The higher Du Bois and higher rational properties for isolated singularities

Introduction

In algebraic geometry, it is often important to identify singularities which are “mild” from an appropriate viewpoint, e.g. that of Hodge theory or birational geometry. From a cohomological perspective, rational and Du Bois singularities are classical examples of such types of singularities, and canonical and log canonical singularities form an important class from an adjunction theoretic perspective. There is a well understood relationship between these classes of singularities (see [Ste83], [Kov99], [KK10]). In particular, if a singularity is normal and Gorenstein, then it is Du Bois if and only if it is log canonical. Recently, M. Mustaţă and M. Popa [MP19] initiated the study of higher adjunction properties and introduced the notion of $k$-log canonical hypersurface singularities. This led to the study of $k$-Du Bois singularities ([MOPW23] and [JKSY22]) and $k$-rational singularities ([KL20], [FL22a, FL22b]). The case $k=0$ recovers the standard notions (of Du Bois and rational singularities), while as $k$ increases the singularities become milder, leading to Hodge theoretic behavior closer to that of the smooth case (cf. [FL22b, Cor. 1.4, Cor. 1.11]).

The purpose of this note is to discuss these notions of singularities and associated results in the case of isolated singularities, with particular attention to isolated lci singularities, using techniques of the “classical” Hodge theory of singularities as developed by Steenbrink in [Ste83] and [Ste97]. Along the way, we find new relations between well-known invariants of singularities and are able to extend some results previously known in the hypersurface case to the case of isolated lci singularities. The starting point of this paper was §3 of the first version of [FL22a], which we expand and streamline here. The subsequent papers [FL22b], [MP22b] and [CDM22] deal with the general case of certain results in this paper, but the discussion of the isolated case helps to clarify and extend many of these statements.

We begin by defining the higher Du Bois and higher rationality properties.

While this definition might seem hard to apply in practice, the situation becomes manageable in the case of isolated singularities. Specifically, assuming that $(X,x)$ is an isolated singularity, we let $X$ be a good Stein representative and $\pi\colon\widehat{X}\to X$ be a log resolution with reduced exceptional divisor $E$. Then the $k$-Du Bois condition reads ([PS08, Ex. 7.25], but see also [FL22b, Rem. 3.19]):

Suppose that $X$ is normal and has an isolated singularity at $x$. Then $X$ has $k$-Du Bois singularities $\iff$ for all $p\leq k$, the map $\Omega^{p}_{X}\to\mathcal{H}^{0}\underline{\Omega}^{p}_{X}$ is an isomorphism and, for all $p\leq k$ and all $q>0$, $H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E)(-E))=0$.

Moreover, if $X$ has an isolated lci singularity, then $X$ has $k$-Du Bois singularities $\iff$ for all $p\leq k$ and all $q>0$, $H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E)(-E))=0$. ∎

As in the case of $k$-Du Bois singularities, the situation for rational singularities is much simpler if $X$ has isolated singularities [FL22b, Cor. 3.17 and Lem. 3.18]:

Suppose that $X$ is normal and has an isolated singularity at $x$. Then $X$ has $k$-rational singularities $\iff$ for all $p\leq k$, the natural map $\Omega_{X}^{p}\to R^{0}\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)$ is an isomorphism, and $H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E))=0$ for all $q>0$.

Moreover, if $X$ has an isolated lci singularity, then $X$ has $k$-rational singularities $\iff$ for all $p\leq k$ and all $q>0$, $H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E))=0$. ∎

In [FL22b], partially generalizing a theorem of Kovács [Kov99], we showed:

Suppose that $X$ is $k$-rational and that $X$ has either isolated or lci singularities. Then $X$ is $k$-Du Bois. ∎

Mustaţă and Popa [MP22b] have given an independent proof of Theorem 1.5 for the case of lci singularities.

