On k-rational and k-Du Bois local complete intersections

A. Introduction

A systematic study of natural refinements of the standard notions of rational and Du Bois singularities of complex algebraic varieties has been taking shape recently, guided especially by developments of a Hodge theoretic and $\mathscr{D}$-module theoretic flavor. On one hand, the papers [MOPW] and [Saito_et_al] introduced and studied the notion of $k$-Du Bois singularities for hypersurfaces (the terminology appeared in the latter) as a natural extension of the concept of Du Bois singularities. The definition, which makes sense for an arbitrary variety $Z$, is that the natural morphisms

are isomorphisms for $0\leq i\leq k$, where $\underline{\Omega}_{Z}^{i}$ is the $i$-th graded piece of the Du Bois complex of $Z$ (suitably shifted) with respect to the Hodge filtration. The results in loc. cit. were extended to local complete intersections in [MP2].

On the other hand, as defined in [FL1] (cf. also [FL3]) for normal isolated singularities, and communicated to us by R. Laza in general, rational singularities also admit a natural refinement: a variety $Z$ has $k$-rational singularities if for any resolution of singularities $\mu\colon\widetilde{Z}\to Z$ that is an isomorphism over the smooth locus of $Z$ and such that the reduced inverse image $D$ of the singular locus is a simple normal crossing divisor, the canonical morphisms

are isomorphisms for all $0\leq i\leq k$.

The study we undertake here is motivated by two points. First, for $k=0$ these are the standard notions of Du Bois and rational singularities, and as it is well known by work of Kovács [Kovacs1], and later Saito [Saito-HC] as well, rational singularities are Du Bois; it is natural to ask whether this persists for the higher versions. Second, in the case of hypersurfaces both notions have other possible (numerical) refinements in terms of minimal exponents; the natural question, already approached for isolated singularities in [FL1], is whether they coincide with the notions defined above.

In this paper we answer some of these questions positively for local complete intersections, and others, when the $V$-filtration and minimal exponents are involved, only in the case of hypersurfaces. While writing it, we learned that some of the results we obtain have also been arrived at independently in [FL2] and [Saito-AppendixFL], following work in the case of isolated singularities in [FL1], [FL3]; see below. At the moment it seems quite hard to say much beyond the case of local complete intersections.

In what follows we always work over ${\mathbf{C}}$, and $X$ is an irreducible $n$-dimensional smooth algebraic variety.

Results for local complete intersections. The main result we obtain in this context is an injectivity theorem for the cohomologies of the Grothendieck duals of the various graded quotients of the Du Bois complex, whose proof relies on the study of the Hodge filtration on local cohomology in [MP2].

Let $Z$ be an algebraic variety which is locally a complete intersection, and let $k$ be a nonnegative integer such that $Z$ has $(k-1)$-Du Bois singularities. Then the morphism

in the derived category of coherent sheaves on $Z$, obtained by dualizing the canonical morphism $\Omega_{Z}^{k}\to\underline{\Omega}_{Z}^{k}$, is injective at the level of cohomology.

When $k=0$ the hypothesis is vacuous, and the statement is known to hold for an arbitrary variety $Z$ (meaning not necessarily a local complete intersection). This result was shown by Kovács and Schwede [KoS, Theorem 3.3], with a somewhat stronger version obtained by different means in [MP2, Theorem A], and has proven to be useful for a whole range of applications.

A quick consequence of Theorem A that we derive here is the natural higher analogue, for local complete intersections, of the well-known fact discussed above that rational singularities are Du Bois.

Let $Z$ be an algebraic variety which is locally a complete intersection. If $Z$ has $k$-rational singularities, then $Z$ has $k$-Du Bois singularities.

This result was also obtained in a different fashion by Friedman and Laza in [FL2], and previously in the case of isolated singularities in [FL1] (cf. also [FL3]).

An interesting consequence of Theorem B regards duality for the graded pieces of the Du Bois complex. It is not hard to show, see Proposition 6.1 below, that for every $d$-dimensional irreducible variety $Z$ and every $k\geq 0$ there is a natural morphism

where $\omega_{Z}^{\bullet}$ is the dualizing complex of $Z$. This is however usually not an isomorphism, and in fact we show that this property is precisely what makes the difference between $k$-Du Bois and $k$-rational singularities:

If $Z$ is an irreducible $d$-dimensional local complete intersection variety, then $Z$ has $k$-rational singularities if and only if it has $k$-Du Bois singularities and the canonical morphism

By duality, in the setting of Corollary C one can also compute $\underline{\Omega}_{Z}^{d-k}$ as the derived dual of $\Omega_{Z}^{k}$; see Corollary 6.3.

On a different note, since the work of Steenbrink, see e.g. [Steenbrink], it has been known that the graded pieces of the Du Bois complex are closely related to direct images of sheaves of forms with log poles on log resolutions. In this direction, we record the following local vanishing theorem, which is mainly a consequence of our results in [MP2], with some additions from [GKKP]; this gives a positive answer to a question of R. Laza. A related result, Theorem G below, will be a crucial technical step towards a finer understanding of $k$-rational singularities of hypersurfaces. We again take $\mu\colon\widetilde{Z}\to Z$ to be a resolution that is an isomorphism over the smooth locus of $Z$, and such that the reduced preimage of the singular locus of $Z$ is a simple normal crossing divisor $D$.

If $Z$ is a $d$-dimensional local complete intersection variety which has $k$-Du Bois singularities and is normal,Note that $Z$ is automatically normal if $k\geq 1$. then

$p\leq k$, $q\geq 1$, and $p+q\geq d-2k+1$;

When the singularities of $Z$ are isolated, a different approach to such vanishing is provided in [FL1, Theorem 3.6].

Results for hypersurfaces. In the case of hypersurfaces, one can obtain stronger results; the key technical tool allowing for this is the minimal exponent, and especially its connection with the $V$-filtration of Kashiwara and Malgrange.

Let $Z$ be a hypersurface in $X$. Its minimal exponent $\widetilde{\alpha}(Z)$ is the negative of the greatest root of the reduced Bernstein-Sato polynomial $b_{Z}(s)/(s+1)$; see e.g. [MP3, §6] for a general discussion of this singularity invariant. The minimal exponent is related to the log canonical threshold by the formula ${\rm lct}(X,Z)=\min\{\widetilde{\alpha}(Z),1\}$. M. Saito showed in [Saito-B] that $Z$ has rational singularities if and only if $\widetilde{\alpha}(Z)>1$. This can be extended to the following:

If $Z$ is a hypersurface in $X$, and $k$ is a nonnegative integer, then $Z$ has $k$-rational singularities if and only if $\widetilde{\alpha}(Z)>k+1$.

