The minimal exponent and $k$-rationality for local complete intersections

Introduction

It is well-known that rational and Du Bois singularities play an important role in the hierarchy of singularities of higher-dimensional algebraic varieties. Recently, definitions of “higher order” versions of these classes of singularities have been proposed, as follows. Suppose that $Z$ is a complex algebraic variety. If $\underline{\Omega}_{Z}^{p}$ is the $p$-th graded piece of the Du Bois complex of $Z$ (suitably shifted), then there is a canonical morphism

that is an isomorphism over the smooth locus of $Z$. Following [Saito_et_al], we say that $Z$ has $k$-Du Bois singularities if this morphism is an isomorphism for $0\leq p\leq k$. For $k=0$, we recover the definition of Du Bois singularities.

On the other hand, if $\mu\colon\widetilde{Z}\to Z$ is a resolution of singularities that is an isomorphism over $Z\smallsetminus Z_{\rm sing}$ and such that $D=\mu^{-1}(Z_{\rm sing})$ is a simple normal crossing divisor on $\widetilde{Z}$, then following [FL1] we say that $Z$ has $k$-rational singularities if the canonical morphism

is an isomorphism for $0\leq p\leq k$. Again, for $k=0$ this is the classical notion of rational singularities. Our main goal in this note is to characterize numerically, in the case when $Z$ is locally a complete intersection, the condition for having $k$-rational singularities. A similar characterization for $k$-Du Bois local complete intersections has been obtained in [MP1], extending work on hypersurfaces in [MOPW] and [Saito_et_al].

Suppose that $X$ is a smooth, irreducible, $n$-dimensional complex algebraic variety and $Z$ is a local complete intersection closed subscheme of $X$, of pure codimension $r$ in $X$. In this setting the minimal exponent $\widetilde{\alpha}(Z)$ was introduced and studied in [CDMO]. In the case $r=1$, this is the invariant introduced by Saito in [Saito_microlocal] as the negative of the largest root of the reduced Bernstein-Sato polynomial of $Z$. In general, $\widetilde{\alpha}(Z)$ can be described in terms of the Kashiwara-Malgrange $V$-filtration associated to $Z$ and it is also related to the Hodge filtration on the local cohomology sheaf $\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$. The minimal exponent can be considered as a refinement of the log canonical threshold of $(X,Z)$: we always have {\rm lct}(X,Z)=\min\big{\{}\widetilde{\alpha}(Z),r\}. Moreover, it is shown in [CDMO] that $\widetilde{\alpha}(Z)>r$ if and only if $Z$ has rational singularities, extending a result due to Saito [Saito-B] in the case of hypersurfaces.

If $Z$ is a local complete intersection subvariety of the smooth, irreducible variety $X$, of pure codimension $r$, then $Z$ has $k$-rational singularities if and only if $\widetilde{\alpha}(Z)>k+r$.

In the case of hypersurfaces, this result was proved independently in [FL2, Appendix] and [MP2]. The proof we give follows the idea in [FL2], making also essential use of results from [CD] on the Kashiwara-Malgrange $V$-filtration in the case of higher codimension subvarieties. A key ingredient in the proof is Saito’s theory of mixed Hodge modules [Saito_MHM].

The characterization of $k$-Du Bois singularities in [MP1] for local complete intersections can also be formulated in terms of the minimal exponent: it says that, with the notation in Theorem 1.1, $Z$ has $k$-Du Bois singularities if and only if $\widetilde{\alpha}(Z)\geq k+r$. In particular, we obtain the following

If $Z$ is a complex algebraic variety which is locally a complete intersection and if $Z$ has $k$-Du Bois singularities, for some $k\geq 1$, then $Z$ has $(k-1)$-rational singularities.

Another consequence of the numerical characterizations of $k$-rational and $k$-Du Bois local complete intersections is that $k$-rational implies $k$-Du Bois. However, this result has already been known (it was proved independently in [FL2] and [MP1]) and we use it in our proof of Theorem 1.1.

As a consequence of the result in Theorem 1.1 and of general properties of the minimal exponent, we obtain an upper bound for the dimension of the singular locus. We note that if in the following corollary we replace “$k$-rational” by “$k$-Du Bois”, then it follows from the results in [MP1] that ${\rm codim}_{Z}(Z_{\rm sing})\geq 2k+1$.

If $Z$ is a complex algebraic variety which is locally a complete intersection and if $Z$ has $k$-rational singularities, then

Let us recall the condition for $k$-Du Bois singularities in terms of the Hodge filtration on local cohomology. For every subvariety $Z$ of a smooth complex algebraic variety $X$ and every $i$, the local cohomology sheaf $\mathcal{H}^{i}_{Z}(\mathcal{O}_{X})$ underlies a mixed Hodge module. As such, it carries a Hodge filtration $F_{\bullet}\mathcal{H}^{i}_{Z}(\mathcal{O}_{X})$, an increasing filtration by coherent $\mathcal{O}_{X}$-submodules. If $Z$ is a local complete intersection of pure codimension $r$, then the only nonzero local cohomology sheaf is $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$. There is another filtration $E_{\bullet}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ on $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$, also by coherent $\mathcal{O}_{X}$-modules, given by

where $I_{Z}$ is the ideal defining $Z$. It is shown in [MP1] that $F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\subseteq E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ for all $p\geq 0$ and equality for $p=k$ implies equality also for $p<k$. One defines the cohomological level of the Hodge filtration on $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ by

with the convention that $p(Z)=-1$ if there are no such $k$. It is then shown in [MP1] that $Z$ has $k$-Du Bois singularities if and only if $p(Z)\geq k$. The condition in terms of the minimal exponent follows from this and the equality p(Z)=\max\big{\{}\lfloor\widetilde{\alpha}(Z)\rfloor-r,-1\big{\}}, proved in [CDMO].

We characterize $k$-rationality in a similar fashion. Recall that if $X$ is a smooth irreducible $n$-dimensional variety and $Z$ is a closed subvariety of $X$ of pure codimension $r$, then $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ also carries a weight filtration and the lowest weight piece is $W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$, which underlies a pure Hodge module of weight $n+r$ (this $\mathcal{D}_{X}$-module is the intersection cohomology $\mathcal{D}_{X}$-module of Brylinski and Kashiwara [BK]). We prove the following result, which in the case of hypersurfaces was proved in [Olano].