For hypersurface singularities, not necessarily isolated, Saito has defined an invariant $\widetilde{\alpha}_{X,x}$ as follows:

The $k$-Du Bois and $k$-rational properties are related to $\widetilde{\alpha}_{X,x}$ as follows [JKSY22, Thm. 1], [MOPW23, Thm. 1.1.]:

The hypersurface $X$ has $k$-Du Bois singularities in a neighborhood of $x$ $\iff$ $\widetilde{\alpha}_{X,x}\geq k+1$. ∎

In the first version of [FL22a], we showed:

If $X$ has an isolated hypersurface singularity at $x$, then $X$ is $k$-rational $\iff$ $\widetilde{\alpha}_{X}>k+1$. ∎

Suppose that $X$ has an isolated hypersurface singularity at $x$. If $X$ is $k$-Du Bois, then $X$ is $(k-1)$-rational. ∎

Saito (appendix to [FL22b]) and Mustaţă and Popa [MP22b] have proved Theorem 1.8 and hence Corollary 1.9 without assuming isolated singularities.

One goal of this paper is to give a short proof of Theorem 1.5 for isolated singularities and to generalize Corollary 1.9 to the isolated lci case. A key point is to find the appropriate analogues of Theorems 1.7 and 1.8 in the lci case. Saito’s invariant $\widetilde{\alpha}_{X}$, which is defined for hypersurfaces and takes on positive rational values, is replaced by two integer invariants, $\sum_{p=0}^{k}s_{p}$ and $\sum_{p=0}^{k}s_{n-p}$. Here, $s_{p}=\dim\operatorname{Gr}^{p}_{F}H^{n}(M)$ is the dimension of the $p^{\text{\rm}{{th}}}$ graded piece of the Hodge filtration on the Milnor fiber $M$ of $X$, and is independent of the choice of a smoothing in the lci case. We show the following (cf. Proposition 2.12 and Corollary 4.3):

$\displaystyle\sum_{p=0}^{k-1}s_{n-p}\leq\sum_{p=0}^{k}s_{p}\leq\sum_{p=0}^{k}s_{n-p}.$

Then Theorems 1.7 and 1.8 are replaced by the following (Theorem 5.1 and Theorem 5.3):

$X$ is $k$-Du Bois $\iff$ $\displaystyle\sum_{p=0}^{k}s_{p}=0$.

$X$ is $k$-rational $\iff$ $\displaystyle\sum_{p=0}^{k}s_{n-p}=0$

The case $k=0$ reads: $X$ is Du Bois $\iff$ $s_{0}=0$, and $X$ is rational $\iff$ $s_{n}=0$. This case is due to Steenbrink and Saito.

Combining the two results, we immediately see:

Suppose that $X$ has an isolated lci singularity at $x$. If $X$ is $k$-Du Bois, then $X$ is $(k-1)$-rational. ∎

Chen-Dirks-Mustaţă have recently proved Corollary 1.14 without assuming that the singularities of $X$ are isolated [CDM22]. The basic idea of the proof is as follows. Mustaţă and Popa [MP22a, Theorem F] have defined an integer invariant $p(X)$ in terms of filtrations and have showed that, for $X$ a local complete intersection, $p(X)\geq k$ $\iff$ $X$ is $k$-Du Bois. There is also the analogue $\widetilde{\alpha}_{X}$ of the minimal exponent as defined by Chen-Dirks-Mustaţă-Olano [CDMO22]. If $X$ is a local complete intersection of codimension $r$, then it follows from [MP22a] that $X$ is $k$-Du Bois $\iff$ $\widetilde{\alpha}_{X}\geq k+r$ and from [CDM22] that $X$ is $k$-rational $\iff$ $\widetilde{\alpha}_{X}>k+r$. Combining these two results gives a proof of Corollary 1.14 in the non-isolated case.

Although the methods of this paper only apply to the case of isolated singularities, they have the added benefit of exhibiting the following strong numerical connection between higher Du Bois and rational singularities in the isolated lci case. For example, we show the following:

Let $(X,x)$ be an isolated lci of dimension $n$.

