal {M})$ is a quasi-coherent sheaf, whose local sections are annihilated by suitable powers of $\mathcal {I}_Z$ . For every affine open subset $U\subseteq X$ , if

$$ \begin{align*}A=\mathscr{O}_X(U), \,\,\,\,\,\,I=\mathcal{I}_Z(U),\,\,\,\,\,\, \mathrm{and} \,\,\,\,\,\,M=\mathcal{M}(U),\end{align*} $$

then ${\mathcal H}^q_Z(\mathcal {M})\vert _U$ is the sheaf associated to the local cohomology module $H^q_I(M)$ . This can be computed as follows: if $I=(f_1,\ldots ,f_r)$ (or, more generally, if I and $(f_1,\ldots ,f_r)$ have the same radical) and for a subset $J\subseteq \{1,\ldots ,r\}$ , we put $f_J:=\prod _{i\in J}f_i$ , then we have the Čech-type complex

(2.4) $$ \begin{align} C^{\bullet} : \quad 0\to C^0\to C^1 \to \cdots\to C^r\to 0, \end{align} $$

with

$$ \begin{align*}C^p=\bigoplus_{|J|=p}A_{f_J},\end{align*} $$

and with the maps given (up to suitable signs) by the localisation homomorphisms. With this notation, we have functorial isomorphisms

(2.5) $$ \begin{align} H^q_I(M)\simeq H^q(C^{\bullet}\otimes_AM). \end{align} $$

The two main cases we will be interested in are those when $\mathcal {M}$ is either $\mathscr {O}_X$ or $\omega _X$ . Note that we have ${\mathcal H}^0_Z(\mathscr {O}_X) = {\mathcal H}^0_Z(\omega _X) = 0$ .

In general, given X and Z as above, we write $U=X\backslash Z$ and let $j\colon U\hookrightarrow X$ be the inclusion map. For every $\mathcal {M}$ , we have a functorial exact sequence

(2.6) $$ \begin{align} 0\to {\mathcal H}^0_Z(\mathcal{M})\to\mathcal{M}\to j_*(\mathcal{M}\vert_U)\to {\mathcal H}^1_Z(\mathcal{M})\to 0 \end{align} $$

and functorial isomorphisms

(2.7) $$ \begin{align} R^qj_*(\mathcal{M}\vert_U)\simeq {\mathcal H}^{q+1}_Z(\mathcal{M})\quad\text{for all}\quad q\geq 1. \end{align} $$

Remark 2.2. Since X is smooth, hence, Cohen-Macaulay, the usual description of depth in terms of local cohomology (see [Reference Hartshorne15, Theorem 3.8]) implies that

$$ \begin{align*}\mathrm{codim}_X(Z)=\min\{q\mid{\mathcal H}_Z^q(\mathscr{O}_X)\neq 0\}.\end{align*} $$

Moreover, from the description via the Čech complex in (2.5), it follows that if Z is locally cut out set-theoretically by $\leq N$ equations, then ${\mathcal H}^q_Z(\mathscr {O}_X)=0$ for $q>N$ . In particular, we conclude that if Z is a local complete intersection of pure codimension r, then ${\mathcal H}^q_Z(\mathscr {O}_X)=0$ for all $q\neq r$ .

If $Z'\subseteq Z$ is another closed subset of X, then for every quasi-coherent sheaf $\mathcal {M}$ , we have natural maps

$$ \begin{align*}{\mathcal H}^q_{Z'}(\mathcal{M})\to {\mathcal H}^q_Z(\mathcal{M}).\end{align*} $$

These are induced by the natural transformation of functors ${\mathcal H}^0_{Z'}(-)\to {\mathcal H}^0_Z(-)$ .

It is a standard fact that if $\mathcal {M}$ is a left $\mathscr {D}_X$ -module, then each ${\mathcal H}_Z^q(\mathcal {M})$ has a canonical structure of left $\mathscr {D}_X$ -module. Locally, this can be seen by using the description in (2.5). Indeed, each localisation $M_{f_J}$ has an induced $\mathscr {D}_X(U)$ -module structure, such that the morphisms in the complex $C^{\bullet }\otimes _AM$ are morphisms of $\mathscr {D}_X(U)$ -modules. Therefore, each cohomology module $H^q_I(M)$ has an induced $\mathscr {D}_X(U)$ -module structure and one can easily check that this is independent of the choice of generators for I. Hence, these structures glue to a $\mathscr {D}_X$ -module structure on ${\mathcal H}^q_Z(\mathcal {M})$ .

Note also that if $\mathcal {M}$ is a left $\mathscr {D}_X$ -module, then the sheaves $R^qj_*(\mathcal {M}\vert _U)$ are left $\mathscr {D}_X$ -modules as well; they are the cohomology sheaves of the $\mathscr {D}$ -module pushforward of $\mathcal {M}$ via j. Moreover, in this case, the exact sequence (2.6) and the isomorphism (2.7) also hold at the level of $\mathscr {D}_X$ -modules.

Similar considerations apply for right $\mathscr {D}_X$ -modules. It is easy to see, using the local description, that for every $q\geq 0$ , we have a canonical isomorphism

$$ \begin{align*}{\mathcal H}^q_Z(\mathcal{M})^r\simeq{\mathcal H}^q_Z(\mathcal{M}^r).\end{align*} $$

In particular, we have a canonical isomorphism of right $\mathscr {D}_X$ -modules

$$ \begin{align*}{\mathcal H}^q_Z(\mathscr{O}_X)^r\simeq{\mathcal H}^q_Z(\omega_X).\end{align*} $$

2.3 The Hodge filtration on local cohomology

We continue to use the notation introduced in the previous sections. Denoting by $i\colon Z\hookrightarrow X$ the inclusion, there are objects $j_*{\mathbf Q}_U^H[n]$ and $i_*i^!{\mathbf Q}_X^H[n]$ in the derived category of Hodge modules and an exact triangle

(2.8) $$ \begin{align} i_*i^!{\mathbf Q}_X^H[n]\longrightarrow {\mathbf Q}_X^H[n]\longrightarrow j_*{\mathbf Q}_U^H[n]\overset{+1}\longrightarrow, \end{align} $$

as shown in [Reference Saito49, § 4]. The cohomologies of these objects are Hodge modules whose underlying $\mathscr {D}_X$ -modules we have already seen:

$$ \begin{align*}{\mathcal H}^q\big(i_*i^!{\mathbf Q}_X^H[n]\big)={\mathcal H}_Z^q(\mathscr{O}_X)\quad\text{and}\quad {\mathcal H}^q\big(j_*{\mathbf Q}_U^H[n]\big)=R^qj_*\mathscr{O}_U.\end{align*} $$

In particular, these $\mathscr {D}_X$ -modules carry canonical Hodge filtrations; note that these only depend on the reduced subscheme $Z_{\mathrm {red}}$ . Moreover, the long exact sequence of cohomology corresponding to (2.8) gives the counterparts of (2.6) and (2.7) in this setting: an exact sequence of filtered $\mathscr {D}_X$ -modules

(2.9) $$ \begin{align} 0\to \mathscr{O}_X\to j_* \mathscr{O}_U \to {\mathcal H}_Z^1(\mathscr{O}_X)\to 0 \end{align} $$

and an isomorphism of filtered $\mathscr {D}_X$ -modules

(2.10) $$ \begin{align} R^qj_* \mathscr{O}_U \simeq{\mathcal H}_Z^{q+1}(\mathscr{O}_U)\quad\text{for all}\quad q\geq 1. \end{align} $$

For example, when $Z = D$ is a reduced effective divisor, we have a canonical Hodge filtration on $\mathscr {O}_X(*D) = j_* \mathscr {O}_U$ , which underlies the Hodge module $j_*{\mathbf Q}_U^H[n]$ ; this is analysed in [Reference Mustaţă and Popa36]. Up to taking the quotient by $\mathscr {O}_X$ , this is therefore equivalent to the Hodge filtration on the local cohomology sheaf ${\mathcal H}^1_D(\mathscr {O}_X)$ . It turns out that for arbitrary Z, the Hodge filtration on the local cohomology sheaves ${\mathcal H}^q_Z(\mathscr {O}_X)$ can be defined using the one on sheaves of the form $\mathscr {O}_X(*D)$ . Note first that it is enough to describe the Hodge filtration in each affine chart U. In this case, if we consider $f_1,\ldots ,f_r\in A=\mathscr {O}_X(U)$ that generate an ideal having the same radical as $\mathcal {I}_Z(U)$ , then each localisation $A_{f_J}$ carries a Hodge filtration, such that the corresponding Čech-type complex (2.4) is a complex of filtered $\mathscr {D}$ -modules. In fact, the maps come from morphisms of mixed Hodge modules: each component is, up to sign, induced by the canonical map

$$ \begin{align*}{j_U}_*{\mathbf Q}^H_U[n]\to {j_V}_*{\mathbf Q}^H_V[n],\end{align*} $$

where $V\subseteq U$ are complements of suitable hypersurfaces and $j_U\colon U\hookrightarrow X$ and $j_V\colon V\hookrightarrow X$ are the inclusion maps. Therefore, the maps in the complex (2.4) are strict and the Hodge filtration on the cohomology sheaves ${\mathcal H}^q_Z(\mathscr {O}_X)$ is the induced filtration.