If $Z$ is a local complete intersection subvariety of the smooth, irreducible, $n$-dimensional variety $X$, of pure codimension $r$, then for every nonnegative integer $k$, we have $\widetilde{\alpha}(Z)>k+r$ if and only if $F_{k}W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=E_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$.

We also show that for singular local complete intersections that have $k$-rational singularities, with $k\geq 1$, some higher cohomology groups of the graded pieces of the Du Bois complex do not vanish. This extends the result from [MOPW, Theorem 1.5] in the case of hypersurfaces.

Let $Z$ be a local complete intersection subvariety of the smooth, irreducible, $n$-dimensional variety $X$. If $Z$ has pure dimension $d$ and $k$-rational singularities, for some $k\geq 1$, then

where $\mathcal{Q}$ is the cokernel of the canonical map $\mathcal{T}_{X}|_{Z}\to\mathcal{N}_{Z/X}$. In particular, if $Z$ is singular at $x$, then \mathcal{H}^{k}\big{(}\underline{\Omega}^{d-k}_{Z}\big{)}_{x}\neq 0.

As observed in [MOPW], such a result imposes restrictions on varieties with quotient or toroidal singularities. Indeed, if $Z$ is a variety with quotient or toroidal singularities, then \mathcal{H}^{i}\big{(}\underline{\Omega}_{Z}^{p})=0 for all $p$ and all $i\geq 1$; for quotient singularities, this follows from [DuBois, Section 5] and for toroidal singularities, it follows from [GNPP, Chapter V.4]. On the other hand, it is well-known that such singularities are rational. By combining Theorems 1.1 and 1.5, we thus obtain

Let $Z$ be a local complete intersection subvariety, of pure codimension $r$, of the smooth, irreducible algebraic variety $X$. If $Z$ is singular, with quotient or toroidal singularities, then $r<\widetilde{\alpha}(Z)\leq r+1$.

Our final result concerns the level of generation of the Hodge filtration on $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$. Recall that if $\mathcal{M}$ is a $\mathcal{D}_{X}$-module endowed with a good filtration, where $\mathcal{D}_{X}$ is the sheaf of differential operators on $X$, then we have $F_{1}\mathcal{D}_{X}\cdot F_{p}\mathcal{M}\subseteq F_{p+1}\mathcal{M}$, with equality for $p\gg 0$ (here $F_{\bullet}\mathcal{D}_{X}$ is the order filtration on $\mathcal{D}_{X}$). If equality holds for $p\geq p_{0}$, we say that the filtration on $\mathcal{M}$ is generated at level $p_{0}$. This definition applies, in particular, for the filtered $\mathcal{D}_{X}$-module underlying a mixed Hodge module on $X$.

If $Z$ is a singular, pure codimension $r$, local complete intersection subvariety of the smooth, irreducible, $n$-dimensional variety $X$, then the Hodge filtration on $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is generated at level $n-\lceil\widetilde{\alpha}(Z)\rceil-1$.

When $r=1$, this is [MP0, Theorem A]. We also note that it follows from [MP1, Theorem 4.2] that the filtration on $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is always generated at level $n-r$, hence the assertion in the above theorem is interesting when $\widetilde{\alpha}(Z)>r-1$. Furthermore, via the equivalence in loc. cit., the assertion in Theorem 1.7 admits the following interpretation in terms of relative vanishing.

Let $Z$ be a singular, pure codimension $r$, local complete intersection subvariety of the smooth, irreducible, $n$-dimensional variety $X$. If $f\colon Y\to X$ is a proper morphism that is an isomorphism over $X\smallsetminus Z$, with $Y$ smooth and $E=f^{-1}(Z)_{\rm red}$ a simple normal crossing divisor, then

Outline of the paper. In the next section, we review some basic notions and results that we will need for the proofs of our main results. Theorem 1.1 and its corollaries, as well as Theorem 1.4 are proved in Section 3. Theorem 1.5 is proved in Section 4, while Theorem 1.7 is proved in Section 5.

Acknowledgments. We would like to thank Sebastián Olano and Mihnea Popa for many helpful discussions. We are also indebted to Christian Schnell for some useful suggestions and to Morihiko Saito for his comments on a previous version of this paper.

Background overview

In this section we recall some definitions and results that we will need. We work over the field ${\mathbf{C}}$ of complex numbers. By a variety we mean a reduced scheme of finite type over ${\mathbf{C}}$, not necessarily irreducible. For a variety $Z$, we denote by $Z_{\rm sing}$ the singular locus of $Z$.

We only give a brief introduction to mixed Hodge modules and refer for proofs and details to [Saito_MHM]. Let $X$ be a smooth, irreducible, $n$-dimensional variety and let $X^{\rm an}$ be the complex manifold corresponding to $X$. We denote by $\mathcal{D}_{X}$ the sheaf of differential operators on $X$. For basic facts about $\mathcal{D}_{X}$-modules, we refer to [HTT]. All the $\mathcal{D}_{X}$-modules we will consider will be left $\mathcal{D}_{X}$-modules. Since some of the results in the literature are stated for right $\mathcal{D}_{X}$-modules, we recall that there is an equivalence of categories between left and right $\mathcal{D}_{X}$-modules such that if $\mathcal{M}^{r}$ is the right $\mathcal{D}_{X}$-module corresponding to the left $\mathcal{D}_{X}$-module $\mathcal{M}$, then we have an isomorphism of $\mathcal{O}_{X}$-modules

When dealing with filtered $\mathcal{D}_{X}$-modules, the filtrations on $\mathcal{M}$ and $\mathcal{M}^{r}$ are indexed such that the above isomorphism maps $F_{p-n}\mathcal{M}^{r}$ to $F_{p}\mathcal{M}\otimes_{\mathcal{O}_{X}}\omega_{X}$ for all $p\in{\mathbf{Z}}$.

All filtrations on $\mathcal{D}_{X}$-modules that we will encounter are assumed to be bounded below, good filtrations compatible with the filtration $F_{\bullet}\mathcal{D}_{X}$ on $\mathcal{D}_{X}$ by order of differential operators. This means that they are increasing, exhaustive filtrations by $\mathcal{O}_{X}$-submodules such that we have

and there is $q_{0}$ such that this inclusion is an equality for all $p\geq 0$ and $q\geq q_{0}$. In this case we say that the filtration is generated at level $q_{0}$.

