\to R^1\pi_*\Omega_{X'}^j({\rm log}\,D')\to 0.$$

Note that $\alpha$ is the composition

$$\Omega^j_X({\rm log}\,G)\to \Omega_H^{j-1}({\rm log}\,D\vert_H)\to \Omega_H^{j-1}\big({\rm log}\,(D\vert_H+W)\big),$$

where the first map is surjective and the second one is injective. We thus see that

$$\pi_*\Omega^j_{X'}({\rm log}\,D')\simeq \Omega^j_X({\rm log}\,D)\quad\text{and}\quad R^1\pi_*\Omega_{X'}^j({\rm log}\,D')\simeq
\Omega_W^{j-2}({\rm log}\,D\vert_W).$$

Since $\pi$ has fibers of dimension $\leq 1$, we deduce that $R^i\pi_*\Omega_{X'}^j({\rm log}\,D')=0$ for $i\geq 2$, which completes the proof for $r=2$.

Suppose now that $r\geq 3$, and that we know the assertion for $r-1$. Pushing forward the exact sequence (<ref>)
and using (<ref>) together with the inductive assumption, we first see that for $i\geq 2$, we have

$$R^i\pi_*\Omega^j_{X'}({\rm log}\,D')\simeq R^{i-1}\varphi_*\Omega^{j-1}_{\widetilde{H}}({\rm log}\,D'\vert_{\widetilde{H}})
\begin{array}{cl}
\Omega_Z^{j-r}({\rm log}\,D\vert_Z), & \text{if}\quad i=r-1; \\[2mm]
0, & \text{otherwise}.
\end{array}\right.

We also get an exact sequence

$$0\to \pi_*\Omega_{X'}^j({\rm log}\,D')\to \Omega_X^j({\rm log}\,G)\overset{\beta}\longrightarrow
\Omega_Z^{j-1}({\rm log}\,D\vert_H)\to R^1\pi_*\Omega_{X'}^j({\rm log}\,D')\to 0.$$

We thus conclude that

$$ \pi_*\Omega_{X'}^j({\rm log}\,D')\simeq {\rm ker}(\beta)\simeq \Omega_X^j({\rm log}\,D)\quad\text{and}\quad
R^1\pi_*\Omega_{X'}^j({\rm log}\,D')\simeq {\rm coker}(\beta)=0.$$

This completes the proof of the lemma.

We may and will assume that $X$ is affine, and fix a linear system of divisors on $X$ obtained by restricting
a complete very ample linear system on a projective completion of $X$.
Suppose first that $r<n$ and that $H$ is a general element in our linear system such that $H\cap\overline{\{\xi\}}$
is nonempty. If we knew the assertions in the theorem for $g$ and for the irreducible components of $H\cap\overline{\{\xi\}}$
(which have codimension $r$ in $H$), then we would get using Lemma <ref> that $R^{r-1}f_*\Omega_Y^j(\log E)\cdot \shO_H$
vanishes at some point of $H\cap\overline{\{\xi\}}$ for all $j$ (we assume $j\leq r-1$ if $\overline{\{\xi\}}$ is an irreducible component of $Z$). By Nakayama's lemma, we then deduce that $R^{r-1}f_*\Omega_Y^j(\log E)$ vanishes at some point of $\overline{\{\xi\}}$ and thus its stalk at $\xi$ is $0$.

After repeating this several times, we reduce to the case when $\xi$ is a closed point, hence $r=n$. Since the case $j=n$ follows from
Corollary <ref>, we only need to prove

\begin{equation}\label{eq_case_closed_point}
R^{n-1}f_*\Omega_Y^j(\log E)=0\quad\text{for all}\quad j<n.
\end{equation}

Suppose that we have a sequence of morphisms:

$$X_N\overset{g_N}\longrightarrow X_{N-1}\overset{g_{N-1}}\longrightarrow\cdots\overset{g_2}\longrightarrow X_1\overset{g_1}\longrightarrow X_0=X$$

with the following properties:

i) For every $i$ with $0\leq i\leq N-1$, we have a smooth irreducible subvariety $Z_i$ of $X_i$ of codimension $\geq 2$ such that $g_{i+1}$ is the blow-up of $X_i$ along $Z_i$,
with exceptional divisor $E_{i+1}$;
in particular, $X_1,\ldots,X_N$ are smooth.

ii) For every $i$ with $1\leq i\leq N$, if

$$D_i=\big(g_i^*(D_{i-1})+E_i\big)_{\rm red}$$

(with the convention $D_0=0$), then $D_i$ has SNC, and also has SNC
with $Z_i$ for $i\leq N-1$.

iii) All $Z_i$ lie inside  the inverse image of $Z$.

For every $i$, let us denote by $f_i$ the composition $X_i\to X$.
In order to prove (<ref>), it is enough to show by induction on $N\geq 0$ that under these assumptions

$$R^{n-1}(f_{N})_*\Omega_{X_N}^j({\rm log}\,D_N)=0 \,\,\,\,\,\,{\rm for} \,\,\,\, 0\leq j\leq n-1.$$

Indeed, we can find a sequence of morphisms as above
such that $f_N$ gives a log resolution of $(X, Z)$ as in the statement of the theorem. Since the sheaf $R^{n-1}f_*\Omega_Y^j({\rm log}\,E)$ is independent of the resolution
(see Remark <ref>), we obtain the desired vanishing.

The case $N=0$, when $f_N$ is the identity, is trivial, hence we only need to prove the induction step.
There are two cases to consider. If $Z_{N-1}\subseteq {\rm Supp}(D_{N-1})$, then it follows from
Lemma <ref> that

$$R^{n-1}(f_{N})_*\Omega_{X_N}^j({\rm log}\,D_N)\simeq R^{n-1}(f_{N-1})_*\Omega_{X_{N-1}}^j({\rm log}\,D_{N-1}),$$

hence we are done by induction.

Suppose now that $Z_{N-1}\not\subseteq {\rm Supp}(D_{N-1})$ and consider the Leray spectral sequence

$$E_2^{p,q}=R^p(f_{N-1})_*R^q(g_N)_*\Omega^j_{X_N}({\rm log}\,D_N)\Rightarrow R^{p+q}(f_N)_*\Omega^j_{X_N}({\rm log}\,D_N).$$

It follows from Lemma <ref> that the only possible nonzero terms $E_2^{p,q}$, with $p+q=n-1$, are

$$E_2^{n-1,0}=R^{n-1}(f_{N-1})_*\Omega^j_{X_{N-1}}({\rm log}\,D_{N-1})\quad\text{and}$$

$$E_2^{n-r,r-1}=R^{n-r}(f_{N-1})_*\Omega_{Z_{N-1}}^{j-r}({\rm log}\,D_{N-1}\vert_{Z_{N-1}}),$$

where $r={\rm codim}_{X_{N-1}}(Z_{N-1})$.
We see that $E_2^{n-1,0}=0$ by induction. Since $Z_{N-1}\not\subseteq {\rm Supp}(D_{N-1})$, the induced morphism
$Z_{N-1}\to f_{N-1}(Z_{N-1})$ is birational. If $\dim(Z_{N-1})=n-r$ is positive, then this morphism has fibers of dimension $\leq n-r-1$,
hence $E_2^{n-r,r-1}=0$. On the other hand, if $r=n$, then $j-r<0$, hence again $E_2^{n-r,r-1}=0$. We thus conclude from the spectral sequence
that $R^{n-1}(f_{N})_*\Omega_{X_N}^j({\rm log}\,D_N)=0$, completing the proof of the theorem.

Cohomological dimension $n-3$.
The pattern of the previous two paragraphs continues for one more step, by means of the following result
of Dao-Takagi <cit.>, conjectured and proved in a more restrictive setting by Varbaro <cit.>.

If $\xi\in Z$ is a point with ${\rm codim}_X(\xi)=r$ and ${\rm depth}(\shO_{Z,\xi}) \ge 3$, then ${\rm lcd}_{\xi}(X, Z) \le r-3$.