Another proof of this result was obtained independently by Saito [Saito-AppendixFL]. For isolated singularities, it had been established in [FL1, Theorem 3.8], cf. also [FL3].

Note. The notion of $k$-rationality for hypersurfaces first appeared in [KL] precisely as the condition $\widetilde{\alpha}(Z)>k+1$. Here we are of course using the more natural definition in terms of log resolutions suggested in [FL1]. The point of the theorem is that these two possible generalizations indeed coincide.

Combining the main results of [MOPW] and [Saito_et_al], we also know that $Z$ is $k$-Du Bois if and only if $\widetilde{\alpha}(Z)\geq k+1$. Together with Theorem E, this provides a numerical strengthening of Theorem B, and has the following immediate consequence (cf. also [FL2, Conjecture 1.8]):

If $Z$ is a hypersurface in $X$ which has $(k+1)$-Du Bois singularities, for some $k\geq 0$, then $Z$ has $k$-rational singularities.

At the moment we do not know how to show this last implication in the case of local complete intersections of higher codimension.

We consider an approach that obtains the “if” part of Theorem E as a quick consequence of the following local vanishing theorem, which is the counterpart to Theorem D, and uses the same notation. The question whether a vanishing result roughly of this type holds was posed to us by R. Laza. Part ii) is in fact one of the main results of [MP1] (the proof we give here is technically similar, but slightly simpler); the main new result is part i). Again, for isolated singularities see also [FL1, Theorem 3.8 and Remark 3.9].

If $Z$ is a hypersurface in $X$, $k$ is a nonnegative integer, and $\widetilde{\alpha}(Z)>k+1$, then

This theorem is in turn a consequence of an analogous result we establish for log resolutions of the pair $(X,Z)$, as opposed to $Z$ itself; see Theorem 8.3. The proofs of Theorems E and G make use of Saito’s theory of mixed Hodge modules [Saito-MHM], especially duality and the description of the Hodge filtration on the local cohomology ${\mathcal{H}}_{Z}^{1}(\mathscr{O}_{X})$ in terms of the $V$-filtration.

Acknowledgements. We especially thank R. Laza for outlining the program we address here, and for asking us many relevant questions. We also thank him and R. Friedman for sharing an early version of the paper [FL1], which discusses many of these topics in the case of isolated singularities.

B. Definitions and background

In what follows we work over the field ${\mathbf{C}}$ of complex numbers. By a variety we mean a reduced, separated scheme of finite type over ${\mathbf{C}}$ (not necessarily irreducible). For a variety $Z$, we denote by $Z_{\rm sing}$ its singular locus and by $Z_{\rm sm}$ the complement $Z\smallsetminus Z_{\rm sing}$.

We will deal with two types of resolutions of singularities that we now describe. If $Z$ is an irreducible variety, then by a log resolution of $Z$ we mean a proper birational morphism $\mu\colon\widetilde{Z}\to Z$, with $\widetilde{Z}$ smooth, such that if $W$ is the complement of the domain of $\mu^{-1}$, then the subset $\mu^{-1}(W)_{\rm red}$ (the exceptional locus of $\mu$) is a divisor $D$ with simple normal crossings. We say that $\mu$ is a strong log resolution of $Z$ if, in addition, we have $W=Z_{\rm sing}$ (in general we only have $Z_{\rm sing}\subseteq W$).

Suppose now that $Z$ is a subvariety of the smooth irreducible variety $X$. A log resolution of $(X,Z)$ is a proper morphism $\pi\colon Y\to X$, with $Y$ smooth, which is an isomorphism over $X\smallsetminus Z$, and such that $\pi^{-1}(Z)_{\rm red}$ is a divisor with simple normal crossings. If $Z$ is a hypersurface in $X$, then we say that $\pi$ is a strong log resolution of $(X,Z)$ if, in addition, it is an isomorphism over $X\smallsetminus Z_{\rm sing}$. In this case, if $Z$ is irreducible and $\widetilde{Z}$ is its strict transform on $Y$, then the induced morphism $\widetilde{Z}\to Z$ is a strong log resolution of $Z$. We note that in both contexts strong log resolutions exist by Hironaka’s fundamental theorem. Moreover, a (strong) log resolution of $(X,Z)$ can be obtained as a composition of blow-ups of smooth centers.

Let’s assume now that $Z$ is an irreducible hypersurface in $X$, and establish a connection between higher direct images of forms with log poles in the two contexts. Suppose that $W$ is a proper closed subset of $Z$ containing $Z_{\rm sing}$, and let $\pi\colon Y\to X$ be a log resolution of $(X,Z)$ that is an isomorphism over $X\smallsetminus W$ and is a composition of smooth blow-ups. More precisely, the morphism $\pi$ factors as

where each $\pi_{j}$ with $1\leq j\leq N$ is the blow-up of a smooth irreducible subvariety $W_{j-1}\subseteq X_{j-1}$ that lies over $W$. We denote by $F_{j}$ the exceptional divisor of $X_{j}\to X$ and by $Z_{j}$ the strict transform of $Z$ on $X_{j}$. Moreover, for each $1\leq j\leq N$, we assume that $W_{j-1}$ has simple normal crossings with $Z_{j-1}+F_{j-1}$. We note that such a resolution exists by Hironaka’s theorem (we can take, for example, $W=Z_{\rm sing}$). We denote

If $\mu\colon\widetilde{Z}\to Z$ is the restriction of $\pi$, then $\mu$ is a strong log resolution of $Z$ and $D=\mu^{-1}(Z_{\rm sing})_{\rm red}$.

With the above notation, if $r={\rm codim}_{X}(W)$, then for every $i$ with $0\leq i\leq r-2$ and every $q\geq 1$, we have an isomorphism

The argument is similar to the one in the proof of [MP1, Theorem D], but we include the details for the benefit of the reader. We first show that for every $i\leq r-1$, we have

Using the Leray spectral sequence, we see that in order to prove (1.4) it is enough to show that for every $1\leq j\leq N$, we have

Let us fix $j$. If $W_{j-1}\subseteq F_{j-1}$, then this follows from [MP0, Theorem 31.1(i)]. On the other hand, if $W_{j-1}\not\subseteq F_{j-1}$, then $W_{j-1}$ is the strict transform of its image in $X$. In particular, we have ${\rm codim}_{X_{j-1}}(W_{j-1})\geq r$ and the assertions in (1.5) for $j$ follow from [MP1, Lemma 7.2].