A mixed Hodge module $M=(\mathcal{M},F_{\bullet}\mathcal{M},{\mathcal{P}},\alpha,W_{\bullet}\mathcal{M})$ on $X$ consists of several pieces of data: $\mathcal{M}$ is a $\mathcal{D}_{X}$-module on $\mathcal{M}$ (holonomic and with regular singularities), $F_{\bullet}\mathcal{M}$ is a good filtration on $\mathcal{M}$ (the Hodge filtration), $W_{\bullet}\mathcal{M}$ is a finite increasing filtration on $\mathcal{M}$ by $\mathcal{D}_{X}$-submodules (the weight filtration), and $\mathcal{P}$ is a perverse sheaf over ${\mathbf{Q}}$ on $X^{\rm an}$ (sometimes written as ${\rm rat}(M)$), whose complexification is isomorphic via $\alpha$ to the perverse sheaf over ${\mathbf{C}}$ that corresponds to $\mathcal{M}$ via the Riemann-Hilbert correspondence. These data are supposed to satisfy a complicated set of conditions that we do not discuss. We refer to $(\mathcal{M},F)$ as the filtered $\mathcal{D}_{X}$-module underlying $M$ (though, with an abuse of notation, we sometimes write $F_{k}M$ and $W_{k}M$ instead of $F_{k}\mathcal{M}$ and $W_{k}\mathcal{M}$, respectively).

The Tate twist $M(k)$ of a mixed Hodge module $M$ as above has the same underlying $\mathcal{D}_{X}$-module, but the two filtrations are shifted by

We note that the mixed Hodge modules on $X$ form an Abelian category and every morphism of mixed Hodge modules is a morphism of $\mathcal{D}_{X}$-modules, which preserves the Hodge and the weight filtration and is strict with respect to both filtrations. There is a duality functor ${\mathbf{D}}$ on this category, lifting the usual duality functor on holonomic $\mathcal{D}_{X}$-modules. All our Hodge modules are polarizable, so the choice of a polarization implies that if $M$ as above is pure of weight $k$ (that is, ${\rm Gr}^{W}_{i}(M)=0$ for $i\neq k$), we have an isomorphism ${\mathbf{D}}(M)\simeq M(k)$. For a general mixed Hodge module $M$ and for every $k\in{\mathbf{Z}}$, the graded piece ${\rm Gr}_{k}^{W}(M)$, with the induced Hodge filtration, is a pure Hodge module of weight $k$.

An important example of a mixed Hodge module (in fact, the only one that is easy to describe explicitly besides the ones with -dimensional support) is ${\mathbf{Q}}_{X}^{H}[n]$, which is a pure Hodge module of weight $n$. The underlying $\mathcal{D}_{X}$-module is $\mathcal{O}_{X}$ and the Hodge filtration is such that ${\rm Gr}^{F}_{i}(\mathcal{O}_{X})=0$ for all $i\neq 0$. The corresponding perverse sheaf is ${\mathbf{Q}}_{X^{\rm an}}[n]$. Note that since ${\mathbf{Q}}_{X}^{H}[n]$ has weight $n$, a choice of polarization gives an isomorphism {\mathbf{D}}({\mathbf{Q}}_{X}^{H}[n]\big{)}\simeq{\mathbf{Q}}_{X}^{H}(n)[n].

Given a mixed Hodge module $M$, with underlying filtered $\mathcal{D}_{X}$-module $(\mathcal{M},F)$, the Hodge filtration makes the de Rham complex of $\mathcal{M}$ a filtered complex. The graded pieces are, in fact, complexes of $\mathcal{O}_{X}$-modules. More precisely, ${\rm Gr}_{p}^{F}{\rm DR}_{X}(M)$ is the complex

placed in cohomological degrees $-n,\ldots,0$. For example, we have

For future reference, we include the following lemma, in which we consider arbitrary filtered $\mathcal{D}_{X}$-modules:

If $f\colon(\mathcal{M},F)\to(\mathcal{N},F)$ is a morphism of filtered $\mathcal{D}_{X}$-modules on $X$ and $k\in{\mathbf{Z}}$, then the induced morphism

is an isomorphism (in the derived category) for all $p\leq k$ if and only if $F_{p}f\colon F_{p}M\to F_{p}N$ is an isomorphism for all $p\leq k+n$.

such that the vertical maps in degrees $\neq 0$ are isomorphisms and such that all induced maps in cohomology are isomorphisms. The first condition implies that the map $\operatorname{coker}(\alpha)\to\operatorname{coker}(\gamma)$ is an isomorphism and since the map induced for the $(-1)$-cohomology is an isomorphism, it follows from the 5-Lemma that the map ${\rm Im}(\beta)\to{\rm Im}(\delta)$ is an isomorphism. Since the map induced for the -cohomology is an isomorphism, it follows from the $5$-Lemma that the map ${\rm Gr}^{F}_{k+n}(f)$ is an isomorphism, and one more application of the 5-Lemma implies that $F_{k+n}f$ is an isomorphism. ∎

One can define mixed Hodge modules also on a singular variety $Z$. In our setting, $Z$ will be embedded in a fixed smooth variety $X$, and we will always view the mixed Hodge modules on $Z$ as mixed Hodge modules on $X$ whose support is contained in $Z$. One can consider the bounded derived category of mixed Hodge modules on $Z$, denoted D^{b}\big{(}{\rm MHM}(Z)\big{)}. One can show that this is equivalent to the subcategory of D^{b}\big{(}{\rm MHM}(X)\big{)} consisting of objects whose cohomology is supported on $Z$ (see [Saito_MHM, Corollary 2.23]). We will denote by $\mathcal{H}^{p}$ the standard $p$-th cohomology functor D^{b}\big{(}{\rm MHM}(Z)\big{)}\to{\rm MHM}(Z). The derived category of mixed Hodge modules satisfies a 6-functor formalism. For example, if $i\colon Z\hookrightarrow X$ is the inclusion, where $X$ is smooth, then the underlying $\mathcal{D}_{X}$-module of the mixed Hodge module \mathcal{H}^{p}\big{(}i^{!}{\mathbf{Q}}_{X}^{H}[n]\big{)} is the local cohomology sheaf $\mathcal{H}^{p}_{Z}(\mathcal{O}_{X})$ of $\mathcal{O}_{X}$ along $Z$. With a slight abuse of notation, from now one we will denote by $\mathcal{H}^{p}_{Z}(\mathcal{O}_{X})$ also the corresponding mixed Hodge module. For every variety $Z$, if $a_{Z}\colon Z\to{\rm pt}$ is the morphism to a point, then one defines ${\mathbf{Q}}_{Z}^{H}:=a_{Z}^{*}({\mathbf{Q}}_{\rm pt}^{H})$ in D^{b}\big{(}{\rm MHM}(Z)\big{)}. If $Z$ is smooth, then this coincides (up to a cohomological shift) with the object that we have already discussed. In general, however, it is a more complicated object. If $X$ is a smooth, irreducible $n$-dimensional variety and $i\colon Z\hookrightarrow X$ is a closed embedding, then by functoriality we have a canonical isomorphism ${\mathbf{Q}}_{Z}^{H}\simeq i^{*}{\mathbf{Q}}_{X}^{H}$, so we have a canonical isomorphism