We will point out below how one can use Theorem <ref> in order to give another proof of this theorem at a closed isolated singular point, as part of studying a more general related question (see Conjecture <ref>  and Theorem <ref>  below) that will also be useful in studying the Du Bois complex of $Z$.

Optimal bounds.
It is known that the ideal pattern

$${\rm depth}(\shO_Z) \ge k  \implies {\rm lcd}(X, Z) \le n-k$$

stops in fact at $k =3$ with the result above; in characteristic $0$ there are examples of determinantal subschemes $Z \subseteq \AAA^n$ for which ${\rm depth}(\shO_Z) \ge 4$ but ${\rm lcd}(X, Z) \le n-3$, see e.g. <cit.>. Note that things are better in positive characteristic, where Peskine and Szpiro <cit.> showed that indeed for any point $\xi\in Z$, we have  ${\rm lcd}_{\xi}(X, Z) \leq n-{\rm depth}(\shO_{Z,\xi})$.

As a consequence of Theorem <ref> or the surrounding circle of ideas, we do however obtain the ideal result for two classes of subschemes: those with quotient singularities, and those given by monomial ideals. We state the next result here, as it fits well with the current discussion, but for its proof it is useful to first read <ref>.

Let $Z$ be a closed subscheme of codimension $r$, with quotient singularities, in the smooth $n$-dimensional variety $X$.

$${\rm lcd}(X, Z) = n - {\rm depth}(\shO_Z) = r.$$

First, since quotient singularities are rational, hence Cohen-Macaulay, the fact that ${\rm lcd}(X, Z) \geq n - {\rm depth}(\shO_Z) = r$
follows from Remark <ref>. Second, the graded pieces of the
Du Bois complex of quotient singularities have a very simple form. Concretely, by <cit.> we have quasi-isomorphisms

$$ \underline{\Omega}^i_{Z} \simeq \Omega_Z^{[i]}$$

for each $i \ge 0$, where the right hand side is the reflexive hull of $\Omega_Z^i$.
It follows from Corollary <ref> below
that in order to show that ${\rm lcd}(X,Z)\leq r$,
it suffices to show

\begin{equation}\label{eq_cor:quotient}
\shE xt^{j+i +1}_{\shO_X} \big(\Omega_Z^{[i]}, \omega_X\big) = 0\quad\text{for all}\quad j \ge r, i \ge 0.
\end{equation}

But for quotient singularities we have

$${\rm depth} \big(\Omega_Z^{[i]} \big) = \dim Z = n-r,$$

which indeed implies (<ref>).
In order to see this, arguing étale locally, we may assume  that we have a morphism $\pi\colon Y \to Z = Y/G$, where $Y$ is smooth and $G$ is a group acting without quasi-reflections. In this case we have

$$\Omega_Z^{[i]}  \simeq (\pi_* \Omega_Y^i)^G$$

by <cit.>. This in turn is a direct summand of $\pi_* \Omega_Y^i$, since we are in characteristic zero,
and therefore has maximal depth $n-r$.

Corollary <ref> says that the only non-trivial local cohomology sheaf is $\cH^r_Z (\shO_X)$,
even though $Z$ is not necessarily a local complete intersection. Due to results of Ogus, this in turn implies that subvarieties in $\PP^n$ with quotient singularities share other nice features of local complete intersections (cf. <cit.>) regarding the global cohomological dimension ${\rm cd}(\cdot)$ of the complement, or of Barth-Lefschetz-type.

Let $Z\subset \PP^n$ be a closed subscheme of codimension $r$, with quotient singularities. Then

* ${\rm cd}(\PP^n \smallsetminus Z) < 2r -1$.

* The restriction maps $H^i (\PP^n, \CC) \to H^i (Z, \CC)$ are isomorphisms for $i \le n - 2r$.

This holds more generally for any $Z$ which is Cohen-Macaulay and satisfies ${\rm lcd}(\PP^n, Z) = r$, as an immediate consequence of <cit.>. The second part is stated for algebraic de Rham cohomology in loc. cit.; over $\CC$ this is identified with singular cohomology by <cit.>.

If $Z$ has toroidal singularities, it is also the case that $ \underline{\Omega}^i_{Z} \simeq \Omega_Z^{[i]}$
for all $i$; see <cit.>. However it is known that the depth of $\Omega_Z^{[i]}$
is not always maximal, and therefore the argument above does not go through.[It is known that the depth is maximal for all $i$ in the case of simplicial toric varieties, but these are exactly the toric varieties that have quotient singularities.] It would be interesting to have a complete answer for the local cohomological dimension of such singularities.

We next consider the case of monomial ideals, for which we recover a result of Lyubeznik (see <cit.>).

If $I \subseteq A = \CC [X_1, \ldots, X_n]$ is a radical monomial ideal (i.e. $A/I$ is a Stanley-Reisner
ring), then

$${\rm lcd}(A, I) = n - {\rm depth}(A/I) = {\rm pd} (A/I).$$

According to Example <ref>, the Hodge filtration on each $H^q_I (A)$ is generated at level $0$.
It suffices then to apply Corollary <ref> and Remark <ref>.
Note that in this case $A/I$ has Du Bois singularities by <cit.>,
as Stanley-Reisner rings have $F$-injective (even $F$-pure) type.

Unlike in the case of quotient singularities, Stanley-Reisner rings are not necessarily Cohen-Macaulay, and therefore
$ {\rm pd} (A/I)$ does not coincide in general with ${\rm codim}(I)$.

§ THE DU BOIS COMPLEX AND REFLEXIVE DIFFERENTIALS

Building on Deligne's work constructing mixed Hodge structures on the cohomology of algebraic varieties, Du Bois introduced
in <cit.> a version of the de Rham complex for singular varieties. Given a (reduced) complex algebraic variety $Z$,
the Du Bois complex of $Z$ is an object $\underline{\Omega}_Z^{\bullet}$ in the derived category of filtered complexes on $Z$.
Its (shifted) graded pieces $\underline{\Omega}_Z^p:={\rm Gr}^p_F\underline{\Omega}_Z^{\bullet}[p]$ are objects in the derived category of coherent sheaves on $Z$.
There are canonical morphisms $\Omega_Z^p\to \underline{\Omega}_Z^p$ that are isomorphisms over the smooth locus of $Z$.  The condition that this is an isomorphism everywhere on $Z$ for $p =0$ defines Du Bois singularities, an important class of singularities (see e.g. <cit.> for a survey) that
has already made an appearance in this paper. The complexes $\underline{\Omega}_X^p$ enjoy, in the proper setting, several important properties of the De Rham complex of smooth varieties.
For an introduction to this circle of ideas, see <cit.> and <cit.>.

§.§ The Du Bois complex and local cohomological dimension

This is a preliminary section, in which we record some basic facts involving the Du Bois complex of $Z$, and in particular
reinterpret the vanishing conditions in Theorem <ref> in these terms. A key ingredient is a description, due to Steenbrink, for the graded pieces of the Du Bois complex of $Z$ via a log resolution of an ambient variety. Suppose that $X$ is an irreducible complex algebraic variety and
$Z$ is a reduced closed subscheme of $X$ such that $X\smallsetminus Z$ is smooth (we will shortly assume that $X$ is smooth, but we want to state the first result in this slightly more general setting).
Let $f \colon Y \to X$ be a proper map that is an isomorphism over $X \smallsetminus Z$, with $Y$ smooth, and such that
$E = f^{-1} (Z)_{{\rm red}}$ is an SNC divisor.