Remark 2.3. We have

$$ \begin{align*}F_p{\mathcal H}_Z^q(\mathscr{O}_X)=0 \,\,\,\,\,\,\mathrm{for ~all}\,\,\,\,p<0 \,\,\,\,\mathrm{and} \,\,\,\, q\geq 0.\end{align*} $$

Indeed, arguing locally and using the computation of local cohomology via the Čech-type complex (2.4), we deduce this assertion from the fact that if D is a reduced effective divisor on X, then $F_p\mathscr {O}_X(*D)=0$ for $p<0$ (see, for example, [Reference Mustaţă and Popa36, § 9]).

Remark 2.4. Suppose that $Z'\subseteq Z$ is another closed subset. If $U'=X\backslash Z'$ , then j factors as

$$ \begin{align*}U\overset{k}\hookrightarrow U'\overset{j'}\hookrightarrow X.\end{align*} $$

In the derived category of Hodge modules on X, we have an isomorphism

$$ \begin{align*}j_*{\mathbf Q}^H_U[n]\simeq j^{\prime}_*\big(k_*{\mathbf Q}^H_U[n]\big).\end{align*} $$

The canonical morphism ${\mathbf Q}^H_{U'}[n]\to k_*{\mathbf Q}^H_U[n]$ induces therefore

$$ \begin{align*}j^{\prime}_*{\mathbf Q}^H_{U'}[n]\to j_*{\mathbf Q}^H_U[n],\end{align*} $$

so that after passing to cohomology, we get morphisms of Hodge modules

$$ \begin{align*}R^qj^{\prime}_*{\mathbf Q}^H_{U'}[n]\to R^qj_*{\mathbf Q}^H_U[n].\end{align*} $$

We deduce using (2.9) and (2.10) that the canonical morphisms

$$ \begin{align*}\big({\mathcal H}_{Z'}^q(\mathscr{O}_X), F\big)\to \big({\mathcal H}_Z^q(\mathscr{O}_X), F\big)\end{align*} $$

are (strict) morphisms of filtered $\mathscr {D}_X$ -modules.

Remark 2.5. It is not hard to compare the Hodge filtrations on the local cohomology sheaves along Z with respect to two different embeddings in smooth varieties. Indeed, suppose that $k\colon X\hookrightarrow X'$ is a closed embedding, with $X'$ smooth, and $\dim (X)=n$ and $\dim (X')=n+d$ . In this case, if $i\colon Z\hookrightarrow X$ and $i'\colon Z\hookrightarrow X'$ are the two embeddings, we have an isomorphism

$$ \begin{align*}i^{\prime}_*{i'}^!{\mathbf Q}_{X'}[n+d] \simeq \big(k_*i_*i^!{\mathbf Q}_X^H[n]\big)[-d](-d),\end{align*} $$

where we recall that $(-d)$ denotes the Tate twist defined in §2.1. By taking cohomology, we see that for every q, we have an isomorphism of filtered $\mathscr {D}_{X'}$ -modules

$$ \begin{align*}\big({\mathcal H}_Z^{q+d}(\mathscr{O}_{X'}), F\big)\simeq \big(k_+{\mathcal H}^{q}_Z(\mathscr{O}_X)(-d), F\big).\end{align*} $$

Explicitly, if $y_1,\ldots ,y_{n+d}$ are local coordinates on $X'$ , such that X is defined by $(y_1,\ldots ,y_d)$ , then we have an isomorphism

$$ \begin{align*}{\mathcal H}_Z^{q+d}(\mathscr{O}_{X'})\simeq\bigoplus_{\alpha_1,\ldots,\alpha_d\geq 0}{\mathcal H}^q_Z(\mathscr{O}_X)\otimes\partial_{y_1}^{\alpha_1}\cdots\partial_{y_d}^{\alpha_d},\end{align*} $$

such that the Hodge filtrations are related by

$$ \begin{align*}F_p{\mathcal H}_Z^{q+d}(\mathscr{O}_{X'})\simeq\bigoplus_{\alpha_1,\ldots,\alpha_r\geq 0} F_{p-\sum_i\alpha_i}{\mathcal H}^q_Z(\mathscr{O}_X)\otimes\partial_{y_1}^{\alpha_1}\cdots\partial_{y_d}^{\alpha_d}.\end{align*} $$

We end this section with two examples: the case of a smooth subvariety and that of subsets defined by monomial ideals. We note, in addition, that the Hodge filtration on the local cohomology Hodge modules with support in generic determinantal ideals is studied extensively in [Reference Perlman and Raicu45] and [Reference Perlman44].

Example 2.7 (Smooth subvarieties)

If Z is a smooth, irreducible subvariety of X of codimension r, then ${\mathcal H}_Z^q(\mathscr {O}_X)=0$ for $q\neq r$ (see Remark 2.2). We claim that

(2.11) $$ \begin{align} F_p{\mathcal H}^r_Z(\mathscr{O}_X)=\{u\in{\mathcal H}^r_Z(\mathscr{O}_X)\mid\mathcal{I}_Z^{p+1}u=0\}. \end{align} $$

In order to see this, we may assume that X is affine with $A=\mathscr {O}_X(X)$ and we have global algebraic coordinates $x_1,\ldots ,x_n$ on X, such that Z is defined by $I=(x_1,\ldots ,x_r)$ . In this case, as we have previously discussed, $H_I^r(A)$ is filtered isomorphic to the cokernel of the map

$$ \begin{align*}\bigoplus_{i=1}^rA_{x_1\cdots \widehat{x_i}\cdots x_r} \to A_{x_1\cdots x_r}.\end{align*} $$

Moreover, since $x_1\cdots x_r$ defines a SNC divisor on X, the Hodge filtration on $A_{x_1\cdots x_r}$ is given by

$$ \begin{align*}F_pA_{x_1\cdots x_r}=F_p\mathscr{D}_X\cdot\frac{1}{x_1\cdots x_r}\end{align*} $$

(see [Reference Mustaţă and Popa36, § 8]). This is generated over A by the classes of $\frac {1}{x_1^{a_1}\cdots x_r^{a_r}}$ , with $a_1,\ldots ,a_r\geq 1$ , such that $a_1+\cdots +a_r=p+r$ . The fact that this is equal to the right-hand side of (2.11) follows easily from the fact that $x_1,\ldots ,x_r$ form a regular sequence in A (for example, by reducing to the case when A is the polynomial ring ${\mathbf C}[x_1,\ldots ,x_r]$ ).

Another way to obtain this description, relying on the formalism of mixed Hodge modules, is the following. If $i\colon Z\hookrightarrow X$ is the inclusion, then

$$ \begin{align*}i^!{\mathbf Q}_X^H[n]=\big({\mathbf Q}_Z^H[n-r]\big)[-r](-r). \end{align*} $$

Applying $i_*$ and taking cohomology, we get an isomorphism of filtered $\mathscr {D}_X$ -modules

(2.12) $$ \begin{align} {\mathcal H}^r_Z(\mathscr{O}_X)\simeq i_+\mathscr{O}_Z(-r). \end{align} $$

Explicitly, if $x_1,\ldots ,x_n$ are local coordinates on X, such that Z is defined by $(x_1,\ldots ,x_r)$ , then we have an isomorphism

$$ \begin{align*}{\mathcal H}^r_Z(\mathscr{O}_X)\simeq\bigoplus_{\alpha_1,\ldots,\alpha_r\geq 0}\mathscr{O}_Z\otimes\partial_{x_1}^{\alpha_1}\cdots \partial_{x_r}^{\alpha_r}, \end{align*} $$

such that the Hodge filtration is given by

$$ \begin{align*}F_p{\mathcal H}^r_Z(\mathscr{O}_X)\simeq \bigoplus_{\alpha_1+\cdots+\alpha_r\leq p}\mathscr{O}_Z\otimes \partial_{x_1}^{\alpha_1}\cdots \partial_{x_r}^{\alpha_r}.\end{align*} $$

Note that the element $1\otimes 1$ corresponds to the class of $\frac {1}{x_1\cdots x_r}$ in the Čech description of local cohomology, hence, we obtain the same description of the Hodge filtration as in (2.11). For completeness, we note that it also follows from (2.12) that ${\mathcal H}^r_Z(\mathscr {O}_X)$ underlies a pure Hodge module of weight $n+r$ .

Example 2.8 (Monomial ideals)

We now consider the case when $X={\mathbf A}^n$ and $I\subseteq A={\mathbf C}[x_1,\ldots ,x_n]$ is a monomial ideal. The $({\mathbf C}^*)^n$ -action on ${\mathbf A}^n$ induces a $({\mathbf C}^*)^n$ -action on each $H^q_I(A)$ , which translates into a ${\mathbf Z}^n$ -grading on $H^q_I(A)$ . The action of every element of $({\mathbf C}^*)^n$ induces an isomorphism of filtered $\mathscr {D}_X$ -modules $H^q_I(A)\to H^q_I(A)$ , hence, every $F_pH^q_I(A)$ is a graded A-submodule of $H^q_I(A)$ .

Let us denote by $e_1,\ldots ,e_n$ the standard basis of ${\mathbf Z}^n$ . It is easy to see that multiplication by $x_i$ induces an isomorphism

$$ \begin{align*}H^q_I(A)_u\to H^q_I(A)_{u+e_i}\end{align*} $$

for every $u\in {\mathbf Z}^n$ , unless $u_i=-1$ . This follows, for example, by computing the local cohomology via the Čech-type complex associated to a system of reduced monomial generators of $\mathrm {Rad}(I)$ (see [Reference Eisenbud, Mustaţǎ and Stillman10, Theorem 1.1]). Similarly, multiplication by $\partial _i$ induces an isomorphism

(2.13) $$ \begin{align} H^q_I(A)_u\to H^q_I(A)_{u-e_i} \end{align} $$

for every $u\in {\mathbf Z}^n$ , unless $u_i=0$ .