We now consider on $Y$ the residue short exact sequence

The long exact sequence for higher direct images together with the formulas in (1.4) imply that for $i\leq r-2$ we have an isomorphism

This completes the proof of the proposition. ∎

Let $Z$ be an irreducible variety, $\mu\colon\widetilde{Z}\to Z$ a log resolution of $Z$ with $W$ the complement of the domain of $\mu^{-1}$, and let $D$ be the simple normal crossing divisor on $\widetilde{Z}$ such that $\mu^{-1}(W)_{\rm red}=D$. For every $p$ and $q$, the sheaves

only depend on $W$ (but not on $\mu$). In particular, these sheaves only depend on $Z$ if $\mu$ is assumed to be a strong log resolution.

Since any two log resolutions with the same $W$ are dominated by a common one, in order to prove the assertion it is enough to show that if $E$ is a reduced simple normal crossing divisor on the smooth $n$-dimensional variety $Y$ and $g\colon\widetilde{Y}\to Y$ is a proper morphism that is an isomorphism over $Y\smallsetminus{\rm Supp}(E)$, with $\widetilde{Y}$ smooth and $F=g^{*}(E)_{\rm red}$ having simple normal crossings, then for every $p\geq 0$, we have canonical isomorphisms

In fact (1.7) is already known; see for example [MP0, Theorem 31.1(i)]. On the other hand, since we have

The isomorphism in (1.8) follows from the fact that $\mathbf{R}{\mathcal{H}om}_{\mathscr{O}_{Y}}(-,\omega_{Y})$ is a duality. ∎

k𝑘k-rational singularities

The following definition is due to Friedman and Laza [FL1] in the case of normal isolated singularities, and was communicated to us by Laza in general.

Let $Z$ be an irreducible variety and $\mu\colon\widetilde{Z}\to Z$ a strong log resolution, with $\mu^{-1}(Z_{\rm sing})_{\rm red}=D$. For an integer $k\geq 0$, we say that $Z$ has $k$-rational singularities if the canonical morphisms

are isomorphisms for all $0\leq i\leq k$. We say that an arbitrary variety $Z$ has $k$-rational singularities if all its connected components are irreducible, with $k$-rational singularities.

By Lemma 1.6, the definition is independent of the choice of strong log resolution.

For $k=0$, we recover the familiar notion of rational singularities.

With the notation in Definition 2.1, the condition that $Z$ has $k$-rational singularities is equivalent to the vanishings

is an isomorphism for all $i\leq k$. Note that for $i=0$, the morphism (2.6) is an isomorphism if and only if $Z$ is normal. On the other hand, for $i\geq 1$, we have

If $Z$ has rational singularities, then the morphism (2.6) is an isomorphism if and only if $\Omega_{Z}^{i}$ is a reflexive sheaf.

The hypothesis implies that $\mu_{*}\omega_{\widetilde{Z}}(D)$ is a reflexive sheaf. Indeed, the fact that $Z$ has rational singularities implies that $Z$ is normal and Cohen-Macaulay and $\mu_{*}\omega_{\widetilde{Z}}\simeq\omega_{Z}$, which is reflexive; see [KM, Theorem 5.10]. Since the inclusion $j\colon\mu_{*}\omega_{\widetilde{Z}}\hookrightarrow\mu_{*}\omega_{\widetilde{Z}}(D)$ is an isomorphism over the smooth locus of $Z$ and $\mu_{*}\omega_{\widetilde{Z}}(D)$ is torsion-free, it is straightforward to deduce that $\mu_{*}\omega_{\widetilde{Z}}(D)$ is reflexive, and in fact isomorphic to $\mu_{*}\omega_{\widetilde{Z}}$. We can thus apply [KS, Theorem 1.5] to deduce that $\mu_{*}\Omega_{\widetilde{Z}}^{i}(\log\,D)$ is reflexive for every $i$. Therefore the morphism (2.6) is an isomorphism if and only if $\Omega_{Z}^{i}$ is reflexive. ∎

It is a natural question whether the morphisms (2.2) are isomorphisms when $X$ has $k$-rational singularities, but $\mu$ is an arbitrary log resolution of $Z$. We now address this.

Let $Z$ be an irreducible variety, $\mu\colon\widetilde{Z}\to Z$ a log resolution of $Z$ with $W$ the complement of the domain of $\mu^{-1}$, and let $D$ be the simple normal crossing divisor on $\widetilde{Z}$ such that $\mu^{-1}(W)_{\rm red}=D$. We assume that $W\cap Z_{\rm sm}\neq\emptyset$ and let $r={\rm codim}_{Z_{\rm sm}}(W\cap Z_{\rm sm})$.

If $R^{q}\mu_{*}\Omega^{i}_{\widetilde{Z}}({\rm log}\,D)=0$ for $q\geq 1$ and $i\leq k$, then $r>k$.

If $Z$ has $k$-rational singularities and $r>k$, then the canonical morphism $\Omega_{Z}^{i}\to\mathbf{R}\mu_{*}\Omega^{i}_{\widetilde{Z}}(\log\,D)$ is an isomorphism for every $i\leq k$.

We first note that since $\mu^{-1}$ is defined in codimension $1$ on $Z_{\rm sm}$, we have $r\geq 2$. In order to prove the assertion in i), given any irreducible component $W_{0}$ of $W$ that intersects $Z_{\rm sm}$, we may replace $Z$ by a suitable smooth open subset $U$ such that $U\cap W_{0}$ is smooth and irreducible. Therefore we may and will assume that both $Z$ and $W$ are smooth and irreducible. By Lemma 1.6, we may assume that $\mu$ is the blow-up of $Z$ along $W$. In this case, we have

(see for example [MP2, Lemma 4.27]). Therefore this is nonzero if $r\leq j\leq\dim(Z)$, hence the assumption in i) implies $r>k$.