With the above notation, for each $p$ we have an isomorphism in ${\bf D}^b \big({\rm Coh} (X)\big)$:

$$\derR f_* \big(\Omega^{n-p}_Y (\log E) (-E)\big) \simeq \underline{\Omega}^{n-p}_{X, Z}.$$

Here $\underline{\Omega}^{n-p}_{X, Z}$ is the $(n-p)$-th du Bois complex of the pair $(X, Z)$,  which sits in an exact triangle

\begin{equation}\label{eqn2}
\underline{\Omega}^{n-p}_{X, Z} \to \underline{\Omega}^{n-p}_{X} \to \underline{\Omega}^{n-p}_{Z} \overset{+1}{\longrightarrow}
\end{equation}

where the other two terms are the usual du Bois complexes.

From now on, we return to our usual assumption that the ambient variety $X$ is smooth.
In this case we of course have
$\underline{\Omega}^{n-p}_{X} \simeq \Omega^{n-p}_{X}$.

We relate $R^q f_* \Omega^p_Y (\log E)$ to the Du Bois complex as follows: by Grothendieck  duality

\begin{equation}\label{GD}
\derR f_* \Omega^p_Y (\log E) \simeq \derR f_*\big(\derR \shH om (\Omega^{n-p}_Y (\log E) (-E), \omega_Y)\big) \simeq
\end{equation}

$$\simeq \derR \shH om \big(\derR f_* \big(\Omega^{n-p}_Y (\log E) (-E)\big), \omega_X\big),$$

and therefore for each $q$ we have

\begin{equation}\label{eqn1}
R^q f_* \Omega^p_Y (\log E) \simeq \shE xt^q_{\shO_X} \big(\derR f_* \big(\Omega^{n-p}_Y (\log E) (-E)\big), \omega_X\big).
\end{equation}

As a consequence, we have:

For every $q \ge 1$ we have

$$R^q f_* \Omega^p_Y (\log E) \simeq \shE xt^{q+1}_{\shO_X} (\underline{\Omega}^{n-p}_{Z}, \omega_X).$$

By ($\ref{eqn1}$), Theorem <ref> and ($\ref{eqn2}$), both sides are isomorphic to
the sheaf $\shE xt^q_{\shO_X} (\underline{\Omega}^{n-p}_{X, Z}, \omega_X)$.

The above lemma leads to a useful equivalent formulation of Theorem <ref>:

For every positive integer $c$, the following are equivalent:

* ${\rm lcd}(X, Z) \le c$.

* $\shE xt^{j+i +1}_{\shO_X}(\underline{\Omega}^i_{Z}, \omega_X) = 0$ for all  $j \ge c$ and $i \ge 0$.

We end this section with the discussion of the example pointed out in Remark <ref>.

Consider the surjection

$$R=\CC[x_1,\ldots,x_4]\to S=\CC[s^4,s^3t,st^3,t^4],$$

with kernel $I$,
and the corresponding embedding $Z={\rm Spec}(S)\hookrightarrow X={\rm Spec}(R)$.
Note that $Z$ is an integral variety, with a unique singular point $0$, defined by the ideal ${\mathfrak m}_0$.
The normalization morphism
$\pi\colon \widetilde{Z}\to Z$ corresponds to the inclusion $S\hookrightarrow \widetilde{S}=\CC[s^4,s^3t,s^2t^2,st^3,t^4]$. Note that
$\widetilde{Z}$ is a toric variety, hence it is Du Bois. The morphism
$\pi$ is an isomorphism over $Z\smallsetminus\{0\}$ and $\pi^{-1}(0)$ consists of one point. It thus follows from <cit.>
that $\underline{\Omega}_Z^p\simeq\pi_*\underline{\Omega}_{\widetilde{Z}}^p$ for all $p$. In particular, we have
$\underline{\Omega}_Z^0\simeq\pi_*\shO_{\widetilde{Z}}$, hence $Z$ is not Du Bois.

On the other hand, we have $F_0H^q_I(R)=E_0H^q_I(R)$ for all $q$. This is clear for $q\neq 2$ since in this case
$H^q_I(R)=0$ (for $q=3$ one can use <cit.>). Using Lemma <ref> and graded local duality,
we can identify the graded Matlis dual of the morphism

\begin{equation}\label{eq_eg_Ma}
F_0H_I^2(R)\hookrightarrow E_0H^2_I(R)
\end{equation}

with the morphism

$$H^2_{{\mathfrak m}_0}(S)\to H^2_{{\mathfrak m}_0}(\widetilde{S}).$$

This is an isomorphism, as can be seen from the long exact sequence of local cohomology associated to

$$0\to S\to \widetilde{S}\to {\mathbf C}\to 0.$$

Therefore (<ref>) is an isomorphism.

§.§ Higher Du Bois singularities for local complete intersections

The Hodge filtration on local cohomology allows us to state and prove a generalization of the vanishing theorem for cohomologies of the Du Bois complex of a hypersurface <cit.>, and of its converse <cit.>, to the case of local complete intersections of arbitrary codimension.

Before proving the main result, Theorem <ref>, we make some preparations.
We begin with a description of the graded pieces of the Du Bois complex in terms of the de Rham complex
of the local cohomology Hodge module (cf. <cit.> for the case of hypersurfaces). This is all we use here, but a stronger result, at the level of the Du Bois complex, was proved by Saito in <cit.>.

If $X$ is a smooth $n$-dimensional complex algebraic variety and $i\colon Z\hookrightarrow X$ is the inclusion map of a closed reduced subscheme $Z$,
then for every integer $p$ we have an isomorphism in ${\bf D}^b \big({\rm Coh} (X)\big)$

$$\underline{\Omega}_Z^p\simeq \derR \shH om_{\shO_X}\big({\rm Gr}^F_{p-n}{\rm DR}_Xi_*i^!\QQ^H_X[n],\omega_X\big)[p].$$

In particular, if $Z$ is a local complete intersection of pure codimension $r$, then

$$\underline{\Omega}_Z^p\simeq \derR \shH om_{\shO_X}\big({\rm Gr}^F_{p-n}{\rm DR}_X\shH_Z^r(\shO_X),\omega_X\big)[p+r].$$

Consider a log resolution
$f\colon Y\to X$ of $(X,Z)$ as in Section <ref>.
Lemma <ref> gives an isomorphism

$${\rm Gr}^F_{p-n}{\rm DR}_X\big(j_*\QQ_U^H[n]\big)\simeq
\derR f_*\Omega_Y^{n-p}({\rm log}\,E)[p].$$

We similarly have an isomorphism

$${\rm Gr}^F_{p-n}{\rm DR}_X\big(\QQ^H_X[n]\big)\simeq\Omega_X^{n-p}[p].$$

Applying $ \derR \shH om_{\shO_X}(\cdot,\omega_X)$ and the isomorphisms (<ref>), we obtain the commutative diagram

\begin{tikzcd}
\derR \shH om_{\shO_X}\big({\rm Gr}^F_{p-n}{\rm DR}_X(j_*\QQ^H_U[n]), \omega_X\big)\dar \rar & \derR \shH om_{\shO_X}\big({\rm Gr}^F_{p-n}{\rm DR}_X\QQ_X^H[n],
\omega_X\big)\dar\\
\derR f_*\big(\Omega_Y^p({\rm log}\,E)(-E)\big)[-p] \rar & \Omega_X^p[-p],
\end{tikzcd}

in which the vertical maps are isomorphisms. On the other hand, the exact triangle (<ref>) in the derived category
of Hodge modules induces the exact triangle

$${\rm Gr}^F_{p-n}{\rm DR}_X\big(i_*i^!\QQ_X^H[n]\big)\longrightarrow {\rm Gr}^F_{p-n}{\rm DR}_X\QQ_X^H[n]\longrightarrow
{\rm Gr}^F_{p-n}{\rm DR}_X\big(j_*\QQ_U^H[n]\big)\overset{+1}\longrightarrow.$$

Applying $\derR\shH om_{\shO_X}(\, \cdot \,, \omega_X)$ and using the above commutative diagram, as well as Theorem <ref>, we obtain
the first assertion in the proposition. The second one is an immediate consequence.