$X$ is $k$-Du Bois $\iff$ $X$ is $(k-1)$-rational and $b^{k,n-k-1}=0$,

The contents of this paper are as follows. In Section 2, we collect some basic facts about the Hodge theory of resolutions of isolated singularities. We further study the invariants $\sum_{p=0}^{k}s_{p}$ and $\sum_{p=0}^{k}s_{n-p}$ in detail and prove Proposition 2.12 as in Steenbrink [Ste97]. Section 3 gives a short proof of Theorem 1.5 in the isolated case, following the strategy of Steenbrink’s proof for the case $k=0$ [Ste83, Proposition 3.7]. The remainder of the paper is concerned with the isolated lci case. Section 4 is devoted to an analysis of Steenbrink’s construction of the mixed Hodge structure on the Milnor fiber. As a corollary, we obtain further relations among the basic numerical invariants (Theorem 4.5). With these preliminaries, we prove the main theorems characterizing $k$-Du Bois and $k$-rational isolated lci singularities in Section 5. We also prove a new inverse of adjunction type result (Corollary 5.7). In Section 6, we specialize to the case of isolated hypersurface singularities. Our main interest is in those which are $k$-Du Bois but not $k$-rational, or equivalently those for which $\widetilde{\alpha}_{X}=k+1$. A result of Dimca-Saito [DS12, §4.11] shows that these singularities, which we term $k$-liminal, have especially appealing properties. These results, for $k=1$, are used in an essential way in [FL22a].

We work in the analytic category. $X$ will always denote a good Stein representative for the germ of the isolated singularity $(X,x)$ of dimension $n\geq 2$, and $\pi\colon\widehat{X}\to X$ will denote a log resolution, with exceptional divisor $E$. In particular, we will always assume that $X$ is contractible and hence that $E$ is a deformation retract of $\widehat{X}$. Let $U=X-\{x\}=\widehat{X}-E$ be the link of the singularity. Then $U$ has the homotopy type of an oriented $(2n-1)$-manifold $L$.

Acknowledgements

It is a pleasure to thank M. Mustaţă, M. Popa, and M. Saito for stimulating correspondence during the preparation of this paper. We would also like to thank the referee for a careful reading of the paper and several helpful suggestions.

Some general results

Following our conventions, $X$ is a good Stein representative of an isolated singularity of dimension $n\geq 2$, $\pi\colon\widehat{X}\to X$ is a log resolution with exceptional divisor $E$, and $U=X-\{x\}=\widehat{X}-E$. If $L$ is the link of the singularity, then $U$ and $L$ have the same homotopy type. We begin by recalling some results which hold in slightly greater generality before specializing to the lci case.

Let $E$ be a simple normal crossing divisor in a complex manifold $\widehat{X}$.

Setting $E^{[k]}$ to be the locus of $k$-fold intersections if the components of $E$, with $E^{}=\coprod_{i}E_{i}$, there is an exact sequence (omitting the inclusion morphisms)

The morphism $\Omega_{\widehat{X}}^{\bullet}\to\Omega_{E}^{\bullet}/\tau_{E}^{\bullet}$ is surjective and its kernel is $\Omega_{\widehat{X}}^{\bullet}(\log E)(-E)$. ∎

From now on, we return to the convention that $X$ is a good Stein representative for the isolated singularity $X$, with log resolution $\widehat{X}$. Thus, for a coherent sheaf $\mathcal{F}$ on $\widehat{X}$ and for $i>0$, we can identify $R^{i}\pi_{*}\mathcal{F}$ with $H^{i}(\widehat{X};\mathcal{F})$. The following is the fundamental vanishing theorem of Guillén, Navarro Aznar, Pascual-Gainza, Puerta and Steenbrink (see e.g. [PS08, p. 181]):

For $p+q>n$, $H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E)(-E))=0$. ∎

The first part of the following lemma is due to Namikawa-Steenbrink [NS95, p. 407], [Ste97, p. 1369]:

Under the convention that $E$ is a deformation retract of $\widehat{X}$, let $L$ be the link of the pair $(\widehat{X},E)$.

where $\operatorname{Gr}^{p}_{F}H^{p+q}(L)$ denotes the associated graded for the Hodge filtration of the mixed Hodge structure on $H^{p+q}(L)$.

which is an exact sequence of mixed Hodge structures. Similarly, there are exact sequences

(i) By Lemma 2.1(iii), there is an exact sequence

Taking hypercohomology, we have a long exact sequence

(ii) We have the long exact hypercohomology sequence associated to

Note that $\Omega^{\bullet}_{\widehat{X}}(\log E)|E$ is independent of the choice of the representative $X$ representing the germ, and the corresponding cohomology and hypercohomology groups are independent of the choice of a resolution. Likewise, the mixed Hodge structure on $H^{k}(L)$ only depends on the germ $(X,x)$. By contrast, the mixed Hodge structure on the cohomology $H^{k}(M)$ of the Milnor fiber depends on the choice of a smoothing, even in the lci case.