For every $u\in {\mathbf Z}^n$ , we write $x^u$ for the corresponding Laurent monomial. Given $J\subseteq \{1,\ldots ,n\}$ , let $u_J=\sum _{i\in J}e_i$ and $x_J=x^{u_J}$ . If for $u=(u_1,\ldots ,u_n)\in {\mathbf Z}^n$ , we put $\mathrm {neg}(u):=\{i\mid u_i<0\}$ , then $A_{x_J}=\bigoplus _{\mathrm {neg}(u)\subseteq J}{\mathbf C} x^u$ . Furthermore, the explicit description of the Hodge filtration on $A_{x_J}$ that we have seen in Example 2.7 can be rewritten as

$$ \begin{align*}F_pA_{x_J}=\bigoplus_u{\mathbf C} x^u,\end{align*} $$

where the direct sum is over those u with $\mathrm {neg}(u)\subseteq J$ and, such that, $\sum _{i\in \mathrm {neg}(u)}(-u_i-1)\leq p$ . Computing the local cohomology via the Čech-type complex, thus, gives

$$ \begin{align*}F_pH^q_I(A)=\bigoplus_uH^q_I(A)_u,\end{align*} $$

where the sum is over those $u\in {\mathbf Z}^n$ , such that $\sum _{i\in \mathrm {neg}(u)}(-u_i-1)\leq p$ . In particular, we see that

$$ \begin{align*}F_0H^q_I(A)=\bigoplus_uH^q_I(A)_u,\end{align*} $$

where the sum is over all $u\in {\mathbf Z}^n$ , such that $u_i\geq -1$ for all i.

2.4 Birational description and strictness

For us, it will be important to have a description of the Hodge filtration on ${\mathcal H}^q_Z(\mathscr {O}_X)$ via a log resolution of the pair $(X,Z)$ . To this end, it is more convenient to use the corresponding right Hodge modules, with corresponding $\mathscr {D}_X$ -modules ${\mathcal H}^q_Z(\omega _X)$ .

Suppose that $f\colon Y\to X$ is such a resolution; more precisely, we require f to be a projective morphism that is an isomorphism over $U=X\backslash Z$ , such that Y is a smooth variety and the reduced inverse image $f^{-1}(Z)_{\mathrm {red}}$ is a SNC divisor E. If $j'\colon V=Y\backslash E\hookrightarrow Y$ is the inclusion map, then we have a commutative diagram

(2.14)

in which the left vertical map is an isomorphism. We, thus, have an isomorphism

$$ \begin{align*}j_*{\mathbf Q}_U^H[n]\simeq f_*j^{\prime}_*{\mathbf Q}_V^H[n]\end{align*} $$

in the derived category of Hodge modules. As indicated in the paragraph after (2.10), since E is a divisor, the underlying filtered right $\mathscr {D}_Y$ -module of the Hodge module $j^{\prime }_*{\mathbf Q}_V^H[n]$ is $\big (\omega _Y (*E), F\big )$ , where F is the Hodge filtration (see [Reference Mustaţă and Popa36, §8] for a more precise description in the present SNC setting). Taking cohomology in the isomorphism above, we, therefore, obtain isomorphisms of filtered right $\mathscr {D}_X$ -modules

(2.15) $$ \begin{align} \mathcal{H}^q f_+\omega_Y(*E) \simeq R^qj_*\omega_U \simeq \begin{cases} {\mathcal H}^{q+1}_Z(\omega_X) \,\,\,\,\mathrm{for} \,\,\,\, q \ge 1 \\ \\ j_* \omega_U \to\hspace{-0.7em}\to {\mathcal H}^{1}_Z(\omega_X) \simeq j_* \omega_U/ \omega_X \,\,\,\,\mathrm{for} \,\,\,\, q =0, \end{cases} \end{align} $$

where for the last isomorphism, we use (2.9) and (2.10).

On the other hand, we have a filtered resolution of $\omega _Y(*E)$ by induced right $\mathscr {D}_Y$ -modules, given by the complex

(2.16) $$ \begin{align} 0\to\mathscr{D}_Y\to\Omega_Y^1(\log E)\otimes_{\mathscr{O}_Y}\mathscr{D}_Y\to\cdots\to\omega_Y(E)\otimes_{\mathscr{O}_Y}\mathscr{D}_Y\to 0, \end{align} $$

placed in degrees $-n,\ldots ,0$ (see [Reference Mustaţă and Popa36, Proposition 3.1]). Using the notation and discussion in §2.1, by tensoring with $\mathscr {D}_{Y \to X}$ over $\mathscr {D}_Y$ , we, thus, obtain a filtered complex

$$ \begin{align*}A^{\bullet}:\quad 0\to f^*\mathscr{D}_X\to \Omega_Y^1(\log E)\otimes_{\mathscr{O}_Y}f^*\mathscr{D}_X\to\cdots\to\omega_Y(E)\otimes_{\mathscr{O}_Y}f^*\mathscr{D}_X\to 0,\end{align*} $$

which is filtered quasi-isomorphic to $\omega _Y (*E) \overset {\mathbf {L}}{\otimes }_{\mathscr {D}_Y} \mathscr {D}_{Y\to X}$ and, therefore, can be used to compute $f_+ \omega _Y(*E)$ . Note that the filtration on $A^{\bullet }$ is, such that, $F_{p-n}A^{\bullet }$ , for $p \ge 0$ , is the subcomplex

$$ \begin{align*}0\to f^*F_{p-n}\mathscr{D}_X\to \Omega_Y^1(\log E)\otimes_{\mathscr{O}_Y}f^*F_{p-n+1}\mathscr{D}_X\to\cdots\to\omega_Y(E)\otimes_{\mathscr{O}_Y}f^*F_p\mathscr{D}_X\to 0.\end{align*} $$

A detailed exposition of all this can be found in [Reference Mustaţă and Popa36, §2 and 3].

Now by (2.15) and the definition of $\mathscr {D}$ -module pushforward, for all $q \ge 0$ , we have

$$ \begin{align*}R^qj_* \omega_U \simeq R^qf_*A^{\bullet}\end{align*} $$

as right $\mathscr {D}_X$ -modules. The filtration on the right-hand side is described at the end of §2.1, and since these filtered $\mathscr {D}$ -modules underlie Hodge modules, the strictness property of the Hodge filtration in (2.3) leads to the following key Hodge-theoretic consequence for this birational interpretation:

Proposition 2.9. With the above notation, for every $p,q\geq 0$ , the inclusion $F_{p-n}A^{\bullet }\hookrightarrow A^{\bullet }$ induces an injective map

$$ \begin{align*}R^qf_*F_{p-n}A^{\bullet}\hookrightarrow R^qf_*A^{\bullet}\simeq R^qj_*\omega_U,\end{align*} $$

whose image is $F_{p-n}R^qj_*\omega _U$ . Moreover, via (2.15), this image coincides with $F_{p-n} {\mathcal H}^{q+1}_Z(\omega _X)$ for $q \ge 1$ , while for $q=0$ , its quotient by $\mathscr {O}_X$ gives $F_{p-n} {\mathcal H}^1_Z(\omega _X)$ .

As a concrete example, we obtain the following description of the lowest term of the filtration (note that by (2.1) and Remark 2.3, we have $F_p{\mathcal H}_Z^q(\omega _X)=0$ for all q and all $p<-n$ ):

Corollary 2.10. For every $q\geq 2$ , we have a canonical isomorphism

$$ \begin{align*}F_{-n}{\mathcal H}^q_Z(\omega_X)\simeq R^{q-1}f_*\omega_Y(E).\end{align*} $$

Moreover, we have a short exact sequence

$$ \begin{align*}0\to\omega_X\to f_*\omega_Y(E)\to F_{-n}{\mathcal H}^1_Z(\omega_X)\to 0.\end{align*} $$

2.5 An injectivity theorem

We continue to use the notation in the previous section. Using the exact sequence

$$ \begin{align*}0 \to \omega_Y \to \omega_Y(E) \to \omega_E \to 0\end{align*} $$

and the Grauert-Riemenschneider vanishing theorem, Corollary 2.10 tells us that for all $q \ge 1$ , as a consequence of strictness for the Hodge filtration, we have an isomorphism

$$ \begin{align*}\gamma_q \colon R^{q-1} f_* \omega_E \to F_{-n}{\mathcal H}^q_Z(\omega_X).\end{align*} $$

We also consider the inclusion maps

$$ \begin{align*}i_q \colon F_{-n}{\mathcal H}^q_Z(\omega_X) \hookrightarrow {\mathcal H}^q_Z(\omega_X).\end{align*} $$

Theorem A in the Introduction follows, therefore, once the following compatibility is established:

Proposition 2.12. For each q, the composition

$$ \begin{align*}i_q \circ \gamma_q \colon R^{q-1} f_* \omega_E \to {\mathcal H}^q_Z(\omega_X)\end{align*} $$

coincides with the morphism on cohomology described in the statement of Theorem A.

Proof. For simplicity, we write down the argument for $q \ge 2$ , when $ R^{q-1} f_* \omega _E \simeq R^{q-1} f_* \omega _Y(E)$ . The argument for $q =1$ is similar, only this time, by definition, one needs to consider $f_* \omega _Y(E) / \omega _Y$ (as opposed to $f_* \omega _Y(E)$ ).