For every $Z$, it is shown in [Saito_MHM, Section 4.5] that ${\mathbf{Q}}_{Z}^{H}$ is of weight $\leq 0$, that is, we have {\rm Gr}^{W}_{i}\big{(}\mathcal{H}^{j}({\mathbf{Q}}_{Z}^{H})\big{)}=0 for $i>j$. Furthermore, if $Z$ has pure dimension $d$, then $\mathcal{H}^{i}({\mathbf{Q}}_{Z}^{H})=0$ for $i>d$ and the intersection complex Hodge module

can be characterized as the unique object of ${\rm MHM}(Z)$ whose restriction to $U=Z\smallsetminus Z_{\rm sing}$ is ${\mathbf{Q}}_{U}^{H}[d]$ and which has no subobject or quotient supported on $Z_{\rm sing}$. The corresponding perverse sheaf is the intersection complex of $Z$; if $Z$ is irreducible, then this is simple, hence so is ${\rm IC}_{Z}{\mathbf{Q}}^{H}$ and we have {\mathbf{Q}}={\rm End}\big{(}{\rm IC}_{Z}{\mathbf{Q}}^{H}\big{)}. In general, if $Z$ has $N$ irreducible components, we have {\rm End}\big{(}{\rm IC}_{Z}{\mathbf{Q}}^{H}\big{)}={\mathbf{Q}}^{N} and a morphism \big{(}{\rm IC}_{Z}{\mathbf{Q}}^{H}\big{)}\to\big{(}{\rm IC}_{Z}{\mathbf{Q}}^{H}\big{)} is uniquely determined by its restriction to the smooth locus of $Z$.

Note that by definition of ${\rm IC}_{Z}{\mathbf{Q}}^{H}$, we have a canonical morphism

Suppose now that $X$ is a smooth, irreducible $n$-dimensional variety and $i\colon Z\hookrightarrow X$ is a closed embedding. Let $r=n-d$. Since ${\rm IC}_{Z}{\mathbf{Q}}^{H}={\rm Gr}^{W}_{d}\mathcal{H}^{d}({\mathbf{Q}}_{Z}^{H})$ and ${\rm Gr}^{W}_{p}\mathcal{H}^{d}({\mathbf{Q}}_{Z}^{H})=0$ for $p>d$, it follows using (2) that

and ${\rm Gr}^{W}_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=0$ for $p<n+r$. We note that this lowest weight piece of $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is the intersection cohomology $\mathcal{D}$-module introduced by Brylinski and Kashiwara in [BK]; if $Z$ is irreducible, then it can be characterized as the unique simple $\mathcal{D}_{X}$-submodule of $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$.

We also consider the shifted dual $\gamma_{Z}^{\vee}={\mathbf{D}}(\gamma)(-d)$ of $\gamma_{Z}$, that can be identified via (2) to

Note that since ${\rm IC}_{Z}{\mathbf{Q}}^{H}$ is pure of weight $d$, the choice of a polarization gives an isomorphism ${\mathbf{D}}({\rm IC}_{Z}{\mathbf{Q}}^{H})(-d)\simeq{\rm IC}_{Z}{\mathbf{Q}}^{H}$.

We will be especially interested in the case when $Z$ is a local complete intersection subvariety of $X$, of pure codimension $r$. In this case $\mathcal{H}^{i}_{Z}(\mathcal{O}_{X})=0$ for all $i\neq r$, hence $i^{!}{\mathbf{Q}}_{X}^{H}[n+r]$ is a mixed Hodge module on $Z$. Duality implies that also ${\mathbf{Q}}_{Z}^{H}[d]$ is a mixed Hodge module on $Z$, hence $\gamma_{Z}$ and $\gamma_{Z}^{\vee}$ are morphisms of mixed Hodge modules.

2. V𝑉V-filtrations

Suppose that $X$ is a smooth, irreducible, $n$-dimensional affine variety and $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)=R$ are nonzero regular functions such that the ideal $(f_{1},\ldots,f_{r})$ defines the closed subscheme $Z$ of $X$. We consider the graph embedding

and the $\mathcal{D}$-module pushforward $B_{{\bf f}}=\iota_{+}\mathcal{O}_{X}$ (where ${\bf f}$ stands for $(f_{1},\ldots,f_{r})$). If $t_{1},\ldots,t_{r}$ denote the standard coordinates on ${\mathbf{A}}^{r}$, then we can write

where for $\alpha=(\alpha_{1},\ldots,\alpha_{r})$, we put $\partial_{t}^{\alpha}=\partial_{t_{1}}^{\alpha_{1}}\cdots\partial_{t_{r}}^{\alpha_{r}}$. The action of $R$ and of $\partial_{t_{i}}$ are the obvious ones, while the actions of $D\in{\rm Der}_{{\mathbf{C}}}(R)$ and of the $t_{i}$ are given by

where $e_{1},\ldots,e_{r}$ is the standard basis of ${\mathbf{Z}}^{d}$. In fact, $B_{{\bf f}}$ underlies the pure Hodge module $\iota_{*}{\mathbf{Q}}_{X}^{H}[n]$, of weight $n$, with the Hodge filtration given by

where for $\alpha=(\alpha_{1},\ldots,\alpha_{r})$, we put $|\alpha|=\alpha_{1}+\ldots+\alpha_{r}$.