In order to prove the assertion in ii), we use again the fact that by Lemma 1.6, we may choose a convenient $\mu$. We thus may and will assume that $\mu=\varphi\circ\psi$, as follows. The morphism $\varphi\colon\widetilde{Z_{0}}\to Z$ is a strong log resolution of $Z$, with $\varphi^{-1}(Z_{\rm sing})_{\rm red}=F$; we denote by $W_{0}\subseteq\widetilde{Z}_{0}$ the union of the strict transforms of the irreducible components of $W$ that intersect $Z_{\rm sm}$. The morphism $\psi\colon\widetilde{Z}\to\widetilde{Z}_{0}$ is a composition of smooth blow-ups

where each $\psi_{j}$ is the blow-up of a smooth, irreducible subvariety $V_{j-1}\subseteq\widetilde{Z}_{j-1}$ that is mapped into $W_{0}$. Moreover, if $E_{j}\subseteq\widetilde{Z}_{j}$ is the exceptional divisor of $\widetilde{Z}_{j}\to\widetilde{Z}_{0}$ and $F_{j}$ is the strict transform of $F$ on $\widetilde{Z}_{j}$, then we assume that $V_{j-1}$ has simple normal crossings with $F_{j-1}+E_{j-1}$ for $1\leq j\leq N$. The existence of such $\psi$ is again a consequence of Hironaka’s theorem. With this notation, note that $D=E_{N}+F_{N}$.

Since $Z$ has $k$-rational singularities, we know that the canonical morphism

is an isomorphism for all $i\leq k$. We deduce using the Leray spectral sequence that in order to prove the assertion in ii), it is enough to show that for every $i\leq k$ and every $j$, with $1\leq j\leq N$, the canonical morphism

is an isomorphism. If $V_{j-1}\subseteq{\rm Supp}(F_{j-1}+E_{j-1})$, then (2.9) is an isomorphism for all $i$, see [MP0, Theorem 31.1(i)]. On the other hand, if $V_{j-1}\not\subseteq{\rm Supp}(F_{j-1}+E_{j-1})$, then (2.9) is an isomorphism if $i\leq{\rm codim}_{\widetilde{Z}_{j-1}}(V_{j-1})-1$, see [MP2, Lemma 4.27]. Note that since $V_{j-1}$ is not contained in the exceptional locus of $\widetilde{Z}_{j-1}\to Z$, it follows that it is the strict transform of its image in $Z$, which is contained in an irreducible component of $W$ meeting $Z_{\rm sm}$. We thus have

hence (2.9) is an isomorphism in this case as well. This completes the proof of the proposition. ∎

Brief review of filtered 𝒟𝒟\mathscr{D}-modules

We will make use of the theory of mixed Hodge modules, for which we refer to [Saito-MHM]. Let $X$ be a smooth, irreducible, $n$-dimensional complex algebraic variety. Recall that every (mixed) Hodge module on $X$ has an underlying filtered $\mathscr{D}_{X}$-module $(\mathcal{M},F)$; the filtration $F$ is called the Hodge filtration. For an integer $q$, the Tate twist $\mathcal{M}(q)$ has the same underlying $\mathscr{D}_{X}$-module, but the filtration is given by $F_{p}\mathcal{M}(q)=F_{p-q}\mathcal{M}$ for all $p\in{\mathbf{Z}}$.

We will use both left and right Hodge modules. Recall that there is an equivalence between the categories of such objects, such that if $(\mathcal{M}^{r},F)$ is the right filtered $\mathscr{D}_{X}$-module that corresponds to $(\mathcal{M},F)$, then there is an isomorphism of $\mathscr{O}_{X}$-modules $\mathcal{M}^{r}\simeq\omega_{X}\otimes_{\mathscr{O}_{X}}\mathcal{M}$ that induces for every $p$ an isomorphism

For example, the left version of the trivial (pure) Hodge module ${\mathbf{Q}}_{X}^{H}[n]$ is $(\mathscr{O}_{X},F)$, where ${\rm gr}^{F}_{i}\mathscr{O}_{X}=0$ for $i\neq 0$, and the corresponding right filtered $\mathscr{D}_{X}$-module is $(\omega_{X},F)$, where ${\rm gr}^{F}_{i}\omega_{X}=0$ for $i\neq-n$.

Recall that for every filtered left Hodge module $(\mathcal{M},F)$ on $X$ and every $i\in{\mathbf{Z}}$, the complex ${\rm Gr}^{F}_{i}{\rm DR}_{X}(\mathcal{M})$ is given by

placed in cohomological degrees $-n,\ldots,0$. The same complex can be written in terms of the corresponding right Hodge module $(\mathcal{M}^{r},F)$ as

The Du Bois complex and k𝑘k-Du Bois singularities

Recall that the Du Bois complex of a complex variety $W$ is an element $(\underline{\Omega}_{W}^{\bullet},F)$ in a suitable filtered derived category. We will only be interested in its shifted graded pieces

which are objects in the bounded derived category of coherent sheaves $D^{b}_{\rm coh}(W)$. For an introduction to the Du Bois complex and for further references, we refer to [GNPP, Chapter V.3] or [PS, Chapter 7.3]. We note that $\mathcal{H}^{i}(\underline{\Omega}_{W}^{p})=0$ for all $i<0$ and $\mathcal{H}^{0}(\underline{\Omega}_{W}^{p})$ is torsion-free; this follows, for example, from the description in [Huber, Theorem 7.12].

For every $W$ as above and every $p$, there is a canonical morphism $\sigma_{p}\colon\Omega_{W}^{p}\to\underline{\Omega}_{W}^{p}$ that is an isomorphism over the smooth locus of $W$. Recall that by definition, $W$ has Du Bois singularities if $\sigma_{0}$ is an isomorphism. More generally, following [Saito_et_al], we will say that $W$ has $k$-Du Bois singularities if $\sigma_{p}$ is an isomorphism for all $p$, with $0\leq p\leq k$.

Suppose that $W$ is an irreducible variety and $\mu\colon\widetilde{W}\to W$ is any resolution of singularities. By functoriality of the Du Bois complex, for every $p$ we have a canonical map

whose composition with $\sigma_{p}$ is the canonical morphism $\Omega_{W}^{p}\to\mathbf{R}\mu_{*}\Omega_{\widetilde{W}}^{p}$. Indeed, this follows using the next remark, since $\mu_{*}\Omega_{\widetilde{W}}^{p}$ is torsion-free and the assertion certainly holds on the locus in $W$ over which $\mu$ is an isomorphism.

The following general remark is used repeatedly throughout the paper.

Let $W$ be a variety and let $u\colon{\mathcal{F}}\to{\mathcal{P}}$ be a morphism in $D^{b}_{\rm coh}(W)$, where ${\mathcal{F}}$ is a coherent sheaf and $\mathcal{H}^{i}({\mathcal{P}})=0$ for all $i<0$. If $\mathcal{H}^{0}({\mathcal{P}})$ is torsion-free and $u|_{V}=0$ for some open dense subset $V$ of $W$, then $u=0$. Indeed, giving such a morphism $u$ is equivalent to giving the corresponding morphism of coherent sheaves ${\mathcal{F}}\to\mathcal{H}^{0}({\mathcal{P}})$, and the assertion is clear.