Assume now that $Z$ is a local complete intersection of pure codimension $r$. We recall from the beginning of <ref> that we have

$$F_k \shH^r_Z(\shO_X) \subseteq E_k \shH^r_Z(\shO_X) \,\,\,\,\,\,{\rm  for~all}\,\,\,\, k.$$

It follows that for every $p\geq 0$, if
we denote by ${\rm Gr}^E_{p-n}{\rm DR}_X\shH^r_Z(\shO_X)$
the corresponding graded piece of the de Rham complex of $\shH^r_Z(\shO_X)$ with respect to the filtration $E_{\bullet}$, we have a morphism
of complexes

$$\varphi_p\colon {\rm Gr}^F_{p-n}{\rm DR}_X\shH^r_Z(\shO_X)\to {\rm Gr}^E_{p-n}{\rm DR}_X\shH^r_Z(\shO_X).$$

By definition, if $p(Z)\geq p$, then $\varphi_p$ is an isomorphism. If we only know that $p(Z)\geq p-1$, then
$\varphi_p$ is an injective morphism of complexes, whose cokernel is concentrated in cohomological degree $0$.

Applying $\derR \shH om_{\shO_X}(\,\cdot\, ,\omega_X)$ to $\varphi_p$ and taking $\cH^{i+p+r}(-)$, we obtain

$$\psi^i_p\colon {\mathcal Ext}^{i+p+r}_{\shO_X}\big({\rm Gr}^E_{p-n}{\rm DR}_X\shH^r_Z(\shO_X),\omega_X\big)\to
{\mathcal Ext}^{i+p+r}_{\shO_X}\big({\rm Gr}^F_{p-n}{\rm DR}_X\shH^r_Z(\shO_X),\omega_X\big)\simeq \cH^i(\underline{\Omega}^p_Z),$$

where the isomorphism on the right is provided by Proposition <ref>.

Using the description of ${\rm Gr}^E_{\bullet}\cH^r_Z(\shO_X)$ in Lemma <ref>, it is easy to describe the domain of
$\psi^i_p$. Indeed, note first that $\shE^{\bullet}:={\rm Gr}^E_{p-n}{\rm DR}_X\shH^r_Z(\shO_X)$ is the complex

0\to \Omega_X^{n-p}\otimes\omega_Z\otimes\omega_X^{-1}\to \Omega_X^{n-p+1}\otimes\shN_{Z/X}\otimes\omega_Z\otimes\omega_X^{-1}\to
\cdots\to \Omega_X^n\otimes {\rm Sym}^p(\shN_{Z/X})\otimes\omega_Z\otimes\omega_X^{-1}\to 0

placed in cohomological degrees $-p,\ldots,0$. Recall now that if $\shF$ is a locally free $\shO_Z$-module, then

\begin{equation}\label{eq_Ext}
{\mathcal Ext}_{\shO_X}^\ell(\shF,\omega_X)
=0\quad\text{for}\quad \ell\neq r\quad \text{and}\quad
{\mathcal Ext}^r_{\shO_X}(\shF,\omega_X)\simeq \shF^{\vee}\otimes_{\shO_Z}\omega_Z.
\end{equation}

Let us consider the hypercohomology spectral sequence

$$E_1^{k,\ell}={\mathcal Ext}_{\shO_X}^\ell(\shE^{-k},\omega_X)\Rightarrow {\mathcal Ext}^{k + \ell}_{\shO_X}(\shE^{\bullet},\omega_X).$$

Note that by (<ref>), we have $E_1^{k,\ell}=0$ unless $\ell=r$ and $0\leq k\leq p$; moreover, if $0\leq k\leq p$, then

$$E_1^{k,r}\simeq \Omega_X^k\otimes_{\shO_X}{\rm Sym}^{p-k}(\shN_{Z/X})^{\vee}.$$

The spectral sequence then implies that for every integer $i$ we have

\begin{equation}\label{eq_Ext2}
{\mathcal Ext}_{\shO_X}^{i+p+r}(\shE^{\bullet},\omega_X)=E_2^{i+p,r}=\cH^i({\mathcal C}_p^{\bullet}),
\end{equation}

where ${\mathcal C}_p^{\bullet}$ is the complex

$$0\to {\rm Sym}^p(\shN_{Z/X})^{\vee}\to \Omega_X^{1}\otimes_{\shO_X}{\rm Sym}^{p-1}(\shN_{Z/X})^{\vee}\to\cdots\to
\Omega_X^{p-1}\otimes_{\shO_X}\shN_{Z/X}^{\vee}\to \Omega_X^p\otimes_{\shO_X}\shO_Z\to 0,$$

placed in cohomological degrees $-p,\ldots,0$. Inspection of the maps in this complex shows that, if $p\leq n-r$, then ${\mathcal C}_p^{\bullet}$
is obtained by truncating the generalized Eagon-Northcott complex ${\mathcal D}_{n-r-p}$ associated to the canonical morphism

$$g\colon {\mathcal T}_X\vert_Z\to \shN_{Z/X},$$

keeping the first $p+1$ terms (and suitably translating). For basic facts about
generalized Eagon-Northcott complexes, we refer to <cit.>.

The exact sequence

$$\shN_{Z/X}^{\vee}\to \Omega^1_X\vert_Z\to \Omega^1_Z\to 0$$

implies that we have a canonical isomorphism

\begin{equation}\label{eq_Ext3}
\cH^0({\mathcal C}_p^{\bullet})\simeq \Omega_Z^p.
\end{equation}

Via the isomorphisms (<ref>) and (<ref>), the morphism $\psi^0_p$ gets identified with the morphism
$\Omega_Z^p\to\cH^0(\underline{\Omega}_Z^p)$ induced by the canonical morphism $\Omega_Z^p\to
\underline{\Omega}_Z^p$.

Finally, we recall the fact that generalized Eagon-Northcott complexes exhibit depth-sensitivity. This implies that if
${\rm depth}(I_r(g)\big)\geq k$, for some $1\leq k\leq p$, then $\cH^i({\mathcal C}^{\bullet})=0$ for $-p\leq i\leq -p+k-1$.
Note that in our case, the ideal $I_r(g)$ defines the singular locus $Z_{\rm sing}$ of $Z$ and since $Z$ is Cohen-Macaulay,
the condition is equivalent to ${\rm codim}_Z(Z_{\rm sing})\geq k$. The depth sensitivity of generalized Eagon-Northcott complexes
is a consequence of the behavior in the case of generic matrices (see <cit.>) and of general properties
of perfect modules (see <cit.>). In fact, the depth sensitivity of a more general class of complexes is proved in

After this preparation, we can now prove our result relating the singularity level of the Hodge filtration $p(Z)$ and higher $p$-Du Bois singularities.

We follow the approach in <cit.> and <cit.>, which treat the two implications in the theorem for hypersurfaces.
We may and will assume that $Z$ is singular, since otherwise the equivalence is trivial. In this case, it follows from
Theorem <ref> that $p(Z)\leq\frac{\dim(Z)-1}{2}$, and it is easy to see that we may assume that $p\leq\dim(Z)=n-r$.

Suppose first that $p(Z)\geq p$.
In this case the morphism $\varphi_p$ is an isomorphism, hence so are the morphisms $\psi_p^i$. Since the complex ${\mathcal C}_p^{\bullet}$ is supported
in nonpositive cohomological degrees, we have $\cH^i({\mathcal C}_p^{\bullet})=0$ for $i>0$. It thus follows from
(<ref>) that

$$\cH^i(\underline{\Omega}_Z^p)\simeq {\mathcal Ext}^{i+p+r}_{\shO_X}({\mathcal E}^{\bullet},\omega_X)\simeq \cH^i({\mathcal C}_p^{\bullet})=0
\quad\text{for}\quad i>0.$$

We also see that $\psi_p^0$ induces an isomorphism

$$\Omega_Z^p\simeq \cH^0({\mathcal C}^{\bullet})\simeq {\mathcal Ext}^{p+r}_{\shO_X}({\mathcal E}^{\bullet},\omega_X)\to
\cH^0(\underline{\Omega}_Z^p).$$

This implies that the canonical morphism $\Omega_Z^p\to \underline{\Omega}_Z^p$ is an isomorphism. Since the same argument applies if we replace
$p$ by any $k$, with $0\leq k\leq p$, we conclude that $Z$ has at most higher $p$-Du Bois singularities.