As a corollary of Lemma 2.3(i), we obtain the following, first noted by Steenbrink in case $k=0$ [Ste97, p. 1369]:

Suppose that the isolated singularity $X$ is $k$-Du Bois. Then $H^{n-k-1}(\widehat{X};\Omega^{k+1}_{\widehat{X}}(\log E)(-E))=0$.

In the spectral sequence of Lemma 2.3(i), $E_{1}^{p,q}=0$ for $p+q>n$ by Theorem 2.2, and $E_{1}^{p,q}=0$ for $p\leq k$ and all $q>0$ by the hypothesis that $X$ is $k$-Du Bois (Theorem 1.3). Then an examination of the spectral sequence shows that all differentials $d_{r}$ with source or target $E_{1}^{k+1,n-k-1}=H^{n-k-1}(\widehat{X};\Omega^{k+1}_{\widehat{X}}(\log E)(-E))$ are . Thus

Since the spectral sequence converges to , we must have $H^{n-k-1}(\widehat{X};\Omega^{k+1}_{\widehat{X}}(\log E)(-E))=0$. ∎

Finally, we note the semipurity theorem of Goresky-MacPherson [Ste83, Theorem 1.11]:

The morphism of mixed Hodge structures $H^{k}_{E}(\widehat{X})\to H^{k}(E)$ is injective for $k\leq n$, surjective for $k\geq n$, and an isomorphism for $k=n$. Thus, for $i\leq n-1$, the map $H^{i}(E)\to H^{i}(L)$ is surjective and hence $\operatorname{Gr}^{W}_{r}H^{i}(L)=0$ for $r>i$. ∎

2. Numerical invariants

Following [Ste97], for $q>0$, we define the Du Bois invariants

The following is [Ste97, Lemma 1] (where we are mainly interested in the case $i=n-1$):

Fix $k$ and $i$ with $0\leq k\leq i\leq n-1$. Then

Note that, if $i-k\leq s\leq i$ and $r+s\leq i$, then $r\leq i-s\leq k$, so the second sum is greater than or equal to the first, giving the inequality. Equality holds for all $k^{\prime}\leq k$ $\iff$ for all $k^{\prime}\leq k$ , $h^{r,s}_{i}=0$ for $r\leq k^{\prime}$ and $s\leq i-k^{\prime}-1$ (and hence automatically $r+s\leq i-1<i$) $\iff$ $h^{r,s}_{i}=0$ for $r\leq k$, $r+s\leq i-1$. This is equivalent to: $\operatorname{Gr}_{F}^{p}W_{j}H^{i}(L)=0$ for $p\leq k$ and $j\leq i-1$.

3. The case of a local complete intersection

In this case, there are stronger conditions on the invariants.

For $X$ an isolated lci singularity, let $M$ be the Milnor fiber. There is a mixed Hodge structure on $H^{n}(M)$, which depends on the choice of a one parameter smoothing, and we set

The integer $s_{p}$ is independent of the choice of a smoothing and is thus an invariant of the germ $(X,x)$. (More generally, for a smoothable isolated singularity $(X,x)$, one can define $s_{p}$ for a given smoothing component of $(X,x)$, compare [Ste85, proof of Theorem 2.9].)

Let $X$ be an isolated lci singularity of dimension $n$.

There is an exact sequence of mixed Hodge structures connecting $M$ and $L$, given by

Moreover, the sequence is self-dual. In particular $H^{n}(M,L)\cong(H^{n}(M))^{\vee}(-n)$ and $H^{n}(L)\cong(H^{n-1}(L))^{\vee}(-n)$, i.e. $H^{n}(M,L)$ and $H^{n}(M)$ are dual mixed Hodge structures up to a Tate twist, and similarly for $H^{n}(L)$ and $H^{n-1}(L)$.