Local complete intersections. We next restrict to the case of a local complete intersection $Z$ of pure codimension $r$ in $X$, and review the connection established in [MP2] between the Du Bois complex of $Z$ and the Hodge filtration on the local cohomology sheaf $\mathcal{H}^{r}_{Z}(\mathscr{O}_{X})$. This sheaf has a $\mathscr{D}_{X}$-module structure which underlies a (left) mixed Hodge module. In particular, it carries a Hodge filtration $F$, and for every integer $k$ we have an isomorphism

The sheaf $\mathcal{H}^{r}_{Z}(\mathscr{O}_{X})$ also carries a more elementary filtration, also compatible with the order filtration on $\mathscr{D}_{X}$, namely the Ext filtration, given for $p\geq 0$ by

where $\mathcal{I}_{Z}$ is the ideal defining $Z$ in $X$. It is known that for each $p$ we have $F_{p}\mathcal{H}^{r}_{Z}(\mathscr{O}_{X})\subseteq E_{p}\mathcal{H}^{r}_{Z}(\mathscr{O}_{X})$; see [MP2, §3.1 and §3.3].

The singularity level of the Hodge filtration on $\mathcal{H}^{r}_{Z}\mathscr{O}_{X}$ is defined as

with the convention that $p(Z)=-1$ if there are no such $k$. For a general study of this invariant, see [MP2, §3.3]. We only mention a few points: first, we have that $p(Z)=\infty$ if and only if $Z$ is smooth. Moreover, when $Z$ is singular, we have

In general, the singularity level $p(Z)$ only depends on $Z$ (not on the ambient smooth variety $X$). If $Z$ is a hypersurface in $X$, then $p(Z)=\lfloor\widetilde{\alpha}(Z)\rfloor-1$, where $\widetilde{\alpha}(Z)$ is the minimal exponent of $Z$, recalled in the Introduction.

The main result proved in loc. cit. relates $p(Z)$ to the complexity of the Du Bois complex of $Z$, extending the hypersurface case treated in [MOPW] and [Saito_et_al].

For every nonnegative integer $p$, we have $p(Z)\geq k$ if and only if $Z$ has $k$-Du Bois singularities.

C. Results for local complete intersections

We recall that $X$ denotes a smooth irreducible $n$-dimensional variety. All throughout this chapter, $Z$ will be a local complete intersection closed subvariety of $X$, of pure codimension $r$.

In this section we prove the main statement for local complete intersections, the injectivity Theorem A, and deduce Theorem B, stating that $k$-rational local complete intersection singularities are $k$-Du Bois.

In preparation for the proof of Theorem A, we consider for each $0\leq k\leq\dim Z$ the complex $C_{k}^{\bullet}$ defined by

and placed in cohomological degrees $-k,\ldots,0$, obtained by truncating the generalized Eagon-Northcott complex $D_{n-r-k}$ (see [Bruns, Chapter 2.C]) associated to the canonical morphism

keeping the first $k+1$ terms, and then suitably translating. We have a canonical isomorphism ${\mathcal{H}}^{0}(C_{k}^{\bullet})\simeq\Omega_{Z}^{k}$ induced by the exact sequence

It is shown in [MP2, §5.2] (right before the proof of Theorem F), that if ${\rm codim}_{Z}(Z_{\rm sing})\geq k$, then $C_{k}^{\bullet}$ is a resolution of $\Omega_{Z}^{k}$.

We are now ready to prove the injectivity theorem for the duals of the graded pieces of the Du Bois complex of a local complete intersection.

Since $p(Z)\geq k-1$, using (4.4) we deduce that

We conclude that the complex $C_{k}^{\bullet}$ is a locally free resolution of $\Omega_{Z}^{k}$ over $\mathscr{O}_{Z}$; for $k=0$ this is obvious, while for $k\geq 1$ the inequality above implies in particular that ${\rm codim}_{Z}(Z_{\rm sing})\geq k$, hence we can apply the discussion before the start of the proof. We can therefore compute $\mathbf{R}\mathcal{H}om_{\mathscr{O}_{Z}}(\Omega_{Z}^{k},\omega_{Z})$ by applying the functor $\mathcal{H}om_{\mathscr{O}_{Z}}(-,\omega_{Z})$ to $C_{k}^{\bullet}$; this leads to the complex

placed in cohomological degrees $-k,\ldots,0$. But this turns out to be precisely the complex ${\rm Gr}^{E}_{k-n}{\rm DR}_{X}\mathcal{H}^{r}_{Z}(\mathscr{O}_{X})$, i.e. the associated graded of the de Rham complex of the $\mathscr{D}_{X}$-module $\mathcal{H}^{r}_{Z}(\mathscr{O}_{X})$ with respect to the Ext filtration; see [MP2, §5.2], especially the discussion before formula (13.2). In other words, under our assumption on the minimal exponent, we have a natural isomorphism

On the other hand, by dualizing the isomorphism in (4.3), we also have

where this time the associated graded is taken with respect to the Hodge filtration. Using Grothendieck duality for the inclusion $Z\hookrightarrow X$, we deduce an isomorphism

in $D^{b}_{\rm coh}(X)$. We claim that via these identifications, the canonical morphism

induced by the inclusion of the Hodge filtration into the Ext filtration is the dual of the canonical morphism $\Omega_{Z}^{k}\to\underline{\Omega}_{Z}^{k}$. Indeed, since $\mathcal{H}^{0}(\underline{\Omega}_{Z}^{k})$ is torsion-free, it follows from Remark 4.2 that it is enough to check this on the complement of the singular locus of $Z$, and this is straightforward (though rather tedious).

Note that $\varphi_{k}$ is a morphism of complexes placed in nonpositive degrees. Moreover, by definition of the singularity level, since $p(Z)\geq k-1$, it follows $\varphi_{k}$ is an injective morphism of complexes and its cokernel is concentrated in degree . It is then immediate to check that ${\mathcal{H}}^{i}(\varphi_{k})$ is an isomorphism for all $i\neq 0$ and ${\mathcal{H}}^{0}(\varphi_{k})$ is injective. ∎

The fact that $k$-rational implies $k$-Du Bois in our setting is now an easy application.