Step 1. We consider the commutative diagram (2.14), and we identify U and V via the left vertical map. Applying $\mathbf {R} f_*$ to the canonical inclusion $\omega _Y (E) \hookrightarrow j^{\prime }_* \omega _U \simeq \omega _Y (*E)$ and passing to cohomology, we obtain morphisms

(2.17) $$ \begin{align} R^{q-1} f_* \omega_Y (E) \to R^{q-1} j_* \omega_U \simeq {\mathcal H}^q_Z (\omega_X) \end{align} $$

for each q, where the last isomorphism is the canonical isomorphism in (2.7). We claim that these morphisms can be identified with the compositions $i_q \circ \gamma _q$ .

To see this, recall that we have identified

$$ \begin{align*}{\mathcal H}^q_Z (\omega_X) \simeq {\mathcal H}^{q-1}f_+ \omega_Y (*E) : = R^{q-1} f_* \big( \omega_Y (*E) \overset{\mathbf{L}}{\otimes}_{\mathscr{D}_Y} \mathscr{D}_{Y\to X} \big).\end{align*} $$

The transfer module admits a canonical morphism $\mathscr {D}_Y \to \mathscr {D}_{Y \to X}$ of left $\mathscr {D}_Y$ -modules (induced by $T_Y \to f^* T_X$ ), which, in turn, induces a morphism

$$ \begin{align*}\rho_q \colon R^{q-1} f_* \omega_Y(*E) \to {\mathcal H}^{q-1}f_+ \omega_Y (*E).\end{align*} $$

Now the morphism $i_q \circ \gamma _q$ , defined using the resolution (2.16) of $\omega _Y(*E)$ , is obtained more precisely by pushing forward the inclusion $\omega _Y (E) \hookrightarrow \omega _Y (*E)$ , and then considering the composition

$$ \begin{align*}R^{q-1} f_* \omega_Y(E) \longrightarrow R^{q-1} f_* \omega_Y(*E) \overset{\rho_q }{\longrightarrow} {\mathcal H}^{q-1}f_+ \omega_Y (*E).\end{align*} $$

But since $f \circ j' = j$ , $\rho _q$ can also be identified canonically with the isomorphism $R^{q-1} j_* \omega _U \to {\mathcal H}^{q-1} j_+ \omega _U$ appearing above, and we are done.

Step 2. In this step, we discuss the following general situation: assume that W is a closed subscheme in X, with ideal sheaf ${\mathcal J}$ . If $j \colon V \hookrightarrow X$ is the inclusion map of the complement $V = X \backslash W$ , we have a diagram of exact triangles

where the vertical map on the right is the canonical morphism, and where the middle vertical map can be described as the canonical morphism

(2.18) $$ \begin{align} \mathbf{R} \mathcal{H} om_{\mathscr{O}_X} ({\mathcal J} , \omega_X) \to \mathbf{R} j_* j^* \mathbf{R} \mathcal{H} om_{\mathscr{O}_X} ({\mathcal J} , \omega_X)=\mathbf{R} j_*\omega_V, \end{align} $$

since ${\mathcal J}_{|V} \simeq \mathscr {O}_V$ .

Moreover, in the presence of a proper morphism $f \colon Y \to X$ , assumed to be an isomorphism over V, we can consider the ideal ${\mathcal J}_Y : = {\mathcal J} \cdot \mathscr {O}_Y$ defining $f^{-1} (W)$ , and, similarly, we have a canonical morphism

$$ \begin{align*}\mathbf{R} \mathcal{H} om_{\mathscr{O}_Y} ({\mathcal J}_Y ,f^{!} \omega_X) \to \mathbf{R} j^{\prime}_*((f^{!} \omega_X)_{|V}) = \mathbf{R} j^{\prime}_* \omega_V,\end{align*} $$

where $j'$ is the inclusion of V in Y. Applying $\mathbf {R} f_*$ to this morphism, and Grothendieck duality to the first term, we obtain a diagram

where we included the natural factorisation of the bottom morphism through the object $\mathbf {R} \mathcal {H} om_{\mathscr {O}_X} ({\mathcal J} , \omega _X)$ , induced by the canonical morphism ${\mathcal J} \to \mathbf {R} f_* {\mathcal J}_Y$ ; this factorisation holds due to the description of the morphism in (2.18) and the fact that Grothendieck duality is compatible with restriction to open subsets.

Step 3. In this step, we apply the constructions in Step 2 to give another description of the morphism (2.17), which will finish the proof. First, (2.17) can be rewritten as the composition

$$ \begin{align*}R^{q-1} f_* \mathbf{R}\mathcal{H} om_{\mathscr{O}_Y}\big(\mathscr{O}_Y (-E), \omega_Y \big) \to R^{q-1} f_* \mathbf{R}\mathcal{H} om_{\mathscr{O}_Y} \big(\mathcal{I}_{f^{-1}(Z)}, \omega_Y \big) \to R^{q-1} j_* \omega_U,\end{align*} $$

where the factorisation through the middle term holds since E is the reduced structure on $f^{-1} (Z)$ . Applying Grothendieck duality, this composition can be rewritten as

$$ \begin{align*}\mathscr{E} xt^{q-1}_{\mathscr{O}_X} \big(\mathbf{R} f_* \mathscr{O}_Y (-E), \omega_X \big) \to \mathscr{E} xt^{q-1}_{\mathscr{O}_X} \big(\mathbf{R} f_* \mathcal{I}_{f^{-1}(Z)}, \omega_X \big) \to R^{q-1} j_* \omega_U,\end{align*} $$

and the map on the right factors further through $\mathscr {E} xt^{q-1} \big ( \mathcal {I}_Z, \omega _X \big )$ , as described in the last diagram in Step 2. Moreover, it is straightforward to see that for $q \ge 2$ , we have canonical isomorphisms

$$ \begin{align*}\mathscr{E} xt^{q-1}_{\mathscr{O}_X} \big(\mathbf{R} f_* \mathscr{O}_Y (-E), \omega_X \big) \simeq \mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathbf{R} f_* \mathscr{O}_E , \omega_X \big)\end{align*} $$

and

$$ \begin{align*}\mathscr{E} xt^{q-1}_{\mathscr{O}_X} \big(\mathcal{I}_Z, \omega_X \big) \simeq \mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathscr{O}_Z , \omega_X \big).\end{align*} $$

Altogether, the morphism (2.17) can be identified with the natural composition

$$ \begin{align*}\mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathbf{R} f_* \mathscr{O}_E , \omega_X \big) \to \mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathscr{O}_Z , \omega_X \big) \to {\mathcal H}^q_Z (\omega_X),\end{align*} $$

which is the same as the map on cohomology described in the statement of Theorem A (note that Grothendieck duality gives $ \mathbf {R} f_* \omega _E^{\bullet } \simeq \mathbf {R} \mathcal {H} om_{\mathscr {O}_X} (\mathbf {R} f_* \mathscr {O}_E, \omega _X[n])$ , while $\omega _Z^{\bullet } \simeq \mathbf {R} \mathcal {H} om_{\mathscr {O}_X} (\mathscr {O}_Z, \omega _X[n])$ ).

The upshot of Theorem A (and Proposition 2.12) is that for each $q\ge 1$ , we have a commutative diagram

where the left vertical map is the isomorphism in Corollary 2.10, the bottom horizontal map is the inclusion, the right vertical map is the natural map to local cohomology, while $\alpha _q$ is the morphism on cohomology induced by

$$ \begin{align*}\alpha \colon \mathbf{R} f_* \omega_E^{\bullet} \to \omega_Z^{\bullet}\end{align*} $$

in $\mathbf {D}^b\big (\mathrm {Coh} (X)\big )$ , obtained, in turn, by dualising the natural morphism $\mathscr {O}_Z \to \mathbf {R} f_* \mathscr {O}_E$ (here, $\omega _E^{\bullet } = \omega _E [n -1]$ since E is Gorenstein).

As a consequence, the top horizontal map is injective. In other words, this gives another proof of the injectivity result of Kovács and Schwede, Corollary B in the Introduction, applied in [Reference Kovács and Schwede29] to the study of deformations of Du Bois singularities.

2.6 A local vanishing theorem

We use the constructions in §2.4 to prove a Nakano-type vanishing result for log resolutions of arbitrary closed subsets. This generalises a result for hypersurfaces due to Saito [Reference Saito52, Corollary 3] (cf. also [Reference Mustaţă and Popa36, Theorem 32.1]).

Proof of Theorem D

We note that by the definition of c and by Remark 2.2, we have

(2.19) $$ \begin{align} {\mathcal H}_Z^{j}(\omega_X)=0\quad\text{if either}\quad j<r\quad\text{or}\quad j>c. \end{align} $$

The vanishing in the statement holds trivially for $p>n$ , hence, we may assume $p\leq n$ . Note that, in this case, the conditions in both 1 and 2 imply $q\geq 1$ . We first check the case $p =n$ .Footnote ² This follows in both cases 1 and 2 from (2.19) and Corollary 2.10.