The $V$-filtration on $B_{{\bf f}}$ has been constructed by Kashiwara [Kashiwara], extending work of Malgrange [Malgrange] in the case $r=1$. (Actually, in both of these references, the $V$-filtration is indexed by integers. The ${\mathbf{Q}}$-indexed version that we discuss below was introduced by Saito [Saito_GM].) It is a decreasing, exhaustive filtration indexed by rational numbers $(V^{\lambda}B_{{\bf f}})_{\lambda\in{\mathbf{Q}}}$. It is discrete and left-continuous and it is characterized by several properties, the most important of these saying that for every $\lambda\in{\mathbf{Q}}$

and if $s=-\sum_{i=1}^{r}\partial_{t_{i}}t_{i}$, then $s+\lambda$ is nilpotent on ${\rm Gr}^{\lambda}_{V}(B_{{\bf f}})=V^{\lambda}B_{{\bf f}}/V^{>\lambda}B_{{\bf f}}$, where $V^{>\lambda}B_{{\bf f}}=\bigcup_{\beta>\lambda}V^{\beta}B_{{\bf f}}$. Note that the Hodge filtration on $B_{{\bf f}}$ induces a Hodge filtration on each ${\rm Gr}_{V}^{\lambda}(B_{{\bf f}})$.

In fact, a $V$-filtration exists on $\iota_{+}\mathcal{M}$, whenever $\mathcal{M}$ underlies a mixed Hodge module. In the case $r=1$, the interplay between the Hodge filtration on $\mathcal{M}$ and $V$-filtrations plays an important role in the definition of mixed Hodge modules. For details about the construction and properties of $V$-filtrations, see [BMS].

Let $i\colon Z\hookrightarrow X$ be the inclusion. For $r=1$, the $V$-filtration is the key ingredient for the definition of $i^{!}(M)$ and $i^{*}(M)$ when $M$ is a mixed Hodge module on $X$. In the case $r>1$, the corresponding description does not follow from the definition of these functors, but it has been recently proved in [CD, Theorem 1.2]. We only state this in the case $M={\mathbf{Q}}_{X}^{H}[n]$.

With the above notation, the following hold: the Koszul-type complex

placed in cohomological degrees $0,\ldots,r$ represents $i^{!}{\mathbf{Q}}_{X}^{H}[n]$ in the derived category of filtered $\mathcal{D}_{X}$-modules and the Koszul-type complex

placed in cohomological degrees $-r,\ldots,0$ represents $i^{*}{\mathbf{Q}}_{X}^{H}[n]$.

3. The minimal exponent

We next discuss the minimal exponent for local complete intersection varieties, following [CDMO]. Let $X$ be a smooth, irreducible, $n$-dimensional variety and $Z$ a (nonempty) closed subscheme of $X$, which is locally a complete intersection of pure codimension $r$. Suppose first that $X={\rm Spec}(R)$ is affine and $Z$ is defined by the ideal generated by $f_{1},\ldots,f_{r}\in R$. The minimal exponent $\widetilde{\alpha}(Z)$ is defined byWe note that what we denote by $F_{p+r}B_{{\bf f}}$ here is denoted by $F_{p}B_{{\bf f}}$ in [CDMO].

In general, we consider a cover $X=U_{1}\cup\ldots\cup U_{N}$, where each $U_{i}$ is an affine open subset as above, and put

It follows from [BMS, Theorem 1] that we always have \min\big{\{}\widetilde{\alpha}(Z),r\big{\}}=\operatorname{lct}(X,Z), the log canonical threshold of the pair $(X,Z)$. Therefore the minimal exponent is interesting precisely when $\operatorname{lct}(X,Z)=r$, in which case $Z$ is automatically reduced (see [CDMO, Remark 4.2]). Moreover, it follows from [CDMO, Corollary 1.7] that $Z$ has rational singularities if and only if $\widetilde{\alpha}(Z)>r$. One can also show (see [CDMO, Remark 4.15]) that $Z$ is smooth if and only if $\widetilde{\alpha}(Z)=\infty$; by definition of the minimal exponent, this can be rephrased as

In fact, if $x\in Z$ is a singular point, then we have the following more precise bound (see [CDMO, Remark 4.21]):

The minimal exponent $\widetilde{\alpha}(Z)$ depends on the ambient variety $X$, but in a predictable way: the difference $\widetilde{\alpha}(Z)-\dim(X)$ only depends on $Z$ (see [CDMO, Proposition 4.14]).

When $r=1$, the minimal exponent was defined by Saito [Saito_microlocal] as the negative of the largest root of the reduced Bernstein-Sato polynomial $\widetilde{b}_{Z}(s)$. For the fact that this agrees with the above definition, see for example [MP5, Lemma 5.3 and Corollary C].

Recall now that the $\mathcal{D}_{X}$-module $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ underlies a mixed Hodge module on $X$, namely \mathcal{H}^{r}\big{(}i^{!}{\mathbf{Q}}_{X}^{H}[n]\big{)}, where $i\colon Z\hookrightarrow X$ is the inclusion. We thus have a canonical filtration on $\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$, the Hodge filtration \big{(}F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\big{)}_{p\geq 0}. We have a second filtration, the order filtration \big{(}E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\big{)}_{p\geq 0}, given by

where $I_{Z}$ is the ideal defining $Z$ in $X$ (see [MP1, Proposition 3.11]). It is a general fact that $F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\subseteq E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ for all $p\geq 0$ (see [MP1, Proposition 3.4]) and the following result shows that the minimal exponent governs how far these two filtrations agree (see [CDMO, Theorem 1.3]):

If $X$ is a smooth, irreducible variety and $Z$ is a local complete intersection subvariety of pure codimension $r$ in $X$, then for a nonnegative integer $k$, we have $F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ for $0\leq p\leq k$ if and only if $\widetilde{\alpha}(Z)\geq r+k$.

4. k𝑘k-Du Bois singularities

To a variety $Z$, Du Bois associated in [DuBois] a complex $\underline{\Omega}_{Z}^{\bullet}$, known now as the Du Bois complex of $Z$. This is a filtered complex that agrees with the de Rham complex $\Omega_{Z}^{\bullet}$, with the “stupid” filtration, when $Z$ is smooth. This allows extending to singular varieties some important cohomological properties of the de Rham complex of smooth varieties, see [PetersSteenbrink, Chapter 7.3] for an introduction to this topic.