Conversely, suppose now that $Z$ has at most higher $p$-Du Bois singularities. Arguing by induction on $p$, it follows that we may assume that
$p(Z)\geq p-1$. In this case the morphism $\varphi_p$ is injective and its cokernel is

placed in cohomological degree $0$. We need to show that $M=0$.
Since the functor $\derR \shH om_{\shO_X}\big(-,\omega_X[n]\big)$ is a duality, it is enough to show that
$\derR \shH om_{\shO_X}\big(M,\omega_X)=0$, that is, we have

$${\mathcal Ext}^{i}_{\shO_X}(M,\omega_X)=0\quad\text{for all}\quad i\in\ZZ.$$

Let us put

$$\shE^{\bullet}:={\rm Gr}^E_{p-n}{\rm DR}_X\shH^r_Z(\shO_X)\quad \text{and}\quad
\shF^{\bullet}:={\rm Gr}^F_{p-n}{\rm DR}_X\shH^r_Z(\shO_X).$$

The exact triangle

$$\derR \shH om_{\shO_X}(M,\omega_X)\longrightarrow \derR \shH om_{\shO_X}(\shE^{\bullet},\omega_X)\longrightarrow  \derR \shH om_{\shO_X}(\shF^{\bullet},\omega_X)\overset{+1}\longrightarrow$$

induces a long exact cohomology sequence

$$\ldots\to {\mathcal Ext}^{i-1}_{\shO_X}(\shF^{\bullet},\omega_X)\to {\mathcal Ext}^{i}_{\shO_X}(M,\omega_X)\to {\mathcal Ext}^{i}_{\shO_X}(\shE^{\bullet},\omega_X)
\to {\mathcal Ext}^{i}_{\shO_X}(\shF^{\bullet},\omega_X)\to\ldots.$$

As we have seen, the fact that the morphism $\Omega_Z^p\to \underline{\Omega}_Z^p$ is an isomorphism translates as saying
that ${\mathcal Ext}^{i}_{\shO_X}(\shF^{\bullet},\omega_X)=0$ for all $i\neq p+r$ and the morphism

$${\mathcal Ext}^{p+r}_{\shO_X}(\shE^{\bullet},\omega_X)\to {\mathcal Ext}^{p+r}_{\shO_X}(\shF^{\bullet},\omega_X)$$

is an isomorphism. It follows from the long exact sequence that
${\mathcal Ext}_{\shO_X}^{p+r}(M,\omega_X)=0$ and

$${\mathcal Ext}_{\shO_X}^{i}(M,\omega_X)\simeq {\mathcal Ext}_{\shO_X}^{i}(\shE^{\bullet},\omega_X)\quad\text{for all}\quad i\neq p+r.$$

Therefore thanks to (<ref>) we are done if we show that $\cH^i({\mathcal C}_p^{\bullet})=0$ for $i\neq 0$.
This is trivial if $p=0$. On the other hand, if $p\geq 1$, recalling that we are assuming $p(Z)\geq p-1$ by induction,
Corollary <ref> gives

$${\rm codim}_Z(Z_{\rm sing})\geq 2p(Z)+1\geq 2p-1\geq p.$$

The depth-sensitivity of the generalized Eagon-Northcott complexes thus implies that $\cH^i({\mathcal C}_p^{\bullet})=0$ for $i\neq 0$,
which completes the proof of the theorem.

We also note the following corollary of the proof of Theorem <ref>:

Suppose  that $Z$ has higher $p$-du Bois singularities, with $p \ge 1$. Then $Z$ is normal and $\Omega_Z^p$ is reflexive, i.e. $\Omega_Z^p\simeq (\Omega_Z^p)^{\vee\vee}$.

Since ${\rm codim}_Z(Z_{\rm sing})\geq 2p+1\geq 3$
by Corollary <ref> and $Z$ is Cohen-Macaulay, it follows that $Z$ is normal.
In order to see that $\Omega_Z^p$ is reflexive, it is enough then to show that ${\rm depth}(J,\Omega_Z^p)\geq 2$,
where $J$ is the ideal defining $Z_{\rm sing}$ in $Z$. Using again the fact that ${\rm codim}_Z(Z_{\rm sing})\geq 2p+1\geq p$,
it follows that the complex ${\mathcal C}_p^\bullet$ used in the proof of Theorem <ref> gives a locally free resolution of $\Omega_Z^p$. The well-known behavior of depth
in short exact sequences together with the fact that ${\rm depth}(J,{\mathcal C}_p^j)={\rm depth}(J,\shO_Z)\geq 2p+1$ for all $j$ implies

$${\rm depth}(J,\Omega_Z^p)\geq (2p+1)-p=p+1\geq 2.$$

We next give a refinement of the vanishing statement for the higher cohomology of the graded pieces of the Du Bois complex in Theorem <ref>,
in terms of the dimension of the locus where $p(Z)$ is small.

If for some $p  \ge 0$ we have $F_p \shH^r_Z \shO_X = E_p \shH^r_Z \shO_X$ away from a closed subset $W\subseteq Z$ of dimension $s$ (with the convention that $s = -\infty$ if $W$ is empty), then

$$\shH^i(\underline{\Omega}_Z^p) = 0 \,\,\,\,\,\,{\rm for ~all} \,\,\,\,\,\,  0< i < \dim Z - p - s  -1 .$$

We keep the setup used in the proof of Theorem <ref>. Note that by assumption, the morphism

$$\varphi_p\colon \shF^{\bullet}={\rm Gr}^F_{p-n}{\rm DR}_X\shH^r_Z(\shO_X)\to \shE^{\bullet}={\rm Gr}^E_{p-n}{\rm DR}_X\shH^r_Z(\shO_X)$$

is an isomorphism away from $W$.
Recall that by (<ref>), we have

\begin{equation}\label{eq1_thm_van_DB}
{\mathcal Ext}^{j+p+r}_{\shO_X}(\shE^{\bullet},\omega_X)=0\quad\text{for all}\quad j>0.
\end{equation}

On the other hand, if $\shG^{\bullet}$ is a complex on $X$ concentrated in degrees $\leq 0$ such that ${\rm Supp}(\shG^q)\subseteq W$ for all $q$, then
$\shE xt_{\shO_X}^m(\shG^{\bullet},\omega_X)=0$ if $m<n-s$. This can be proved inductively by considering the long exact sequences of
$\shE xt$ sheaves associated to the short exact sequences of complexes

$$0\to \sigma^{\leq j-1}(\shG^{\bullet})\to \sigma^{\leq j}(\shG^{\bullet})\to \shG^j[-j]\to 0$$

and the fact that $\shExt_{\shO_X}^m(\shG^j,\omega_X)=0$ for $m<n-s$ (see Lemma <ref>). Here we denote by
$ \sigma^{\leq j}(\shG^{\bullet})$ the “stupid" truncation of $\shG^{\bullet}$ consisting of the terms placed in cohomological degrees $\leq j$.