We next prove the theorem by descending induction on p. Let $p < n$ , and consider the complex

$$ \begin{align*}C^{\bullet} : = F_{-p} A^{\bullet} [p -n],\end{align*} $$

placed in cohomological degrees $0,\ldots ,n-p$ , where $A^{\bullet }$ is as in § 2.4. Note that since $p+q\geq n+1$ , we deduce from Proposition 2.9 that

$$ \begin{align*}R^{p+q-n} f_* F_{-p} A^{\bullet} \hookrightarrow R^{p+q-n}f_* A^{\bullet} \simeq {\mathcal H}_Z^{p+q-n+1} (\omega_X)\quad\text{for all}\quad q,\end{align*} $$

hence, using again (2.19), we see that in both cases 1 and 2, we have

(2.20) $$ \begin{align} R^q f_* C^{\bullet} = R^{p+q-n} f_* F_{-p} A^{\bullet}=0. \end{align} $$

Consider now the hypercohomology spectral sequence

$$ \begin{align*}E_1^{i,j}=R^jf_*C^i\Rightarrow R^{i+j}f_*C^{\bullet}.\end{align*} $$

By definition, we have

$$ \begin{align*}C^i= \Omega_Y^{p+i}(\log E)\otimes_{\mathscr{O}_Y}f^*F_i\mathscr{D}_X\quad\text{for}\quad 0\leq i\leq n-p,\end{align*} $$

hence, we want to show that $E_1^{0,q}=0$ . First, (2.20) implies that in both cases 1 and 2, we have $E_{\infty }^{0,q}=0$ .

Now for every $k \ge 1$ , we clearly have $E_k^{-k,q+k-1}=0$ , since this is a first-quadrant spectral sequence. On the other hand, we also have $E_k^{k,q-k+1}=0$ by induction. Indeed, this is a subquotient of

$$ \begin{align*}E_1^{k,q-k+1}=R^{q-k+1}f_*C^k=R^{q-k+1}f_*\Omega^{p+k}_Y(\log E)\otimes_{\mathscr{O}_X}F_k\mathscr{D}_X,\end{align*} $$

and the right-hand side vanishes by induction. We, thus, conclude that $E_1^{0,q}=E_{\infty }^{0,q}=0$ , completing the proof.

Remark 2.13. (1) In the statement of the theorem, we may replace $c = \mathrm {lcd}(Z, X)$ by any s, such that Z is locally cut out by s equations. Indeed, the fact that $c\leq s$ follows from Remark 2.2.

(2) There exist further useful upper bounds on c that depend only on the codimension r of Z. We only list a couple here. In complete generality, Faltings [Reference Faltings13] showed that

$$ \begin{align*}c \le n - \left[ \frac{n-1}{r} \right].\end{align*} $$

Among many other improvements, Huneke and Lyubeznik [Reference Huneke and Lyubeznik21] showed that if Z is normal, then

$$ \begin{align*}c \le n - \left[ \frac{n}{r + 1} \right] - \left[ \frac{n-1}{r + 1} \right].\end{align*} $$

Further results along these lines, assuming $S_k$ conditions on Z, appear in [Reference Dao and Takagi8]. We obtain, in particular:

Corollary 2.14. Let Z be a closed subscheme of codimension r in a smooth, irreducible n-dimensional variety X. If $f\colon Y \to X$ is a log resolution of $(X, Z)$ , which is an isomorphism away from Z, and $E = f^{-1}(Z)_{\mathrm {red}}$ , then

$$ \begin{align*}R^qf_* \Omega_Y^p (\log E) = 0 \,\,\,\,\,\,\mathrm{for} \,\,\,\, p + q \ge 2n - \left[ \frac{n-1}{r} \right].\end{align*} $$

If, moreover, Z is assumed to be normal, then the same holds for

$$ \begin{align*}p + q \ge 2n - \left[ \frac{n}{r + 1} \right] - \left[ \frac{n-1}{r + 1} \right].\end{align*} $$

We conclude by noting that in [Reference Mustaţă and Popa36, Theorem 32.1], it is shown that when Z is a Cartier divisor, in order to have local vanishing as in Theorem D, it is enough to assume only that X is smooth away from Z. It is, therefore, natural to ask:

3.1 Order and Ext filtration

We now aim to define analogues of the pole order filtration associated to hypersurfaces and compare them with the Hodge filtration. We thank C. Raicu, whose answers to our questions have helped shape the material in this section. We start by recording the following basic property of the Hodge filtration:

Proposition 3.1. For every $p,q\geq 0$ , we have

$$ \begin{align*}\mathcal{I}_Z\cdot F_p{\mathcal H}_Z^q(\mathscr{O}_X)\subseteq F_{p-1}{\mathcal H}_Z^q(\mathscr{O}_X).\end{align*} $$

In particular, we have

$$ \begin{align*}\mathcal{I}_Z^{p+1}F_p{\mathcal H}^q_Z(\mathscr{O}_X)=0\quad\text{for every}\quad p\geq 0.\end{align*} $$

Definition 3.3. The order filtration on ${\mathcal H}^q_Z(\mathscr {O}_X)$ is the increasing filtration given by

$$ \begin{align*}O_k{\mathcal H}^q_Z(\mathscr{O}_X):=\{u\in {\mathcal H}^q_Z(\mathscr{O}_X)\mid \mathcal{I}_Z^{k+1}u=0\},\,\,\,\,\,\, k \ge 0.\end{align*} $$

Note that we have a canonical isomorphism

$$ \begin{align*}O_k{\mathcal H}^q_Z(\mathscr{O}_X) \simeq {\mathcal H} om_{\mathscr{O}_X} \big(\mathscr{O}_X /\mathcal{I}_Z^{k+1}, {\mathcal H}^q_Z(\mathscr{O}_X)\big).\end{align*} $$

If Z is a reduced divisor, then

$$ \begin{align*}O_k{\mathcal H}^1_Z(\mathscr{O}_X) = \mathscr{O}_X \big((k+1)Z\big) / \mathscr{O}_X,\end{align*} $$

and the inclusions $F_k \mathscr {O}_X(*Z) \subseteq \mathscr {O}_X \big ((k+1)Z\big )$ in Remark 3.2 say that

$$ \begin{align*}F_k{\mathcal H}^1_Z(\mathscr{O}_X) \subseteq O_k{\mathcal H}^1_Z(\mathscr{O}_X) \,\,\,\,\,\,\mathrm{for~all} \,\,\,\,k \ge 0.\end{align*} $$

This last fact continues to be true, in general:

Proposition 3.4. For arbitrary Z, and for every k and q, we have

$$ \begin{align*}F_k{\mathcal H}^q_Z(\mathscr{O}_X) \subseteq O_k{\mathcal H}^q_Z(\mathscr{O}_X).\end{align*} $$

Remark 3.5. The order filtration on ${\mathcal H}^q_Z(\mathscr {O}_X)$ is compatible with the filtration on $\mathscr {D}_X$ by order of differential operators:

$$ \begin{align*}F_\ell \mathscr{D}_X \cdot O_k{\mathcal H}^q_Z(\mathscr{O}_X) \subseteq O_{k + \ell} {\mathcal H}^q_Z(\mathscr{O}_X)\quad\text{for all}\quad k,\ell\geq 0.\end{align*} $$

However, unless $q = r = \mathrm {codim}_X(Z)$ , the sheaves $O_k{\mathcal H}^q_Z(\mathscr {O}_X)$ are not coherent as long as ${\mathcal H}^q_Z(\mathscr {O}_X)\neq 0$ . This makes the order filtration less suitable for $q> r$ ; this is a rather deep result of Lyubeznik [Reference Lyubeznik32, Corollary 3.5].Footnote ³ For $q = r$ , the situation is better (see Proposition 3.11 below).

We also consider a related filtration. Recall that a well-known characterisation of local cohomology (see [Reference Hartshorne15, Theorem 2.8]) is

$$ \begin{align*}{\mathcal H}_Z^q(\mathscr{O}_X) = \underset{\underset{k}{\longrightarrow}}{\mathrm{lim}} ~\mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{I}_Z^k, \mathscr{O}_X\big),\end{align*} $$

where the morphisms

(3.1) $$ \begin{align} \mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{I}_Z^k, \mathscr{O}_X\big) \longrightarrow \mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{I}_Z^{k+1}, \mathscr{O}_X\big) \end{align} $$

between the terms in the direct limit are induced from the short exact sequence

$$ \begin{align*}0 \longrightarrow \mathcal{I}_Z^k /\mathcal{I}_Z^{k+1} \longrightarrow \mathscr{O}_X /\mathcal{I}_Z^{k+1} \longrightarrow \mathscr{O}_X/\mathcal{I}_Z^k \longrightarrow 0.\end{align*} $$

Definition 3.6. The Ext filtration on ${\mathcal H}^q_Z(\mathscr {O}_X)$ is the increasing filtration given by

$$ \begin{align*}E_k{\mathcal H}^q_Z(\mathscr{O}_X):=\mathrm{Im} ~\big[ \mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{I}_Z^{k+1}, \mathscr{O}_X\big) \to {\mathcal H}_Z^q(\mathscr{O}_X) \big],\,\,\,\,\,\, k \ge 0.\end{align*} $$

Remark 3.8. If $q = r = \mathrm {codim}_X(Z)$ , then

$$ \begin{align*}E_k{\mathcal H}^q_Z(\mathscr{O}_X) = \mathscr{E} xt^r_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{I}_Z^{k+1}, \mathscr{O}_X\big),\end{align*} $$

that is, the maps in the above definition are injective. Indeed, in this case, the maps in (3.1) are all injective, since

$$ \begin{align*}\mathscr{E} xt^{r-1}_{\mathscr{O}_X} \big(\mathcal{I}_Z^k /\mathcal{I}_Z^{k+1} , \mathscr{O}_X\big) =0.\end{align*} $$