We are interested in the shifted truncations $\underline{\Omega}_{Z}^{p}:={\rm Gr}_{F}^{p}(\underline{\Omega}_{Z}^{\bullet})[p]$, which are objects in the bounded derived category $D^{b}_{\rm coh}(Z)$ of coherent sheaves on $Z$. For every $p$, there is a canonical morphism $\Omega_{Z}^{p}\to\underline{\Omega}_{Z}^{p}$ that is an isomorphism over the smooth locus of $Z$. Following [Saito_et_al], we say that $Z$ has $k$-Du Bois singularitiesStrictly speaking, one should say “has at most $k$-Du Bois singularities”, since we do not require $Z$ to be singular. However, we trust that the simplified formulation will not lead to confusion., for a nonnegative integer $k$, if these morphisms are isomorphisms for all $0\leq p\leq k$. Note that for $k=0$, we recover the familiar notion of Du Bois singularities.

As we have mentioned in the Introduction, it was shown in [MP1, Theorem F] that if $X$ is a smooth, irreducible variety and $Z$ is a local complete intersection subvariety of $X$, of pure codimension $r$, then $Z$ has $k$-Du Bois singularities if and only if $F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ for $p\leq k$. In terms of minimal exponents, this condition can be rephrased as $\widetilde{\alpha}(Z)\geq r+k$. The proof of this result in loc. cit. extends the argument in the case of hypersurfaces, for which the two implications had previously been proved in [MOPW] and [Saito_et_al].

If $Z$ is a local complete intersection variety with $k$-Du Bois singularities, then ${\rm codim}_{Z}(Z_{\rm sing})\geq 2k+1$. Indeed, this is a local statement, hence we may assume that $Z$ has pure dimension (we use the fact that $Z$ is Cohen-Macaulay) and that it is a closed subvariety of the smooth irreducible variety $X$. In this case the assertion follows by combining [MP1, Corollary 3.40 and Theorem F].

The connection between the Du Bois complex and mixed Hodge modules is provided by the following result of Saito. If $Z$ is a closed subvariety of the smooth, irreducible, $n$-dimensional variety $X$ and $i\colon Z\hookrightarrow X$ is the inclusion, then it is a consequence of [Saito-HC, Theorem 4.2] that for every $p$, we have an isomorphism

in $D^{b}_{\rm coh}(X)$. In light of (1) and (2), this is equivalent to

For an easy proof of this isomorphism, see [MP1, Proposition 5.5].

5. k𝑘k-rational singularities

Given a variety $Z$, by a strong log resolution of $Z$ we mean a proper morphism $\mu\colon\widetilde{Z}\to Z$ that is an isomorphism over $Z\smallsetminus Z_{\rm sing}$, such that $\widetilde{Z}$ is smooth and $E=\mu^{-1}(Z_{\rm sing})$ is a simple normal crossing divisor. For a nonnegative integer $k$, following [FL1], we say that $Z$ has $k$-rational singularities if the canonical morphism

is an isomorphism for all $p\leq k$. This is easily seen to be independent of the log resolution (see for example [MP2, Lemma 1.6]). Note that for $k=0$ we recover the classical notion of rational singularities. This condition implies that $Z$ is normal, hence in particular, every connected component of $Z$ is irreducible. The notion of $k$-rational singularities has been extensively studied in [FL1], [FL2], [FL3], [MP2].

For our purpose it will be convenient to consider a different description of $k$-rational singularities. Recall from [MP2, Section 6] that for every variety $Z$ of pure dimension $d$ and every nonnegative integer $k$, we have a canonical morphism

where $\omega_{Z}^{\bullet}$ is the dualizing complex of $Z$. This is defined as follows: suppose that $\mu\colon Y\to Z$ is an arbitrary resolution of singularities (we only require that $Y$ is smooth and $\mu$ is proper and an isomorphism over a dense open subset of $Z$). By functoriality of the Du Bois complex, for every nonnegative integer $k$, we have a canonical morphism $\alpha_{k}\colon\underline{\Omega}_{Z}^{k}\to{\mathbf{R}}\mu_{*}\Omega_{Y}^{k}$. On the other hand, on $Y$ we have a canonical isomorphism

By pushing this forward and using Grothendieck duality for $\mu$, we obtain an isomorphism $\beta_{k}$ as the composition

The morphism $\psi_{k}$ is obtained as the composition $\alpha_{d-k}^{\vee}\circ\beta_{k}\circ\alpha_{k}$, where we put \alpha_{d-k}^{\vee}={\mathbf{R}}\mathcal{H}om_{\mathcal{O}_{Z}}\big{(}\alpha_{d-k},\omega_{Z}^{\bullet}[-d]\big{)}. It is shown in [MP2, Proposition 6.1] that this definition does not depend on the choice of resolution of singularities.

With this notation, we have the following characterization of $k$-rational singularities in the local complete intersection case, see [MP2] (in loc. cit. one assumes that $Z$ is irreducible, but the argument works in general):

If $Z$ is a local complete intersection variety of pure dimension $d$ and $k$ is a nonnegative integer, then $Z$ has $k$-rational singularities if and only if $Z$ has $k$-Du Bois singularities and the morphism \psi_{k}\colon\underline{\Omega}_{Z}^{k}\to{\mathbf{R}}\mathcal{H}om_{\mathcal{O}_{Z}}\big{(}\underline{\Omega}_{Z}^{d-k},\omega_{Z}\big{)} is an isomorphism.

It will be important for us to use an interpretation of the morphism $\psi_{k}$ from [FL2, Appendix], as the graded de Rham of a morphism of mixed Hodge modules. Let $Z$ be a variety of pure dimension $d$ and $\mu\colon Y\to Z$ any resolution of singularities, with $\mu$ a projective morphism. Note that by functoriality we have a canonical morphism of mixed Hodge modules $\alpha\colon{\mathbf{Q}}_{Z}^{H}[d]\to\mu_{*}{\mathbf{Q}}^{H}_{Y}[d]$. On the other hand, since ${\mathbf{Q}}_{Y}^{H}[d]$ is pure of weight $d$, on $Y$ we have a canonical isomorphism {\mathbf{Q}}_{Y}^{H}[d]\to{\mathbf{D}}\big{(}{\mathbf{Q}}_{Y}^{H}[d]\big{)}(-d), which after pushing forward to $Z$ and using the compatibility of pushforward with duality, gives an isomorphism

We then obtain a morphism $\psi_{Z}$ in the derived category of mixed Hodge modules on $Z$ as the following composition

where $\alpha^{\vee}={\mathbf{D}}(\alpha)(-d)$.