This applies in particular to the complexes ${\rm ker}(\varphi)$ and ${\rm coker}(\varphi)$. Since $i+p+r+1<n-s$, we conclude that

\begin{equation}\label{eq2_thm_van_DB}
\shE xt^{i+p+r+1}_{\shO_X}\big({\rm coker}(\varphi), \omega_X\big)=0=\shE xt^{i+p+r}_{\shO_X}\big({\rm ker}(\varphi), \omega_X\big).
\end{equation}

The exact sequences

$$\shE xt^{i+p+r}_{\shO_X}(\shE^{\bullet},\omega_X)\to \shE xt^{i+p+r}_{\shO_X}\big({\rm im}(\varphi),\omega_X\big)\to
\shE xt^{i+p+r+1}_{\shO_X}\big({\rm coker}(\varphi),\omega_X\big)=0$$

$$ \shE xt^{i+p+r}_{\shO_X}\big({\rm im}(\varphi),\omega_X\big)\to
\shE xt_{\shO_X}^{i+p+r}(\shF^{\bullet},\omega_X)\to
\shE xt^{i+p+r}_{\shO_X}\big({\rm ker}(\varphi), \omega_X)=0,$$

together with the vanishing in (<ref>) imply

$$\cH^i(\underline{\Omega}_Z^p)\simeq \shE xt_{\shO_X}^{i+p+r}(\shF^{\bullet},\omega_X\big)=0,$$

where the first isomorphism follows from Proposition <ref>.
This completes the proof of the theorem.

We conclude that there is a range of automatic vanishing in terms of the dimension of the singular locus of $Z$; when $Z$
is a hypersurface, this is <cit.>.

If the singular locus of the local complete intersection $Z$ has dimension $s$, then for all $p \ge 0$ we have

$$\cH^i(\underline{\Omega}_Z^p) =0 \,\,\,\,\,\,{\rm for} \,\,\,\,\,\,1\leq i < \dim Z - s - p -1.$$

Does this result continue to hold when $Z$ is an arbitrary (or at least Cohen-Macaulay) closed subscheme whose singular locus has dimension $s$?

§.§ Depth and local vanishing

We return to the general setting of a reduced closed subscheme $Z$ of a smooth variety $X$.
We have seen in Theorem <ref> that if $n\geq 2$, then
$R^{n-1} f_* \Omega^j_Y (\log E) = 0$ for all $j<n$, and related this to the condition ${\rm lcd}(X, Z) \le n-2$. For lower values of  ${\rm lcd}(X, Z)$ it is also important to understand the
vanishing of $R^{n-2}f_*\Omega_Y^j(\log E)$.  We make the following conjecture based on the depth of $\shO_Z$
at its closed points.

If ${\rm depth} (\shO_Z) \ge k + 2$, then $R^{n-2} f_* \Omega_Y^{n-k} (\log E) = 0$.

Conjecture <ref> holds for $k = 0, 1$.
Indeed, by Corollary <ref> we have an inclusion

$$R^{n-2} f_* \omega_Y (E) \hookrightarrow \shE xt^{n-1}_{\shO_X} (\shO_Z, \omega_X),$$

which gives the assertion for $k =0$. For $k =1$, it is a consequence of Theorem <ref> and Theorem <ref>.

The main result of this section is a proof of Conjecture <ref>
when $Z$ has isolated singularities. Reversing the use of Theorem <ref> in the above remark,
this gives an alternative proof of Theorem <ref> at closed isolated singular points.  Further consequences are explained in <ref>.

If ${\rm depth} (\shO_Z) \ge k + 2$ and $Z$ has isolated singularities, then

$$R^{n-2} f_* \Omega_Y^{n-k} (\log E) = 0.$$

We first  need some preparations, starting with a lemma that will also be used in the next section:

If $Z$ is reduced and $\dim Z \ge 2$, then for any $k \ge 0$ we have an equivalence

$$R^{n-2} f_* \Omega_Y^{n-k} (\log E) = 0 \iff \depth (\underline{\Omega}^k_{Z}) \ge 2,$$

and either condition implies

$$\depth(\cH^0 \underline{\Omega}^k_{Z}) \ge 2.$$

Since $n$ must be at least $3$, by Lemma <ref> we have

$$R^{n-2} f_* \Omega_Y^{n-k} (\log E)  \simeq \shE xt^{n-1} (\underline{\Omega}^k_{Z}, \omega_X),$$

and the vanishing of the latter is equivalent to  $\depth(\underline{\Omega}^k_{Z}) \ge 2$.
For the last statement, we consider the spectral sequence

$$E^{i,j}_2 = \shE xt^i _{\shO_X}(\shH^{-j} \underline{\Omega}^k_{Z}, \omega_X) \implies \shE xt_{\shO_X}^{i + j} (\underline{\Omega}^k_{Z}, \omega_X).$$

Note that $E^{i,j}_2=0$ if $i>n$ or $j>0$; in the latter case, this follows from the
well known fact that $\shH^\ell \underline{\Omega}^k_{Z} = 0$ for $\ell < 0$. The term
$E^{n-1,0}_2 = \shE xt_{\shO_X}^{n-1} (\shH^ 0 \underline{\Omega}^k_{Z}, \omega_X)$ contributes to computing
$\shE xt_{\shO_X}^{n-1} (\underline{\Omega}^k_{Z}, \omega_X)$, and since the differentials at each level are

$$E^{n-1-r , r-1}_r \to  E^{n-1,0}_r \to E^{n-1 + r, 1-r}_r,$$

it follows that $E^{n-1,0}_2 = E^{n-1,0}_{\infty}$. Therefore

$$ \shE xt^{n-1}_{\shO_X} (\shH^ 0 \underline{\Omega}^k_{Z}, \omega_X) =0,$$

which is equivalent to $ \depth(\cH^0 \underline{\Omega}^k_{Z}) \ge 2$.

We now set up some notation under the hypothesis of Theorem <ref>. Since $Z$
has isolated singularities and ${\rm depth}(\shO_Z)\geq 2$, it follows that $Z$ is normal. Without loss of generality, we may assume that $Z$ is irreducible
and $P\in Z$ is a point such that $Z\smallsetminus\{P\}$ is smooth. Let $g\colon \widetilde{Z}\to Z$ be a projective morphism that is an isomorphism over
$Z\smallsetminus\{P\}$, with $\widetilde{Z}$ smooth, and such that $D:=g^{-1}(P)_{\rm red}$ is an SNC divisor.

According to the first statement the Lemma <ref>, in order to prove Theorem <ref>, it suffices to show that under its hypotheses we have

$$\depth (\underline{\Omega}^k_{Z}) \ge 2.$$

We already know that this depth is $\geq 1$, hence it is enough to show that
$H^1_{{\mathfrak m}_P}(\underline{\Omega}^k_{Z})=0$, where ${\mathfrak m}_P$ is the maximal ideal defining $P$.
We may further assume that $Z$ is affine. By Remark <ref>, we may assume that $k\geq 1$.
Using Theorem <ref> and the exact triangle (<ref>), we have an isomorphism

$$\underline{\Omega}_Z^k\simeq \derR g_*\Omega_{\widetilde{Z}}^k({\rm log}\,D)(-D),$$

hence in order to prove the theorem it suffices to show that

\begin{equation}\label{eqn:reduction}
H^1_D\big(\widetilde{Z}, \Omega_{\widetilde{Z}}^k({\rm log}\,D)(-D)\big) = 0.
\end{equation}

The rest of the section is devoted to proving this statement. The proof is independent from the rest of the paper, and involves some basic mixed Hodge theory.

The condition on ${\rm depth}(\shO_Z)$ will be used via the following lemma.