Since the connected components of $Z$ are irreducible, we may and will assume that $Z$ is irreducible. We prove the result by induction on $k\geq 0$. The canonical morphism $\Omega_{Z}^{p}\to\mathbf{R}\mu_{*}\Omega_{\widetilde{Z}}^{k}(\log D)$ factors as

(see Remark 4.1). Dualizing this, we obtain the composition

Our hypothesis gives that $\tau_{k}\circ\sigma_{k}$ is an isomorphism, and therefore so is $\sigma_{k}^{\prime}\circ\tau^{\prime}_{k}$. On the other hand, since $Z$ is also $(k-1)$-rational, it is $(k-1)$-Du Bois by induction (this is vacuous if $k=0$). Therefore Theorem A applies, to the effect that $\sigma^{\prime}_{k}$ induces injective maps in cohomology. These maps are also surjective, hence $\sigma^{\prime}_{k}$ is an isomorphism, and so is $\sigma_{k}$. Therefore $Z$ has $k$-Du Bois singularities. ∎

Duality

The $k$-rationality condition has interesting consequences regarding duality for the graded pieces of the Du Bois complex. We start with a general construction:

For every irreducible variety $Z$ of dimension $d$ and every $k\geq 0$, there is a canonical morphism

in the bounded derived category of coherent sheaves on $Z$.

Let $f\colon Y\to Z$ be any resolution of singularities. The functoriality of the Du Bois complex gives a morphism

for each $k$. On the other hand, on $Y$ we have an isomorphism

and we get an isomorphism $\beta_{k}$ on $Z$ as the composition

where the first isomorphism is $\mathbf{R}f_{*}(\tau^{Y}_{k})$ and the second isomorphism is provided by relative duality for $f$. Finally, taking the Grothendieck dual of $\alpha_{d-k}$ provides a morphism

We need to show that this is independent of the choice of resolution. Since any two resolutions are dominated by a third one, it is enough to show that if $f$ is as above and $g\colon W\to Y$ is a proper birational morphism, with $W$ smooth, then the morphisms $\psi_{k}$ corresponding to $f$ and $h=f\circ g$ coincide. This follows easily from the definitions once we know that the following diagram is commutative

in which the top horizontal map is the $\beta_{k}$ with respect to $f$ and the bottom map is $\beta_{k}$ with respect to $h$. This commutativity follows from the functoriality of relative duality and its compatibility with composition of proper morphisms, together with the commutativity of the diagram

The latter follows in turn from the fact that it trivially holds over any open subset over which $g$ is an isomorphism (note that we are comparing two morphisms between vector bundles). This completes the proof of the proposition. ∎

It is interesting to understand under what assumptions $\psi_{k}$ is an isomorphism, as in the case of smooth varieties. Note that this requires assumptions on the singularities. For instance, when $k=0$, even when $Z$ is Cohen-Macaulay and Du Bois, $\psi_{0}$ being an isomorphism is equivalent to the condition $\mu_{*}\omega_{\widetilde{Z}}\simeq\omega_{Z}$, where $\mu\colon\widetilde{Z}\to Z$ is a resolution of singularities; in other words, it is equivalent to $Z$ having rational singularities. More generally, we now show that the condition that $\psi_{k}$ is an isomorphism is precisely the difference between having $k$-rational and $k$-Du Bois singularities.

By Theorem B, we may assume that $Z$ has $k$-Du Bois singularities, hence $p(Z)\geq k$ by Theorem 4.5. We consider a strong log-resolution $\mu\colon\widetilde{Z}\to Z$ as in Definition 2.1. We put $W=Z_{\rm sing}$, and $D=\mu^{-1}(W)_{\rm red}$. By [Steenbrink, Proposition 3.3], we have an exact triangle

Since $p(Z)\geq k$, using (4.4) we deduce that

It follows that $\underline{\Omega}_{W}^{d-k}=0$, hence the above triangle implies that the canonical morphism

is an isomorphism. Equivalently, using Grothendieck duality, the induced morphism

is an isomorphism. On the other hand, since $\Omega_{Z}^{k}\to\underline{\Omega}_{Z}^{k}$ is an isomorphism, it follows that the canonical morphism $\Omega_{Z}^{k}\to\mathbf{R}\mu_{*}\Omega_{\widetilde{Z}}^{k}({\rm log}\,D)$ can be identified with the composition $\nu_{k}\circ\psi_{k}$. Thus by definition $Z$ has $k$-rational singularities if and only if $\psi_{k}$ is an isomorphism. ∎

Note that in the situation of Corollary C, it follows by duality that one can also compute $\underline{\Omega}_{Z}^{d-k}$ in terms of the sheaf of Kähler differentials $\Omega_{Z}^{k}$.

If $Z$ is has $k$-rational singularities, then we have an isomorphism

The proof of Corollary C shows that if ${\rm codim}_{Z}(Z_{\rm sing})>k$, then we have a natural isomorphism

In particular, this is the case if $2p(Z)+1>k$.

It is well known (see e.g. [PS, Theorem 7.29]) that for any variety $W$ and any $p$, we have $\mathcal{H}^{i}(\underline{\Omega}_{W}^{p})=0$ for $i>\dim W-p$. Thus a numerical consequence of Corollary 6.3 is that if $Z$ has $k$-rational singularities, then ${\rm depth}(\Omega_{Z}^{k})\geq d-k$. However, this holds even if we only assume that $Z$ has $k$-Du Bois singularities: this follows from the Auslander-Buchsbaum formula and the fact that in this case $C_{k}^{\bullet}$ gives a locally free resolution of $\Omega_{Z}^{k}$ (see the discussion before the proof of Theorem A).

We also note that the duality morphism $\psi_{k}$ has a more familiar interpretation for an arbitrary $d$-dimensional irreducible variety $Z$ whose singular locus $W$ has small dimension. Indeed, it follows from the exact triangle (6.2) that the canonical morphism

is an isomorphism when $\dim W<k$, while just as in the proof of Corollary C, the canonical morphism

We conclude that, in general, under the assumption $\dim W<{\rm min}\{k,d-k\}$, the duality morphism $\psi_{k}$ is naturally identified with the morphism

obtained by pushing forward the canonical inclusion on $\widetilde{Z}$. This holds for instance when $Z$ has isolated singularities and $1\leq k\leq d-1$; in this case, at the level of cohomology the map can be studied as in [FL1, §3], using classical Hodge theory arguments involving the mixed Hodge structure on the cohomology of the link of the singularity.

Local vanishing for k𝑘k-Du Bois singularities

We continue to assume that $Z\subseteq X$ is a local complete intersection closed subvariety of pure codimension $r$, and dimension $d=n-r$. For completeness, we start by recording a consequence of results we established in [MP2, §5.2]. Let $f\colon Y\to X$ be a log resolution of the pair $(X,Z)$ and let $E=f^{-1}(Z)_{\rm red}$ (recall that we always assume that $f$ is an isomorphism over $X\smallsetminus Z$).