If $Z$ is a closed subvariety of the smooth, irreducible variety $X$ and $i\colon Z\hookrightarrow X$ is the inclusion, then using the compatibility of the graded de Rham complex with direct image and duality, we see that for every $k\in{\mathbf{Z}}$ we have

hence $\psi_{k}={\rm Gr}^{F}_{-k}{\rm DR}_{X}(\psi_{Z})[k-d]$.

It follows from the definition of $\psi_{Z}$ that $\psi_{Z}^{\vee}:={\mathbf{D}}(\psi_{Z})(-d)$ can be identified with $\psi_{Z}$.

Suppose now that $X$ is a smooth, irreducible, $n$-dimensional variety and $i\colon Z\hookrightarrow X$ is a closed embedding, where $Z$ is a local complete intersection subvariety of $X$, of pure codimension $r$. Let $d=n-r$. As we have already mentioned, in this case, the morphisms

are morphisms of mixed Hodge modules, with $\gamma_{Z}$ surjective and $\gamma_{Z}^{\vee}$ injective.

Since $\psi_{Z}$ is a morphism between two mixed Hodge modules on $Z$, we obtain the same morphism if we take $\mathcal{H}^{0}(-)$; in other words, $\psi_{Z}$ agrees with the composition

We also note that the intermediate composition

For every $k$, it follows from Lemma 2.1 that ${\rm Gr}_{p}^{F}{\rm DR}_{X}(\psi_{Z})$ is an isomorphism for all $p\leq k$ if and only if $F_{p}\psi_{Z}$ is an isomorphism for every $p\leq k+n$. Since $\gamma_{Z}$ is surjective and $\gamma_{Z}^{\vee}$ is injective, it follows from the above discussion that ${\rm Gr}_{p}^{F}{\rm DR}_{X}(\psi_{Z})$ is an isomorphism for all $p\leq k$ if and only if

are isomorphisms for all $p\leq k+n$ (recall that every morphism of mixed Hodge modules preserves the Hodge filtration and it is strict).

We note that in [FL2] one says that a variety $Z$ of pure dimension $d$ has $k$-rational singularities if the composition

is an isomorphism for all $p\leq k$. It is shown in [FL2, Corollary 3.17] that this definition is equivalent to the definition we use in this paper if ${\rm codim}_{Z}(Z_{\rm sing})\geq 2k+1$. Furthermore, it is shown in [FL2, Theorem 3.20] that with their definition as well, if $Z$ is a local complete intersection and has $k$-rational singularities, then $Z$ has Du Bois singularities, and thus ${\rm codim}_{Z}(Z_{\rm sing})\geq 2k+1$ by Remark 2.4. We thus conclude that for local complete intersection varieties, the two definitions of $k$-rational singularities agree.

Characterizations of k𝑘k-rationality for local complete intersections

Let $X$ be a smooth, irreducible variety of dimension $n$ and $Z$ be a local complete intersection subvariety of pure codimension $r$ in $X$. Let $d=n-r$ be the dimension of $Z$ and $i\colon Z\hookrightarrow X$ the inclusion. We will freely use the notation introduced in the previous section. The following is the main result of this section, which implies several of the statements in the introduction.

With the above notation, for every nonnegative integer $k$, the following conditions are equivalent:

$F_{k}W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=E_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$;

induced by $\gamma_{Z}$ and the composition

induced by $\gamma_{Z}^{\vee}$, are isomorphisms for $p\leq k$.

$Z$ has $k$-Du Bois singularities and the morphism

is an isomorphism (in the derived category) for $p\leq k$.

We note that the morphism in (e) is the composition

Note that condition (d) in the theorem is equivalent to $Z$ having $k$-rational singularities by Theorem 2.5. Therefore the equivalence (a)$\Leftrightarrow$(d) is the content of Theorem 1.1, while the equivalence (a)$\Leftrightarrow$(b) is the content of Theorem 1.4.

We proceed with the proof of Theorem 3.1 in several steps, by showing the following implications:

$(a)\Rightarrow(b)\Rightarrow(c)\Rightarrow(d)+(e)$, $(d)\Rightarrow(c)$, and $(e)\Rightarrow(b)\Rightarrow(a)$.

Since all assertions are local, we may and will assume that $X$ is affine and $Z$ is defined in $X$ by $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$. In particular, we will be able to consider the $V$-filtration corresponding to these functions. We denote by $I_{Z}$ the ideal defining $Z$ in $X$.

$(s+\alpha)\cdot W_{\bullet}{\rm Gr}_{V}^{\alpha}B_{{\bf f}}\subseteq W_{\bullet-2}{\rm Gr}_{V}^{\alpha}B_{{\bf f}}$ and

Recall that we know that $W_{i+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=0$ for $i<n$, hence

Since $\widetilde{\alpha}(Z)>k+r$, it follows from the definition of the minimal exponent that $F_{k+r+1}B_{\bf f}\subseteq V^{>r-1}B_{\bf f}$, hence

where the last equality follows from Theorem 2.3.

Step 2. Proof of (b)$\Rightarrow$(c). We first prove the following

The equality $F_{k}W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=E_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ implies

Recall first that $F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\subseteq E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ for all $p$ by [MP1, Proposition 3.4], hence $F_{p}W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ if and only if the inclusion “$\supseteq$” holds. Since $W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is a Hodge module supported on $Z$, we have

(see [Saito_MHP, Lemme 3.2.6]). On the other hand, it follows easily from the definition of the filtration $E_{\bullet}\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})$ that we have

The assertion in the lemma now follows by decreasing induction on $p$. ∎

We next use duality to prove the following

If $F_{p}W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ for some $p\in{\mathbf{Z}}$, then the surjective map

induced by $\gamma_{Z}$ is an isomorphism.

Combining the two equations (23) and (24) yields

as filtered $\mathcal{D}_{X}$-modules, which implies

Returning to the proof of the implication (b)$\Rightarrow$(c), note that the assertion in Lemma 3.4 gives the fact that the morphism (18) is an isomorphism for $p\leq k$. Similarly, by combining Lemmas 3.4 and 3.5 we conclude that the morphism (17) is an isomorphism for $p\leq k$ (in fact, for $p\leq k+1$). We thus have the assertion in (c).

Similarly, once we know that $Z$ is $k$-Du Bois, the condition in (e) is equivalent with ${\rm Gr}^{F}_{-p}{\rm DR}_{X}(\gamma_{Z})$ being an isomorphism for $p\leq k$, which is equivalent by (1) with ${\rm Gr}^{F}_{p-d}{\rm DR}_{X}(\gamma_{Z}^{\vee})$ being an isomorphism for all $p\leq k$. This is implied by $F_{p+r}\gamma_{Z}^{\vee}$ being an isomorphism for all $p\leq k$, but this is precisely the morphism $F_{p}W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\to F_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$. Therefore the condition in (e) holds as well.