This last fact follows from the following well known (see, e.g. [Reference Bănică and Stănăşilă3, Proposition 1.17]):

Lemma 3.9. If $\mathscr {F}$ is a coherent sheaf on a smooth variety, then

$$ \begin{align*}\mathscr{E} xt^i_{\mathscr{O}_X} (\mathscr{F}, \mathscr{O}_X) = 0, \,\,\,\,\,\,\mathrm{for~ all}\,\,\,\,i < \mathrm{codim}~\mathrm{Supp}(\mathscr{F}).\end{align*} $$

In terms of comparing these two natural filtrations, we clearly have

(3.2) $$ \begin{align} E_k{\mathcal H}^q_Z(\mathscr{O}_X) \subseteq O_k{\mathcal H}^q_Z(\mathscr{O}_X), \,\,\,\,\,\, \mathrm{for~all} \,\,\,\, k \ge 0. \end{align} $$

Let’s try to understand the inclusion (3.2) more canonically. According to the proof of [Reference Huneke and Koh20, Proposition 3.1(i)], for any ideal sheaf $\mathcal {J}\subseteq \mathscr {O}_X$ , such that $\mathrm {Supp} \left (\mathscr {O}_X/ \mathcal {J} \right ) = Z$ , there is a spectral sequence

(3.3) $$ \begin{align} E^{p,q}_2 = \mathscr{E} xt^p_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{J}, {\mathcal H}^q_Z(\mathscr{O}_X)\big) \implies H^{p+q} = \mathscr{E} xt^{p+q}_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{J}, \mathscr{O}_X\big), \end{align} $$

which is simply the spectral sequence of the composition of the functors ${\mathcal H}^0_Z (-)$ and ${\mathcal H} om_{\mathscr {O}_X} (\mathscr {O}_X/\mathcal {J}, \cdot )$ , since

$$ \begin{align*}{\mathcal H} om_{\mathscr{O}_X} \big(\mathscr{O}_X/\mathcal{J}, {\mathcal H}^0_Z(\mathcal{M})\big) \simeq {\mathcal H} om_{\mathscr{O}_X} (\mathscr{O}_X/\mathcal{J}, \mathcal{M}),\end{align*} $$

for every $\mathscr {O}_X$ -module $\mathcal {M}$ , and ${\mathcal H}^0_Z (-)$ takes injective objects to injective objects.

Hence, taking $\mathcal {J} = \mathcal {I}_Z^{k+1}$ , in general, the picture is this: $O_k{\mathcal H}^q_Z(\mathscr {O}_X)$ is the $E^{0,q}_2$ -term of this spectral sequence, we have $E^{0,q}_\infty \hookrightarrow E^{0,q}_2$ (as there are no nontrivial differentials coming into $E^{0,q}_r$ ), while $E^{0,q}_\infty $ is a quotient of $H^q = \mathscr {E} xt^{q}_{\mathscr {O}_X} \big (\mathscr {O}_X/ \mathcal {I}_Z^{k+1}, \mathscr {O}_X\big )$ , which is identified with $E_k{\mathcal H}^q_Z(\mathscr {O}_X)$ .

The drawbacks for both the order and the Ext filtration disappear when $q =r=\mathrm {codim}_X(Z)$ due to the following:

Proof. Using the spectral sequence (3.3), since ${\mathcal H}^q_Z(\mathscr {O}_X) = 0$ for $q < r$ , for any ideal sheaf $\mathcal {J}\subseteq \mathscr {O}_X$ , such that $\mathrm {Supp} \left (\mathscr {O}_X/ \mathcal {J} \right ) = Z$ , we have

$$ \begin{align*}{\mathcal H} om_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{J}, {\mathcal H}^r_Z(\mathscr{O}_X)\big) \simeq \mathscr{E} xt^r_{\mathscr{O}_X} \big(\mathscr{O}_X/ \mathcal{J}, \mathscr{O}_X\big).\end{align*} $$

The statement follows again by taking $\mathcal {J} = \mathcal {I}_Z^{k+1}$ (see also Remark 3.8).

In particular, for $q =r$ , Proposition 3.4 can be reinterpreted as saying that

(3.4) $$ \begin{align} F_k{\mathcal H}^r_Z(\mathscr{O}_X) \subseteq E_k{\mathcal H}^r_Z(\mathscr{O}_X) \,\,\,\,\,\,\mathrm{for ~all}\,\,\,\,k\ge0. \end{align} $$

A natural question, potentially interesting for the study of the singularities of Z, is whether this extends to higher values of q as well.

We will answer this question positively for $k =0$ and all q in Proposition 3.15 below. However, we first note that in the smooth case, we, indeed, have equality between all three filtrations, as expected. If Z is a smooth, irreducible subvariety of X of codimension r, then ${\mathcal H}_Z^q(\mathscr {O}_X)=0$ for $q\neq r$ (just as for any local complete intersection; see Remark 2.2).

Example 3.12. If Z is a smooth, irreducible subvariety of X of codimension r, then

$$ \begin{align*}F_k{\mathcal H}^r_Z(\mathscr{O}_X) = O_k {\mathcal H}_Z^r(\mathscr{O}_X) = E_k {\mathcal H}_Z^r (\mathscr{O}_X) \,\,\,\,\,\,\mathrm{for~all}\,\,\,\,k \ge 0.\end{align*} $$

Indeed, we have seen the first equality in Example 2.7 and the second equality follows from Proposition 3.11.

Remark 3.13. We will see in Corollary 3.26 below that if Z is a singular local complete intersection in X, of pure codimension r, then $F_k{\mathcal H}^r_Z(\mathscr {O}_X) \neq O_k {\mathcal H}_Z^r(\mathscr {O}_X)$ for $k\gg 0$ . However, this can fail beyond the local complete intersection case, even for nice varieties. For example, C. Raicu pointed out to us that if X is the variety of $m\times n$ matrices, with $m>n$ , and Z is the subset consisting of matrices of rank $\leq p$ (so that $\mathrm {codim}_X(Z)=r=(m-p)(n-p)$ , then one can show using [Reference Perlman44, Corollary 1.6] that

$$ \begin{align*}F_k{\mathcal H}^r_Z(\mathscr{O}_X) = O_k {\mathcal H}_Z^r(\mathscr{O}_X)\quad\text{for all}\quad k\in{\mathbf Z}.\end{align*} $$

Remark 3.14. Another way of thinking about the isomorphism

$$ \begin{align*}F_0 {\mathcal H}_Z^r(\mathscr{O}_X) \simeq \mathscr{E} xt^r_{\mathscr{O}_X} (\mathscr{O}_Z, \mathscr{O}_X) \simeq \omega_Z \otimes \omega_X^{-1}\end{align*} $$

when Z is smooth is in terms of the description in Corollary 2.10. Indeed, considering the log resolution of $(X, Z)$ to be the blow up of X along Z, it translates into the isomorphism $R^{r-1}f_* \omega _E \simeq \omega _Z$ , which is well known.

As promised, in general, we have a positive answer to Question 3.5 for the lowest piece of the Hodge filtration.

Proof. Equivalently, the statement says that there is an inclusion

$$ \begin{align*}F_{-n}{\mathcal H}^q_Z(\omega_X) \subseteq E_0{\mathcal H}^q_Z(\mathscr{O}_X)\otimes_{\mathscr{O}_X}\omega_X= \mathrm{Im} ~\big[ \mathscr{E} xt^q_{\mathscr{O}_X} \big(\mathscr{O}_Z, \omega_X\big)\to {\mathcal H}_Z^q(\omega_X) \big].\end{align*} $$

This is an immediate consequence of Theorem A, in which we established the existence of commutative diagrams

(3.6)

where the diagonal map is injective, identified with the inclusion $F_{-n}{\mathcal H}^q_Z(\omega _X) \hookrightarrow {\mathcal H}^q_Z(\omega _X)$ via Corollary 2.10.

Remark 3.16 (Normal schemes)

If Z is normal of codimension r, then the inclusion

$$ \begin{align*}F_0 {\mathcal H}_Z^r(\mathscr{O}_X) \subseteq E_0 {\mathcal H}_Z^r(\mathscr{O}_X) \simeq \omega_Z \otimes \omega_X^{-1}\end{align*} $$

has an alternative interpretation: since Z is normal, the dualising sheaf

$$ \begin{align*}\omega_Z=\mathscr{E} xt^r_{\mathscr{O}_X} (\mathscr{O}_Z, \mathscr{O}_X) \otimes \omega_X\end{align*} $$

and the canonical sheaf $i_* \omega _U$ , where $i\colon U\hookrightarrow Z$ is the inclusion of the smooth locus of Z, are isomorphic. Since $\omega _Z \otimes \omega _X^{-1}$ is reflexive, and coincides with $F_0 {\mathcal H}_Z^r(\mathscr {O}_X)$ on U, the conclusion follows.

3.2 The lowest term and Du Bois singularities

When Z is a reduced divisor, it is known that the pair $(X,Z)$ is log canonical if and only if Z has du Bois singularities (see [Reference Kovács and Schwede28, Corollary 6.6]). On the other hand, the condition of being log canonical is equivalent to the equality $F_0 \mathscr {O}_X(*Z) = P_0 \mathscr {O}_X(*Z)$ , where $P_\bullet $ is the pole order filtration or, in other words, $I_0 (Z) = \mathscr {O}_X$ in the language of Hodge ideals (see [Reference Mustaţă and Popa36, Corollary 10.3]).