If $k$ is a nonnegative integer such that $p(Z)\geq k$, then

The hypothesis implies that $\mathcal{H}^{q}(\underline{\Omega}_{Z}^{p})=0$ in the range of the conclusion, by Theorem 4.5. The proof is then identical to that of [MOPW, Corollary 1.2]. ∎

We next establish the generalization of [MOPW, Theorem 1.4] to local complete intersections.

unless $q=d$ (in which case $Z$ is either smooth or a curve with nodal singularities). In particular, we have

unless $q=d$ or $q=d+1$ (in the latter case $Z$ being smooth).

Note first that by [Steenbrink, Proposition 3.3], we have an exact triangle

This immediately implies that (7.3) holds if $Z$ is smooth and $q\neq d+1$. The first assertion in the theorem is also clear if $Z$ is smooth, hence from now on we assume that $Z$ is singular.

In this case it follows from (4.4) that $p(Z)\leq\frac{d-1}{2}$, hence

In particular, we have $q\neq d+1$. We thus have $\mathcal{H}^{d+1-q}(\Omega_{X}^{q})=0$, and the vanishing in (7.3) follows if we know that $\mathcal{H}^{d-q}(\underline{\Omega}_{Z}^{q})=0$.

Therefore it is enough to prove the first assertion in the theorem. We know that $q\leq d$ and it follows from (7.5) that if $q=d$, then $d=1$. Hence $Z$ is a locally complete intersection curve with Du Bois singularities, and it is known that this implies that $Z$ has nodal singularities. (Note that in this case we clearly have ${\mathcal{H}}^{0}(\underline{\Omega}_{Z}^{1})\neq 0$.)

From now on we assume that $q\leq d-1$. We use the notation in the proof of Theorem A. By (4.3) we have

The hypothesis $p(Z)+1\geq q$ implies that $F_{i}\mathcal{H}^{r}(\mathscr{O}_{Z})=E_{i}\mathcal{H}^{r}(\mathscr{O}_{Z})$ for all $i\leq q-1$, hence the morphism of complexes

is injective and its cokernel is a sheaf $\mathscr{F}$ (supported in cohomological degree ). We thus have an exact sequence

and we see that it is enough to show that ${\mathcal{E}xt}^{n}_{\mathscr{O}_{X}}(B_{q}^{\bullet},\omega_{X})=0$.

Recall from the proof of Theorem A that $B_{q}^{\bullet}$ can be described as the complex

placed in cohomological degrees $-q,\ldots,0$. The spectral sequence

This is a consequence of the fact that $q\leq d-1$, combined with the fact that if $r=n-d$, then for every locally free sheaf $\mathscr{E}$ on $Z$ we have

We are now able to prove the local vanishing theorem for $k$-Du Bois singularities.

By a general vanishing result due to Steenbrink (see [Steenbrink, Theorem 2]), we always have $R^{q}\mu_{*}\Omega^{p}_{\widetilde{Z}}(\log\,D)(-D)=0$ if $p+q>d$, i.e. the statement in i) (for an arbitrary variety $Z$). From now on we assume that $p+q\leq d$.

By [Steenbrink, Proposition 3.3], if $W=Z_{\rm sing}$, then we have an exact triangle

Taking the long exact sequence in cohomology, we obtain the exact sequence

By Theorem 4.5, $Z$ having $k$-Du Bois singularities is equivalent to the fact that $p(Z)\geq k$. This in turn implies by (4.4) that $\dim(W)\leq d-2k-1$. Using a general result on the vanishing of the cohomologies of the graded pieces of the Du Bois complex (see [PS, Theorem 7.29]), we then have

On the other hand, if $p\leq k$ and $q\geq 1$, we have ${\mathcal{H}}^{q}(\underline{\Omega}_{Z}^{p})=0$ by definition of $k$-Du Bois singularities. The assertion in ii) thus follows from the exact sequence (7.6).

The assertion in iii) follows directly from Theorem 7.2; note that we avoid the exceptional cases in that theorem since $q\leq d-1$. Let’s now prove the assertion in iv).

We assume that $q\geq\max\{d-k-1,1\}$. Suppose first that $p+q\geq d-2k+1$. If $p\leq k$, then we are done by ii). Hence we may and will assume that $p\geq k+1$. By i), we may also assume that $p+q\leq d$, hence $q\leq d-p\leq d-k-1$. Since we are assuming $q\geq d-k-1$, it follows that $q=d-k-1$ and $p=k+1$, hence we are done by iii).

Suppose now that $p+q\leq d-2k$. Since $q\geq d-k-1$, it follows that $p+k\leq 1$, hence we are left with 3 possible case, when $(k,p)=(1,0)$, $(0,0)$, or $(0,1)$.

If $k=1$, then the condition $p+q\leq d-2k$ gives $d\geq 3$, while the inequality (4.4) gives $\dim(W)\leq d-3$. If $k=0$, then we only get $\dim(W)\leq d-2$, using the fact that $Z$ is normal. Since the case when $Z$ is a smooth curve is obvious, we assume $d\geq 2$. Using the fact that $Z$ is normal, with Du Bois singularities, we can then apply [GKKP, Theorem 13.3] which says that $R^{i}\mu_{*}\mathscr{O}_{\widetilde{Z}}(-D)=0$ for $i>\max\{\dim(W),0\}$. Therefore we are done in this case.

If $p=1$ and $k=0$, then we have $d\geq 2$ and we need to show that

This follows by applying [GKKP, Theorem 14.1].Note that the result we refer to is stated for log canonical pairs $(Z,E)$. However, when $E=0$, the same proof works when $Z$ is only assumed to have Du Bois singularities; this is the only condition that is used in the proof, via [GKKP, Theorem 13.3]. This completes the proof of the theorem. ∎

D. Results for hypersurfaces

In this chapter we show that $k$-rational singularities are characterized numerically in terms of their minimal exponent; see the discussion around Theorem E and Corollary F in the introduction.

All throughout this chapter, $Z$ will be a (nonempty) hypersurface, meaning a closed subscheme of pure codimension $1$, in the smooth irreducible $n$-dimensional variety $X$.

𝑘1>k+1 implies $k$-rational singularities Our goal in this section is to prove one implication in Theorem E, namely to show the following:

If $k$ is a nonnegative integer such that $\widetilde{\alpha}(Z)>k+1$, then $Z$ has $k$-rational singularities.