Step 4. Proof of (d)$\Rightarrow$(c). It follows from Theorem 2.5 that the conditions in (d) are equivalent to $Z$ having $k$-rational singularities. In particular, since we have these conditions for $k$, we also have them for $k-1$. In particular, we know that $\psi_{k}={\rm Gr}^{F}_{p-d}{\rm DR}_{X}(\psi_{Z})[p-d]$ is an isomorphism for all $p\leq k$. Using (1) and the fact that ${\mathbf{D}}(\psi_{Z})=\psi_{Z}(d)$, we conclude that ${\rm Gr}^{F}_{p-d}{\rm DR}_{X}(\psi_{Z})$ is an isomorphism for all $p\leq k$. Lemma 2.1 thus implies that $F_{p+r}\psi_{Z}$ is an isomorphism for all $p\leq k$. As we have seen in Remark 2.7, this implies that the morphisms (17) and (18) are isomorphisms for all $p\leq k$ (for the latter morphism, we also use the fact that $F_{p}\mathcal{H}_{Z}^{r}(\mathcal{O}_{X})=E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ for $p\leq k$, due to the fact that $Z$ has $k$-Du Bois singularities). We thus have the condition in (c).

Step 5. Proof of (e)$\Rightarrow$(b). If $k=0$, applying {\mathbf{R}}\mathcal{H}om_{\mathcal{O}_{X}}\big{(}-,\omega_{X}[r]\big{)} to the isomorphism in (e) gives via (1) an isomorphism

(note that ${\mathcal{E}xt}^{i}_{\mathcal{O}_{X}}(\mathcal{O}_{Z},\omega_{X})=0$ for $i\neq r$ since $Z$ is a local complete intersection of pure codimension $r$). We have

while the image of the inclusion ${\mathcal{E}xt}^{r}_{\mathcal{O}_{X}}(\mathcal{O}_{Z},\omega_{X})\hookrightarrow\omega_{X}\otimes_{\mathcal{O}_{X}}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is $\omega_{X}\otimes_{\mathcal{O}_{X}}E_{0}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$. We thus obtain the condition in (b) in this case.

From now on we assume $k\geq 1$. Arguing by induction on $k$, we may and will assume that $F_{p}W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})=E_{p}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ for $p\leq k-1$. In particular, we know that $Z$ has $(k-1)$-Du Bois singularities, and thus ${\rm codim}_{Z}(Z_{\rm sing})\geq 2k-1\geq k$ by [MP1, Corollary 3.40]. We only need to prove that the injection $F_{k}W_{n+r}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})\hookrightarrow E_{k}\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ is indeed an isomorphism. Moreover, because we know the corresponding assertions for the lower pieces of the filtrations, it follows from Lemma 2.1 that it is enough to show that the induced morphism

is an isomorphism (in the derived category).

Applying {\mathbf{R}}\mathcal{H}om_{\mathcal{O}_{X}}\big{(}-,\omega_{X}[r+k]\big{)} to the isomorphism in (e) implies via (1) that the composition

is an isomorphism. On the other hand, since ${\rm codim}_{Z}(Z_{\rm sing})\geq k$, it follows from [MP1, Section 5.2] that the second map in (26) gets identified with the canonical morphism

We thus conclude that indeed (25) is an isomorphism.

Claim. The condition in (b) implies that the composition of the canonical morphisms

By [CD, Theorem 1.2], we have an isomorphism of filtered $\mathcal{D}_{X}$-modules

Indeed, the morphsim is well-defined because

We deduce that the canonical map $\ker\delta\to\operatorname{coker}\sigma$ factors as

is an isomorphism for $p\leq k+1$ by Lemma 3.5. Therefore the claim is now reduced to the assertion that (29) is a filtered isomorphism. Clearly, (29) is a filtered isomorphism over the complement $V=X\smallsetminus Z_{\rm sing}$ of the singular locus of $Z$, due to

To conclude the proof, note that the condition in (b) implies that $\widetilde{\alpha}(Z)\geq k+r$ by Theorem 2.3. Therefore we have

where the first equality follows from the fact that $F_{k+r}V^{>r}B_{{\bf f}}=\sum_{i=1}^{r}t_{i}\cdot F_{k+r}V^{>r-1}B_{{\bf f}}$ by [CD, Theorem 1.1]. The claim implies that the composition

where the second inclusion comes from the fact that $\widetilde{\alpha}(Z)\geq k+r$. This implies $F_{k+r+1}B_{{\bf f}}\subseteq V^{>r-1}B_{{\bf f}}$, which is equivalent to $\widetilde{\alpha}(Z)>k+r$. This completes the proof of this step and thus the proof of the theorem. ∎

We next prove the two corollaries stated in the Introduction:

The assertion follows from the fact that $Z$ has $k$-Du Bois singularities if and only if $\widetilde{\alpha}(Z)\geq k+r$, while by Theorem 1.1, $Z$ has $(k-1)$-rational singularities if and only if $\widetilde{\alpha}(Z)>k+r-1$. ∎

We may assume that $Z$ is irreducible and affine and let $Z\hookrightarrow X$ be a closed embedding, of codimension $r$, with $X$ a smooth, irreducible variety. The assertion to prove is trivial if $Z$ is smooth (with the convention that the empty set has infinite codimension), hence we may and will assume that $Z$ is singular. If $s=\dim(Z_{\rm sing})$ and $H$ is the intersection of general hyperplanes sections in $X$, then $Z^{\prime}:=Z\cap H$ is a local complete intersection variety with nonempty, -dimensional singular locus, and $\widetilde{\alpha}(Z^{\prime})=\widetilde{\alpha}(Z)$ by [CDMO]*Theorem 1.2. In particular, it follows from Theorem 1.1 that $\widetilde{\alpha}(Z^{\prime})>k+r$. Since ${\rm codim}_{Z}(Z_{\rm sing})={\rm codim}_{Z^{\prime}}(Z^{\prime}_{\rm sing})$, we may replace $Z$ by $Z^{\prime}$ to assume that $Z_{\rm sing}$ is nonempty and zero-dimensional. We then need to show that $d:=\dim(Z)\geq 2k+2$.