If $Z$ is a variety with isolated singularities and ${\rm depth}(\shO_Z)\geq k+2$, then

$$R^qg_*\shO_{\widetilde{Z}}=0\quad\text{for}\quad 1\leq q\leq k.$$

We follow the approach of <cit.>, but include the proof since in loc. cit. the authors make the stronger assumption that $Z$ is Cohen-Macaulay.[Note that in the Cohen-Macaulay case an even stronger statement appears in <cit.>.] Since $Z$ has isolated singularities, it is easy to see that we may assume that $Z$ is projective. In this case, if $\shL$ is an ample
line bundle on $Z$, then Kawamata-Viehweg vanishing gives

\begin{equation}\label{eq1_conseq_depth}
H^i\big(\widetilde{Z},g^*\shL^{-m}\big)=0\quad\text{for all}\quad i<d=\dim(Z),~ m>0.
\end{equation}

For a given $m>0$, consider the Leray spectral sequence

\begin{equation}\label{eqn:SS}
E_2^{p,q}=H^p\big(Z,R^qg_*\shO_{\widetilde{Z}} \otimes\shL^{-m}\big)\implies H^{p+q}\big(\widetilde{Z},g^*\shL^{-m}\big).
\end{equation}

If $q>0$, then $R^qg_*(\shO_{\widetilde{Z}})$ has $0$-dimensional support, hence $E_2^{p,q}=0$ for $p>0$.
On the other hand, since our assumptions imply that $Z$ is normal, we have $g_* \shO_{\widetilde{Z}}=\shO_Z$, and since
${\rm depth}(\shO_Z)\geq k+2$, it follows that if $m\gg 0$, then for all $p\leq k+1$ we have

$$E_2^{p,0}=H^p\big(Z,\shL^{-m})\simeq H^{d-p}\big(Z, \shL^m \otimes \omega_Z^\bullet [-d]\big)^{\vee}=0.$$

Indeed, for the vanishing on the right, note that the depth hypothesis implies that the object $A^\bullet = \omega_Z^\bullet [-d]$ can have cohomologies only in degrees $0, \ldots, d - k -2$. On the other hand, the hypercohomology group $H^{d-p}\big(Z, \shL^m \otimes \omega_Z^\bullet [-d]\big)$ is computed by a spectral sequence whose contributing $E_2$-terms are
$H^i (Z, \shL^m \otimes\cH^j A^\bullet)$, with $i + j = d - p$. Since $m \gg 0$, the only term that could contribute corresponds to
$i =0$, but in this case $\cH^{d - p} A^\bullet = 0$ since $p \le k+1$.

Going back to the spectral sequence in ($\ref{eqn:SS}$), we keep the assumption that $m\gg 0$.
Note that if $r\geq 2$, then $d_r\colon E_r^{0,q}\to E_r^{r,q-r+1}$
is $0$ unless $r=q+1>k+1$, while
$d_r\colon E_r^{-r,q+r-1}\to E_r^{0,q}$ is always $0$. We thus conclude that if $q\leq k$, then
$E_{\infty}^{0,q}=E_2^{0,q}$ and this vanishes because of (<ref>).
If we assume in addition that $q\geq 1$, then we conclude that $R^qg_*\shO_{\widetilde{Z}}=0$, since this sheaf
has $0$-dimensional support.

Under the assumptions in Lemma <ref>, we have

$$H^i(D,\shO_D)=0\quad\text{for}\quad 1\leq i\leq k.$$

It is shown in the proof of <cit.> that the vanishing of $R^ig_* \shO_{\widetilde{Z}}$ implies
$H^i(D,\shO_D)=0$ (the statement in loc. cit. is for rational singularities, but the proof gives in fact this implication in general).

We can now prove the main result; part of the proof follows an argument in <cit.>.

We write $D=\sum_{i=1}^ND_i$, and define

$$\Omega^k_{D^{(q)}}:=\bigoplus_{i_1<\cdots<i_q}\Omega^k_{D_{i_1}\cap\ldots\cap D_{i_q}}.$$

We then have
an exact complex on $\widetilde{Z}$:

$$C_k^{\bullet}:\,\,\,\,0\to \Omega_{\widetilde{Z}}^k({\rm log}\,D)(-D)\to\Omega_{\widetilde{Z}}^k\to \Omega^k_{D^{(1)}}\to \Omega^k_{D^{(2)}}\to\cdots$$

(for example, the case $k=1$ is treated in <cit.>, but the proof therein extends to arbitrary $k$).

$$\Mmod_k :={\rm ker}\big(\Omega^k_{D^{(1)}}\to \Omega^k_{D^{(2)}}\big).$$

Claim: We have

$$H^1_D\big(\widetilde{Z},\Omega_{\widetilde{Z}}^k({\rm log}\,D)(-D)\big)=0 \iff  H^0(\widetilde{Z}, \Mmod_k)=0.$$

In order to prove this, note first that the short exact sequence

$$0\to \Omega_{\widetilde{Z}}^k({\rm log}\,D)(-D)\to\Omega_{\widetilde{Z}}^k\to\Mmod_k\to 0$$

gives a long exact sequence of local cohomology

$$0=H^0_D(\widetilde{Z},\Omega_{\widetilde{Z}}^k)\to H^0_D(\widetilde{Z},\Mmod_k)\to H^1_D\big(\widetilde{Z}, \Omega_{\widetilde{Z}}^k({\rm log}\,D)(-D)\big)
\to H_D^1(\widetilde{Z},\Omega_{\widetilde{Z}}^k)\to H_D^1(\widetilde{Z},\Mmod_k).$$

Of course, $\Mmod_k$ is supported on $D$, hence

The claim thus follows if we show that the map

\begin{equation}\label{eq1_prop_equiv1}
H_D^1(\widetilde{Z},\Omega_{\widetilde{Z}}^k)\to H_D^1(\widetilde{Z},\Mmod_k)
\end{equation}

is injective. This is essentially shown in the proof of <cit.>, but since it is not stated there in this form, we recall the main steps in the argument.

First, Steenbrink's vanishing theorem (see <cit.>) gives

$$R^{d-1}g_* \big( \Omega_{\widetilde{Z}}^{d-k}({\rm log}\,D)(-D)\big)=0,$$

where $d=\dim(Z)$ (note that $k\leq d-2$). Using the Local Duality theorem and relative duality, we deduce from this that

\begin{equation}\label{eq1_prop_equiv2}
H_D^1\big(\widetilde{Z},\Omega^k_{\widetilde{Z}}({\rm log}\,D)\big)=H_{{\mathfrak m}_P}^1\big(\derR g_*\Omega_{\widetilde{Z}}^k({\rm log}\,D)\big)=0.
\end{equation}

Second, it is shown in <cit.> via an argument using mixed Hodge structures that if

$$\Nmod_k=\Omega^k_{\widetilde{Z}}({\rm log}\,D)/\Omega^k_{\widetilde{Z}}({\rm log}\,D)(-D),$$

then the inclusion $\Mmod_k\hookrightarrow \Nmod_k$ induces an isomorphism $H^0(\widetilde{Z},\Mmod_k)\simeq
H^0(\widetilde{Z},\Nmod_k)$. Note that we have a commutative diagram with exact rows

\begin{tikzcd}
0 \rar & \Omega_{\widetilde{Z}}^k\rar \dar& \Omega_{\widetilde{Z}}^k({\rm log}\,D)\rar\dar & {\mathcal Q}_k\dar{\rm Id}\rar & 0\\
0 \rar & \Mmod_k \rar & \Nmod_k \rar & {\mathcal Q}_k \rar & 0.
\end{tikzcd}

By considering the connecting homomorphisms in the long exact sequences of cohomology and local cohomology, we obtain
the commutative diagram

\begin{tikzcd}
H_D^0(\widetilde{Z},{\mathcal Q}_k) \dar{\alpha}\rar{\beta} & H^1_D(\widetilde{Z}, \Omega_{\widetilde{Z}}^k)\dar{\gamma} \\
H^0(\widetilde{Z},{\mathcal Q}_k)\dar{\rm Id} \rar & H^1(\widetilde{Z}, \Omega_{\widetilde{Z}}^k)\dar{\delta}\\
H^0(\widetilde{Z},{\mathcal Q}_k)\rar{\rho} & H^1(\widetilde{Z},\Mmod_k).
\end{tikzcd}

Note that $\alpha$ is an isomorphism since ${\mathcal Q}_k$ is supported on $D$ and $\beta$ is an isomorphism by (<ref>). Since $H^0(\widetilde{Z},\Mmod_k)\to
H^0(\widetilde{Z},\Nmod_k)$ is an isomorphism, it follows that $\rho$ is injective. We thus conclude from the above diagram that the composition
$\delta\circ\gamma$ is injective and this gets identified with the map (<ref>) since ${\mathcal M}_k$ is supported on $D$.
This completes the proof of the claim.