We now show that Theorem A (and the discussion in §2.5), together with a criterion for Du Bois singularities due to Steenbrink and Schwede, imply that, in general, if Z has Du Bois singularities, then $F_0 {\mathcal H}_Z^q(\mathscr {O}_X) = E_0 {\mathcal H}_Z^q(\mathscr {O}_X)$ for all q; moreover, the converse holds if Z is Cohen-Macaulay.

Proof of Theorem C

It follows from work of Steenbrink [Reference Steenbrink59] (see Theorem 5.1 below) that Z has Du Bois singularities if and only if the canonical morphism $\mathscr {O}_Z\to \mathbf {R} f_*\mathscr {O}_E$ is an isomorphism (see also Schwede’s [Reference Schwede55, Theorem 4.6] for a more general criterion). Via duality, this is equivalent to the map

$$ \begin{align*}\alpha \colon \mathbf{R} f_* \omega_E^{\bullet} \to \omega_Z^{\bullet}\end{align*} $$

in the statement of Theorem A being an isomorphism, hence, to the horizontal map in (3.6) being an isomorphism for each q. This shows the first assertion.

Under the extra assumption of Z being Cohen-Macaulay of pure codimension r, we have

$$ \begin{align*}\mathscr{E} xt^q_{\mathscr{O}_X}(\mathscr{O}_Z,\omega_X) = 0 \,\,\,\,\mathrm{for} \,\,\,\,q \neq r,\end{align*} $$

so by Proposition 3.15, we also have $F_0{\mathcal H}^q_Z(\mathscr {O}_X) = 0$ for all $q \neq r$ . Now as explained in Remark 3.8, for $q =r$ , the vertical map in (3.6) is injective, so $F_{-n}{\mathcal H}^r_Z(\omega _X)= E_{-n}{\mathcal H}^r_Z(\omega _X)$ is equivalent (cf. Proposition 2.12) to the canonical map

$$ \begin{align*}\mathscr{E} xt^r_{\mathscr{O}_X}(\mathbf{R} f_*\mathscr{O}_E,\omega_X)\to \mathscr{E} xt^r_{\mathscr{O}_X}(\mathscr{O}_Z,\omega_X)\end{align*} $$

being an isomorphism. Since all the other $\mathscr {E} xt$ sheaves are zero, it follows (using Grothendieck duality) that the morphism $\alpha $ is an isomorphism, which, as noted, is equivalent to Z being Du Bois.

On a related note, the following corollary of Theorem A recovers [Reference Ma, Schwede and Shimomoto33, Theorem B] in the case when the ambient space X is smooth (see also Remark 3.19 below for the general case of this result).

Corollary 3.18. If $Z\subseteq X$ is a closed subscheme with Du Bois singularities, then the natural maps

$$ \begin{align*}{\mathcal Ext}^q_{\mathscr{O}_X}(\mathscr{O}_Z,\mathscr{O}_X) \to {\mathcal H}^q_Z(\mathscr{O}_X)\end{align*} $$

are injective for all q.

Proof. As above, if Z is Du Bois, the natural morphism $\mathscr {O}_Z \to \mathbf {R} f_* \mathscr {O}_E$ is an isomorphism, and, therefore, the canonical maps

$$ \begin{align*}\mathscr{E} xt^q_{\mathscr{O}_X}(\mathbf{R} f_*\mathscr{O}_E,\omega_X)\to \mathscr{E} xt^q_{\mathscr{O}_X}(\mathscr{O}_Z,\omega_X)\end{align*} $$

are isomorphisms for each q. The result then follows from Theorem A.

Remark 3.19. The question of when this injectivity holds is asked in [Reference Eisenbud, Mustaţǎ and Stillman10] (see [Reference Ma, Schwede and Shimomoto33] for further discussion and applications). In fact, as L. Ma has pointed out, one can deduce from Corollary 3.18 the full statement of [Reference Ma, Schwede and Shimomoto33, Theorem B] (in which the ambient variety X is only assumed to be Gorenstein). Indeed, we may assume that $X=\mathrm {Spec}(R)$ and $Z=\mathrm {Spec}(S)$ are affine, with $S=R/I$ . The assertion in the corollary implies via [Reference Dao, De Stefani and Ma7, Proposition 2.1] that S is i-cohomologically full for every i (equivalently, in the language of [Reference Kollár and Kovács26], S has liftable local cohomology). This implies that for every maximal ideal ${\mathfrak m}$ of R and every i and k, the natural map $H_{{\mathfrak m}}^i(R/I^k)\to H^i_{{\mathfrak m}}(R/I)$ is surjective. If R is Gorenstein, then $\omega _{R_{{\mathfrak m}}}\simeq R_{\mathfrak m}$ , and local duality implies that the natural map

$$ \begin{align*}\mathrm{Ext}_R^{n-i}(R/I,R)\to \mathrm{Ext}_R^{n-i}(R/I^k,R)\end{align*} $$

is injective, where $n=\mathrm {dim}(R_{\mathfrak m})$ . By taking the direct limit over k, we obtain the injectivity of

$$ \begin{align*}\mathrm{Ext}_R^{n-i}(R/I,R)\to H_I^{n-i}(R).\end{align*} $$

We conclude this section by noting that a study of the equality $F_1 {\mathcal H}_Z^q(\mathscr {O}_X) = E_1 {\mathcal H}_Z^q(\mathscr {O}_X)$ should also be very interesting. Recall that in [Reference Mustaţă and Popa36, Theorem C], it is shown that for a reduced hypersurface D, the equality $F_1 \mathscr {O}_X(*D) = P_1 \mathscr {O}_X(*D)$ (or, equivalently, $I_1 (D) = \mathscr {O}_X$ ) implies that D has rational singularities. By analogy we make the following:

We will show in Lemma 3.23 below that in the setting of the conjecture, the condition $F_1=E_1$ implies that $F_0=E_0$ also holds. We also note that the assertion in the conjecture follows from the stronger Conjecture 3.31 below. It is an interesting question whether a similar condition on the Hodge filtration on all local cohomology sheaves ${\mathcal H}^q_Z(\mathscr {O}_X)$ would imply the fact that Z has rational singularities for any reduced Z.

3.3 The case of local complete intersections

All throughout this section, Z is assumed to be a local complete intersection subscheme of X, of pure codimension r, defined by the ideal $\mathcal {I}_Z$ . In this case, we have ${\mathcal H}^q_Z(\mathscr {O}_X)=0$ for $q\neq r$ by Remark 2.2. We have seen in Propositions 3.4 and 3.11 that

$$ \begin{align*}F_k{\mathcal H}^r_Z(\mathscr{O}_X)\subseteq E_k {\mathcal H}^r_Z(\mathscr{O}_X) = O_k{\mathcal H}^r_Z(\mathscr{O}_X)\end{align*} $$

for all $k \ge 0$ .

We start by giving more precise descriptions of the order and Ext filtrations; even though they coincide, each of the two filtration provides interesting information. We denote by $\mathscr {N}_{Z/X}$ the normal sheaf $(\mathcal {I}_Z/\mathcal {I}_Z^2)^{\vee }$ .

Lemma 3.21. For every $k\geq 0$ , the quotient

$$ \begin{align*}\mathrm{Gr}_k^E{\mathcal H}^r_Z(\mathscr{O}_X)=E_k {\mathcal H}^r_Z(\mathscr{O}_X)/E_{k-1} {\mathcal H}^r_Z(\mathscr{O}_X)\end{align*} $$

is a locally free $\mathscr {O}_Z$ -module; in fact, it is isomorphic to $\mathrm {Sym}^{k}(\mathscr {N}_{Z/X})\otimes \omega _Z\otimes \omega _X^{-1}$ .

Proof. By Remark 3.8, we have $E_k{\mathcal H}^r_Z(\mathscr {O}_X) = {\mathcal Ext}_{\mathscr {O}_X}^r(\mathscr {O}_X/\mathcal {I}_Z^{k+1},\mathscr {O}_X)$ , and, moreover, each exact sequence

$$ \begin{align*}0\to \mathcal{I}_Z^{k}/\mathcal{I}_Z^{k+1}\to\mathscr{O}_X/\mathcal{I}_Z^{k+1}\to\mathscr{O}_X/\mathcal{I}_Z^{k}\to 0\end{align*} $$

induces an exact sequence

$$ \begin{align*}0\to {\mathcal Ext}^r_{\mathscr{O}_X}(\mathscr{O}_X/\mathcal{I}_Z^{k},\mathscr{O}_X)\to {\mathcal Ext}^r_{\mathscr{O}_X}(\mathscr{O}_X/\mathcal{I}_Z^{k+1},\mathscr{O}_X)\to {\mathcal Ext}^r_{\mathscr{O}_X}(\mathcal{I}_Z^{k}/\mathcal{I}_Z^{k+1},\mathscr{O}_X)\to 0.\end{align*} $$