Under the hypotheses in Theorem 8.1, we have $\widetilde{\alpha}(Z)>1$, hence $Z$ is normal; in particular, it is reduced and all its connected components are irreducible.It follows from [Saito-B, Theorem 0.4] that $\widetilde{\alpha}(Z)>1$ implies that $Z$ has rational singularities, hence in particular it is normal. However, we prefer to give a direct argument for this elementary fact. Indeed, we have ${\rm lct}(X,Z)=1$, hence $Z$ is reduced and in codimension $1$ its singularities are at worst nodal singularities. Since the Bernstein-Sato polynomial of a union of two smooth divisors meeting transversally is $(s+1)^{2}$, we conclude that if $\widetilde{\alpha}(Z)>1$, we have $Z$ smooth in codimension $1$. Therefore $Z$ is normal by Serre’s criterion since it is Cohen-Macaulay, being a hypersurface.

In order to prove the theorem, we may clearly treat each connected component of $Z$ separately. In light of Remark 8.2, we thus may and will assume from now on that $Z$ is normal. We choose a log resolution $\pi\colon Y\to X$ of $(X,Z)$ and write $\pi^{-1}(Z)_{\rm red}=E$. The morphism $\pi$ may be taken to be projective, and we will do so in what follows. The key input for the proof of Theorem 8.1 is the following vanishing result, interesting in its own right.

If $k$ is a nonnegative integer and $\widetilde{\alpha}(Z)>k+1$, then

In particular, it holds for all $p$ and all $q\geq\max\{n-k-2,1\}$.

Part i) is known to hold for an arbitrary hypersurface $Z$ in $X$; cf. (8.6) below. The case when $p=k+2$ and $q=n-k-2$ is the content of [MP1, Corollary C]. In what follows we will show that (a simplified version of) the argument in loc. cit. is enough to give the vanishing in ii) as well.

We note that if $Z$ is smooth, then all the vanishings in Theorem 8.3 are well known; see for example [MP0, Theorem 31.1(i)]. On the other hand, if $Z$ is singular, we have $\widetilde{\alpha}(Z)\leq\tfrac{n}{2}$ by [Saito_microlocal, Theorem 0.4]. If $\widetilde{\alpha}(Z)>k+1$, then $k\leq\tfrac{n-3}{2}$, hence if $q\geq n-k-2$, we automatically have $q\geq 1$.

We begin by relating the higher direct images of the sheaves of log differentials on $Y$ to certain mixed Hodge modules on $X$. Recall that $\omega_{X}(*Z)$ is the right Hodge module whose underlying $\mathscr{D}_{X}$-module is the module of top rational differentials on $X$ with poles along $Z$. It is a basic fact that for every $p$ we have an isomorphism

One way to see this is by using the explicit filtered resolution of $\omega_{Y}(*E)$ in [MP0, Proposition 3.1] to get an isomorphism

The isomorphism in (8.5) then follows from the fact that $\omega_{X}(*Z)$ is the push-forward of $\omega_{Y}(*E)$ (in the category of Hodge modules), using Saito’s Strictness Theorem [Saito-MHP, Section 2.3.7]; cf. [MP0, Section C.4].

Since the complex on the right-hand side of (8.5) is placed in nonpositive degrees, we immediately deduce the following vanishing result

(cf. [Saito-LOG, Corollary 3] and [MP0, Theorem 32.1]). In other words, part i) in Theorem 8.3 holds for an arbitrary hypersurface $Z$ in $X$.

With the notation above, we have $R^{q}\pi_{*}\Omega_{Y}^{p}(\log\,E)=0$ for $q\geq 1$ and arbitrary $p$, provided that

We have a short exact sequence of filtered right $\mathscr{D}_{X}$-modules underlying mixed Hodge modules

where the underlying $\mathscr{D}_{X}$-module of ${\mathcal{H}}_{Z}^{1}(\omega_{X})$ is the first local cohomology module of $\omega_{X}$ along $Z$. By taking ${\rm Gr}^{F}_{-p}{\rm DR}_{X}(-)$, using (8.5) we obtain an exact triangle

This immediately implies the statement by passing to cohomology. ∎

The key point for the proof of Theorem 8.3 is the use of duality. Recall that by the compatibility between duality and the graded de Rham complex (see [Saito-MHP, Sections 2.4.5 and 2.4.11]), for every $p\in{\mathbf{Z}}$, we have an isomorphism

We compute ${\mathbf{D}}({\mathcal{H}}^{1}_{Z}(\omega_{X}))$ using the $V$-filtration. We work locally and suppose that $Z$ is a singular hypersurface in $X$ defined by $f\in\mathscr{O}_{X}(X)$. Recall that if $\iota\colon X\to X\times{\mathbf{C}}$ is the graph embedding \iota(x)=\big{(}x,f(x)\big{)}, then the $V$-filtration of Malgrange and Kashiwara is an increasing, exhaustive, discrete, and right-continuous filtration on

indexed by rational numbers and characterized by a few conditions (see for example [Saito-MHP, Section 3.1]). For example, if $t$ is the coordinate on ${\mathbf{C}}$, then $V_{\alpha}B_{f}\cdot t\subseteq V_{\alpha-1}B_{f}$ and $V_{\alpha}B_{f}\cdot\partial_{t}\subseteq V_{\alpha+1}B_{f}$. For every $\alpha\in{\mathbf{Q}}$, we put ${\rm Gr}^{V}_{\alpha}B_{f}=V_{\alpha}B_{f}/V_{<\alpha}B_{f}$. We note that the Hodge filtration on $B_{f}$ is given by

By [Saito-MHM, Section 2.24], we have a short exact sequence

which after taking duals gives the exact sequence

On the other hand, by [Saito_duality, Theorem 1.6] we have isomorphisms

such that the exact sequence (8.9) becomes

By taking the corresponding graded piece of the de Rham complex, we obtain the short exact sequence of complexes

The connection between the minimal exponent and the $V$-filtration is provided by the following result due to Saito, see [Saito-MLCT, (1.3.8)]: if $q$ is a nonnegative integer and $\alpha\in(0,1]$ is a rational number, then

Moreover, it was shown in [MP1, Proposition 4.5] that (after translating from left to right $\mathscr{D}_{X}$-modules) the condition $\widetilde{\alpha}(Z)>k+1$ implies

where $J=\{h\in\mathscr{O}_{X}\mid\omega_{X}\cdot h\partial_{t}^{k+1}\subseteq V_{-1}B_{f}\}$. We also note that