Our goal is therefore to show that $H^0(\widetilde{Z}, \Mmod_k)=0$.
If we apply $H^q(-)$  to the complex $C_p^{\bullet}$ and ignore the first two terms, we obtain the following complex

$$0\longrightarrow H^q\big(\widetilde{Z}, \Omega^p_{D^{(1)}}\big)\overset{d^{p,q}_1}\longrightarrow H^q\big(\widetilde{Z}, \Omega^p_{D^{(2

With this notation, for every $p$ and $q$, the $(p,q)$ component of the pure Hodge structure on the graded piece
${\rm Gr}^W_{p+q}H^{p+q+i}(D,{\mathbf C})$  is given by

$${\rm Gr}^W_{p+q}H^{p+q+i}(D,{\mathbf C})^{p,q}={\rm ker}(d^{p,q}_{i+1})/{\rm im}(d^{p,q}_i)$$

(with the convention that $d^{p,q}_i=0$ for $i\leq 0$);
see <cit.> for a detailed description of the mixed Hodge structure
on the cohomology of $D$.

We thus see that in order to complete the proof, it is enough to show that

$${\rm Gr}^W_kH^k(D,{\mathbf C})^{k,0}=0.$$

By Hodge symmetry, this is equivalent to

\begin{equation}\label{eq_depth_vanishing}
{\rm Gr}^W_kH^k(D,{\mathbf C})^{0,k}=0.
\end{equation}

On the other hand, we have

$${\rm Gr}_F^0H^k(D,{\mathbf C})\simeq H^k(D,\shO_D)=0,$$

where the vanishing follows from Corollary <ref>.
Since $D$ is compact, we have the identification $W_kH^k(D,\CC)=H^k(D,\CC)$,
so that we have a surjective morphism

$${\rm Gr}_F^0H^k(D,{\mathbf C})\to {\rm Gr}_F^0{\rm Gr}^W_kH^k(D,{\mathbf C})={\rm Gr}^W_kH^k(D,{\mathbf C})^{0,k}.$$

We thus obtain the vanishing in (<ref>) and this completes the proof.

§.§ $h$-differentials and reflexive differentials

The characterization of local cohomological dimension and the local vanishing results in the previous sections have consequences regarding $h$-differentials on singular spaces.  Recall that a recent result of Kebekus-Schnell <cit.>
states that if $Z$ is a variety with rational singularities, then for all $p$ the $h$-differentials ${\Omega_h^p}|_{Z}$ of <cit.>  coincide with the reflexive differentials $\Omega_Z^{[p]} := (\Omega_Z^p)^{\vee \vee}$.

Here we show, using Theorem <ref>, that when $Z$ has isolated singularities this holds under a weaker hypothesis, at least for forms of low degree.

The sheaves of $h$-differentials ${\Omega_h^k}|_{Z}$ are identified in <cit.> with $\cH^0 \underline{\Omega}^k_{Z}$, hence equivalently we will show that the natural morphism

$$\cH^0 \underline{\Omega}^k_{Z} \to \Omega_Z^{[k]}$$

in an isomorphism under the hypothesis of the theorem. Since the two sheaves are isomorphic on the smooth locus of
$Z$, it suffices in turn to show that the sheaf $\cH^0 \underline{\Omega}^k_{Z}$ satisfies the $S_2$ property. Equivalently,
if $\I$ is the ideal sheaf of the singular locus of $Z$, we need to show that

\begin{equation}\label{desire}
\depth (\I, \cH^0 \underline{\Omega}^k_{Z}) \ge 2.
\end{equation}

Since $Z$ has isolated singularities, using Theorem <ref> we deduce that the hypothesis implies
$R^{n-2} f_* \Omega_Y^{n-k} (\log E) = 0$. Lemma <ref> then gives

\depth (\cH^0 \underline{\Omega}^k_{Z}) \ge 2.

By Auslander-Buchsbaum it follows that for each
$x\in Z$, in some open neighborhood of $x$ we can find a resolution with locally free $\shO_X$-modules of finite rank

$$0 \to \shF_{n-2} \to \cdots \to \shF_0 \to \cH^0 \underline{\Omega}^k_{Z} \to 0.$$

Since the singular locus of $Z$ is $0$-dimensional, for each $i$ we have $\depth (\I, \shF_i) = n$.
If we denote

$$\shG_{i-1} : = {\rm Coker} ( \shF_i \to \shF_{i-1}),$$

then it follows using basic properties of depth that $\depth (\I, \shG_{i-1}) \ge i +1$ for $0\leq i\leq n-1$. In particular we obtain that
$\cH^0 \underline{\Omega}^k_{Z} = \shG_0$ satisfies ($\ref{desire}$).
This completes the proof of the theorem.

It is known that if $Z$ is normal, then the isomorphism  ${\Omega_h^k}|_{Z} \simeq \Omega_Z^{[k]}$ implies that
the $k$-forms on $Z_{\rm reg}$ extend to $k$-forms on any resolution of singularities $\pi \colon \widetilde{Z} \to Z$.
This consequence of Theorem <ref> is known in greater generality: van Straten
and Steenbrink <cit.> have shown that if $Z$ is any variety with isolated singularities, then $k$-forms extend for $k \le \dim Z - 2$.

When $k = 0$, we have in fact that if ${\rm depth} (\shO_Z) \ge 2$ and $Z$ is Du Bois away from a finite set of points, then the canonical morphism

$$\shO_Z \to \cH^0 \underline{\Omega}^0_{Z} \simeq  {\Omega_h^0}|_{Z}$$

is an isomorphism, or in other words $Z$ is weakly normal. Indeed, in this case we have a short exact sequence

$$0 \to \shO_Z \to \cH^0 \underline{\Omega}^0_{Z} \to \tau \to 0,$$

where $\tau$ is supported in dimension $0$, and hence has depth $0$ if it is nonzero. On the other hand, if ${\rm depth} (\shO_Z) \ge 2$, then $R^{n-2} f_* \omega_Y (E) = 0$ (see Remark <ref>), hence again by Lemma <ref> we have ${\rm depth}(\cH^0 \underline{\Omega}^0_{Z}) \ge 2$. The only way this can happen is to have $\tau = 0$.

Another source for the type of vanishing needed in the proof of Theorem <ref>  is Corollary <ref>. We record the special case $p = n-k$ and $q = n-2$ needed here:

Assuming that $Z$ has codimension $r$, we have

$$R^{n-2} f_* \Omega_Y^{n-k} (\log E) = 0 \,\,\,\,\,\,{\rm for}\,\,\,\, k \le \left[ \frac{n-1}{r}\right] - 2.$$

Moreover, if $Z$ is normal, the same holds for

$$k \le \left[ \frac{n}{r+1}\right]  +  \left[ \frac{n-1}{r+1}\right] - 2.$$

In general this only tells us about the depth of ${\Omega_h^k}|_{Z}$ at closed points, which is only a step towards the stronger $S_2$-property.  When $Z$ has isolated singularities however, the stronger statement is true, as in
Theorem <ref>.

With the notation above we have the following:

(i) If ${\rm depth} (\shO_Z) \ge 3$, then  $\depth({\Omega_h^1}|_{Z}) \ge 2$.

(ii) If $Z$ has codimension $r$ and $k \le \left[ \frac{n-1}{r}\right] - 2$, then $\depth({\Omega_h^k}|_{Z}) \ge 2$. The same conclusion holds if $Z$ is in addition normal, and $k \le \left[ \frac{n}{r+1}\right]  +  \left[ \frac{n-1}{r+1}\right] - 2$.

(iii) Under the assumptions of $(ii)$, if $Z$ has isolated singularities, then ${\Omega_h^k}|_{Z} \simeq \Omega_Z^{[k]}$.

This follows from Lemma <ref>, Remark <ref> and Lemma <ref>. For (iii) we use in addition the same argument as in the proof of Theorem <ref>.

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