We, thus, see that

$$ \begin{align*}\mathrm{Gr}_k^E{\mathcal H}^r_Z(\mathscr{O}_X)\simeq {\mathcal Ext}^r_{\mathscr{O}_X}(\mathcal{I}_Z^{k}/\mathcal{I}_Z^{k+1},\mathscr{O}_X)\end{align*} $$

$$ \begin{align*}\simeq \mathrm{Sym}^k(\mathscr{N}_{Z/X}^{\vee})^{\vee}\otimes {\mathcal Ext}^r_{\mathscr{O}_X}(\mathscr{O}_Z,\omega_X)\otimes\omega_X^{-1} \simeq \mathrm{Sym}^k(\mathscr{N}_{Z/X})\otimes \omega_Z\otimes\omega_X^{-1}.\\[-38pt] \end{align*} $$

Furthermore, working locally, we may assume that Z is the closed subscheme of X defined by $f_1,\ldots ,f_r\in \mathscr {O}_X(X)$ , with $f_1,\ldots ,f_r$ forming a regular sequence at every point of Z. Given this, an easy computation shows that as in the case of smooth subvarieties in Example 2.7, we have:

By analogy with the case of hypersurfaces [Reference Mustaţă and Popa36], one of the main questions to understand is when, given $p\geq 0$ , we have

$$ \begin{align*}F_k{\mathcal H}_Z^r(\mathscr{O}_X)= O_k{\mathcal H}^r_Z(\mathscr{O}_X) \,\,\,\,\,\,\mathrm{for} \,\,\,\,k\leq p.\end{align*} $$

We first note that it suffices to check the equality for $k = p$ :

Lemma 3.23. If $F_p {\mathcal H}_Z^r(\mathscr {O}_X)= O_p {\mathcal H}^r_Z(\mathscr {O}_X)$ , then

$$ \begin{align*}F_k {\mathcal H}_Z^r(\mathscr{O}_X)= O_k {\mathcal H}^r_Z(\mathscr{O}_X) \,\,\,\,\,\,\mathrm{for ~all}\,\,\,\,k \le p.\end{align*} $$

Proof. It suffices to check this for $k = p-1$ . We use the notation $F_k$ and $O_k$ for simplicity. Note first that by Propositions 3.1 and 3.4, we have

$$ \begin{align*}I_Z\cdot O_p = I_Z \cdot F_p \subseteq F_{p -1} \subseteq O_{p-1}.\end{align*} $$

On the other hand, a brief inspection of the concrete description of $O_k$ given in Lemma 3.22 shows that $\mathcal {I}_Z \cdot O_p = O_{p-1}$ . We conclude by combining these two facts.

The next lemma shows that this question regarding the comparison between the Hodge and order filtration is interesting only if we assume that Z is reduced.

Proof. Let $O^{\prime }_k{\mathcal H}^r_Z(\mathscr {O}_X)$ be the order filtration on ${\mathcal H}^r_Z(\mathscr {O}_X)$ corresponding to $Z_{\mathrm {red}}$ . Since we have the inclusions

$$ \begin{align*}F_0{\mathcal H}^r_Z(\mathscr{O}_X)\subseteq O^{\prime}_0{\mathcal H}^r_Z(\mathscr{O}_X)\subseteq O_0 {\mathcal H}^r_Z(\mathscr{O}_X),\end{align*} $$

it is enough to show that $O^{\prime }_0{\mathcal H}^r_Z(\mathscr {O}_X)\neq O_0{\mathcal H}^r_Z(\mathscr {O}_X)$ .

Note that Z is Cohen-Macaulay, being a local complete intersection; since it is not reduced, it is not generically reduced. After restricting to a suitable open subset, we may, thus, assume that X is affine, with coordinates $x_1,\ldots ,x_n$ , such that the ideal of $Z_{\mathrm {red}}$ is generated by $x_1,\ldots ,x_r$ , and if we denote by $f_1,\ldots ,f_r$ the generators of $\mathcal {I}_Z$ , and write $f_i=\sum _{j=1}^ra_{i,j}x_j$ , then $\mathrm {det}(a_{i,j})\in (x_1,\ldots ,x_r)$ . The assertion in the lemma follows from the fact that via the isomorphisms

$$ \begin{align*}{\mathcal H}_Z^r(\mathscr{O}_X)\simeq\mathscr{O}(X)_{f_1\cdots f_r}/\sum_{i=1}^r\mathscr{O}(X)_{f_1\cdots\widehat{f_i}\cdots f_r}\simeq \mathscr{O}(X)_{x_1\cdots x_r}/\sum_{i=1}^r\mathscr{O}(X)_{x_1\cdots\widehat{x_i}\cdots x_r}\end{align*} $$

given by the Čech-complex description in §2.2, the class of $\frac {1}{x_1\cdots x_r}$ , which generates $O^{\prime }_0{\mathcal H}_Z^r(\mathscr {O}_X)$ , corresponds to $\mathrm {det}(a_{i,j})\frac {1}{f_1\cdots f_r}$ . On the other hand, $O_0{\mathcal H}^r_Z(\mathscr {O}_X)$ is generated by the class of $\frac {1}{f_1\cdots f_r}$ , hence, it is different from $O^{\prime }_0{\mathcal H}_Z^r(\mathscr {O}_X)$ .

Before stating the next result, we note that a stronger bound will be obtained in Theorem 3.39, however, with much more work; the simple argument here is sufficient for establishing Corollary 3.26.

Proposition 3.25. If Z is not smooth, then

$$ \begin{align*}F_k{\mathcal H}_Z^r(\mathscr{O}_X)\subsetneq O_k{\mathcal H}^r_Z(\mathscr{O}_X)\quad\text{for every}\quad k\geq n-r+1.\end{align*} $$

Proof. We may assume that X is affine and $\mathcal {I}_Z$ is generated by $f_1,\ldots ,f_r$ . Since Z is not smooth, it follows that there is a point $Q\in Z$ defined by the ideal $\mathcal {I}_Q$ , such that, after possibly renumbering and replacing $f_1$ by a linear combination of $f_1,\ldots ,f_r$ , we have $f_1\in \mathcal {I}_Q^2$ . We now need to appeal to a result that will be proved later, Theorem 4.2, saying that the Hodge filtration on ${\mathcal H}_Z^r(\mathscr {O}_X)$ is generated at level $n-r$ . If $k\geq n-r+1$ , we, thus, have

$$ \begin{align*}F_k{\mathcal H}_Z^r(\mathscr{O}_X)\subseteq F_1\mathscr{D}_X\cdot F_{k-1}{\mathcal H}_Z^r(\mathscr{O}_X).\end{align*} $$

Recall that $O_{k-1}{\mathcal H}_Z^r(\mathscr {O}_X)$ is generated by the classes of $\frac {1}{f_1^{a_1}\cdots f_r^{a_r}}$ , with $a_i\geq 1$ for all i and $\sum _ia_i\leq k-1+r$ . Moreover, using the fact that $f_1,\ldots ,f_r$ form a regular sequence in $\mathscr {O}_{X,Q}$ , it is easy to see that these elements form a minimal system of generators of $O_{k-1}{\mathcal H}_Z^r(\mathscr {O}_X)$ at Q. A similar assertion holds for $O_k{\mathcal H}_Z^r(\mathscr {O}_X)$ .

Since $f_1\in \mathcal {I}_Q^2$ , a straightforward calculation shows that

$$ \begin{align*}F_k{\mathcal H}_Z^r(\mathscr{O}_X)\subseteq F_1\mathscr{D}_X\cdot O_{k-1}{\mathcal H}_Z^r(\mathscr{O}_X)\subseteq \mathcal{I}_Q\cdot\frac{1}{f_1^{k+1}f_2\cdots f_r}+\sum_{a_1,\ldots,a_r} \mathscr{O}_X\cdot\frac{1}{f_1^{a_1}\cdots f_r^{a_r}},\end{align*} $$

where the last sum is over those $a_1,\ldots ,a_r$ , such that $a_i\geq 1$ for all i, with the inequality being strict for some $i\geq 2$ , and, such that, $\sum _ia_i=k+r$ . This shows that $F_k{\mathcal H}_Z^r(\mathscr {O}_X)$ is a proper subset of $O_k{\mathcal H}_Z^r(\mathscr {O}_X)$ at Q.

In particular, the coincidence of the two filtrations characterises smoothness, similar to [Reference Mustaţă and Popa36, Theorem A] for hypersurfaces (for another approach to the same result, see Theorem 3.39 below).

We formalise the discussion above by introducing a measure of singularities that will figure prominently in the study of the Du Bois complex of Z.

Definition 3.27. The singularity level of the Hodge filtration on $\mathcal {H}^r_Z \mathscr {O}_X$ is

$$ \begin{align*}p (Z) := \mathrm{sup}\{~k ~| ~ F_k \mathcal{H}^r_Z \mathscr{O}_X = O_k \mathcal{H}^r_Z \mathscr{O}_X \},\end{align*} $$

with the convention that $p (Z) = -1$ if equality never holds.

Remark 3.28. The invariant $p(Z)$ only depends on Z and not on the embedding in a smooth, ambient variety. In order to see this, let us temporarily denote by $p(Z\hookrightarrow X)$ the invariant corresponding to a closed embedding in a smooth variety X. Given a closed immersion of smooth varieties $X\hookrightarrow Y$ , by Remark 2.5, we have

$$ \begin{align*}p(Z\hookrightarrow X)=p(Z\hookrightarrow X\hookrightarrow Y).\end{align*} $$

Suppose next that Z is affine, and consider closed immersions $i\colon Z\hookrightarrow X$ and $i'\colon Z\hookrightarrow X'$ , with X and $X'$ smooth and affine. In order to show that $p(Z\hookrightarrow X)=p(Z\hookrightarrow X')$ , by the identity above, we may assume that