(i) is a standard result. Then (ii) and (iii) are easy consequences of the exactness of the sequence in (i) and the fact that morphisms of mixed Hodge structures are strict with respect to the Hodge filtration. ∎

We then have the following [Ste97, p. 1372]:

For $X$ an isolated lci singularity of dimension $n$, in the above notation,

k𝑘k-rational implies k𝑘k-Du Bois

The goal of this section is to give a quick proof of Theorem 1.5 in the case of an isolated singularity, motivated by Steenbrink’s proof for the case $k=0$ [Ste83, Proposition 3.7]. As noted in the introduction, this result has already been proved in [FL22b] (by a different method, inspired by [Kov99]), and proofs in the non-isolated lci case have been given in [FL22b] and [MP22b]. We include this proof as some evidence that the result might hold in general without the lci assumption. We begin with a result that holds for a general isolated singularity, and which is an analogue of [Ste83, Lemma 2.14]:

This holds $\iff$ for all $i\geq 0$, the natural map

is surjective. Consider the commutative diagram

where the upper horizontal arrow is an isomorphism by Lemma 2.3(ii). By Hodge theory, the right hand vertical arrow is surjective. Hence

Let $X$ be an isolated $k$-rational singularity. Then $X$ is $k$-Du Bois.

The proof is by induction on $k$. The case $k=0$ is an argument of Steenbrink (and is a special case of the arguments below). Assume inductively that the theorem has been proved for all $j<k$. In particular, we assume that $\Omega_{X}^{p}\to R^{0}\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)$ is an isomorphism, $H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E))=0$ for all $p\leq k$ and all $q>0$, and that, for all $p\leq k-1$, $\Omega_{X}^{p}\to R^{0}\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)(-E)$ is an isomorphism and $H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E)(-E))=0$ all $q>0$. We have to show that $\Omega_{X}^{k}\to R^{0}\pi_{*}\Omega^{k}_{\widehat{X}}(\log E)(-E)$ is an isomorphism and that $H^{q}(\widehat{X};\Omega^{k}_{\widehat{X}}(\log E)(-E))=0$ for all $q>0$. Note that the statement on $R^{0}\pi_{*}$ is automatic: the isomorphism $\Omega_{X}^{k}\to R^{0}\pi_{*}\Omega^{k}_{\widehat{X}}(\log E)$ factors as $\Omega_{X}^{k}\xrightarrow{f}R^{0}\pi_{*}\Omega^{k}_{\widehat{X}}(\log E)(-E)\xrightarrow{g}R^{0}\pi_{*}\Omega^{k}_{\widehat{X}}(\log E)$, where $g$ is injective. Since $g\circ f$ is an isomorphism, $f$ is an isomorphism.

The mixed Hodge structure on the Milnor fiber

We begin by recalling some standard results about smoothings of isolated singularities, for which a general reference is [Loo13]. In particular, we note the following:

If $\rho\colon(\mathcal{X},x)\to\Delta$ is a smoothing of the germ $(X,x)$, then it is possible to choose good Stein representatives for both $X$ and $\mathcal{X}$.

If $X$ has an isolated lci singularity at $x$, then the singularity of $\mathcal{X}$ is also lci [Loo13, Proposition 6.10].

Let $M$ be the Milnor fiber, so that $M$ is identified as a topological space with $\mathcal{X}_{t}$, $t\neq 0$. Moreover, $H^{k}(M)\neq 0$ for $k\neq 0,n$. Let $\mathcal{L}$ be the link of $\mathcal{X}$, i.e. $\mathcal{L}=\mathcal{X}-\{x\}=\widehat{\mathcal{X}}-\mathcal{E}$. Then $H^{i}(M)$ and $H^{i}(\mathcal{L})$ carry mixed Hodge structures, and we want to describe the corresponding Hodge filtrations in more detail. The following lemma is just an exposition of [Ste83, (2.6)(b)]:

There is an exact sequence of mixed Hodge structures:

For clarity, first consider the Wang sequence for the fibration $\widehat{\mathcal{X}}^{*}\to\Delta^{*}$, where $\widehat{\mathcal{X}}^{*}=\widehat{\mathcal{X}}-\widehat{\mathcal{X}}_{0}$. This arises by considering the sequence of relative differentials

Then taking hypercohomology essentially gives the Wang sequence for the fiber bundle over $\Delta^{*}$, which is homotopy equivalent to $S^{1}$, with fiber $\widehat{\mathcal{X}}_{t}$:

As it stands, the Wang sequence does not yield an exact sequence of mixed Hodge structures because $\widehat{\mathcal{X}}_{0}$ is not compact. Instead, define

Note that these complexes are supported on $\mathcal{E}$, and there is a perfect pairing

The associated long exact sequence of hypercohomology then gives the long exact sequence of Lemma 4.1. ∎

Suppose that $X$ is an lci of dimension $n$. Then there is a self-dual exact sequence of mixed Hodge structures

The first statement is clear since $H^{i}(M)=0$ for $i\neq 0,n$. Then, as in the proof of Proposition 2.11, the remaining statements are easy consequences of the exactness of the sequence in (i) and the fact that morphisms of mixed Hodge structures are strict with respect to the Hodge filtration, together with the fact that $H^{n}(M)$ and $H^{n}_{c}(M)(-1)$ are dual mixed Hodge structures. ∎

The proof is essentially the same as that for Proposition 2.12. Since $\dim\mathcal{X}=n+1$,

Let $X$ be an isolated lci of dimension $n$. Suppose that there exists a $k>\frac{1}{2}(n-1)$ such that $s_{p}=0$ for all $p\leq k$. Then $X$ is smooth.

The given inequality on $k$ implies that $n-(k-1)=n-k+1\leq k+1$. By Corollary 4.3, $s_{n-p+1}=0$ for all $p\leq k$. Thus $s_{p}=0$ for all $p$, $0\leq p\leq n$. Hence the Milnor number $\mu=\dim H^{n}(M)=0$, so $X$ is smooth. ∎

As another application of the exact sequence in Corollary 4.2, we have the following formula, which is an identity involving the Du Bois invariants, the $s_{p}$ and the link invariants:

If $X$ is an isolated lci of dimension $n$, then for all $k\leq n-2$,

If $0\to\mathcal{F}^{\prime}\to\mathcal{F}\to\mathcal{F}^{\prime\prime}\to 0$ is an exact sequence such that $H^{a}(Z;\mathcal{F}^{\prime})\to H^{a}(Z;\mathcal{F})$ is injective, in particular if $H^{a}(Z;\mathcal{F}^{\prime})=0$ or if $H^{a-1}(Z;\mathcal{F}^{\prime\prime})=0$, then

We want to apply Definition 4.6 for the case $a=n-k-1$ to the following three exact sequences:

twisted by $\mathcal{O}_{\widehat{\mathcal{X}}}(-\mathcal{E})$, and the Poincaré residue sequence

In what follows, we assume that $k\leq n-2$, so that $n-k-1\geq 1$. Then $\chi_{\geq n-k-1}$ is defined for all of the sheaves in question. We claim that $\chi_{\geq n-k-1}$ is additive for the exact sequences (4.1)–(4.3). To simplify the notation, we write $\overline{\chi}$ for $\chi_{\geq n-k-1}$. By Theorem 2.9 applied to $X$ and to $\mathcal{X}$,

Thus $\overline{\chi}$ is additive for the exact sequences (4.1) and (4.3). Moreover, by the exact sequence (4.3), the fact that $\mathcal{O}_{\widehat{\mathcal{X}}}(-\widehat{X}-\mathcal{E})\cong\mathcal{O}_{\widehat{\mathcal{X}}}$, and the vanishing of $H^{n-k-1}(X;\Omega^{k-1}_{\widehat{X}}(\log E))$,

so that $\overline{\chi}$ is additive for the exact sequence (4.2) as well.

and $H^{n-k-1}(\widehat{X};\Omega^{k-1}_{\widehat{X}}(\log E)(-E))=0$ by Theorem 2.9. Thus

The proof of the theorem then follows by induction on $k$. ∎

(ii) We will apply Theorem 4.5 in the case that $s_{p}=0$ for all $p\leq k$. In this case, by Corollary 4.4, we may as well assume that $k\leq(n-1)/2$. Note that $(n-1)/2\leq n-2$ $\iff$ $n\geq 3$. However, for $n=2$, if $k\leq(n-1)/2=1/2$, then $k=n-2=0$ and so Theorem 4.5 applies in this case as well.

Proofs of the main theorems

In this subsection, we prove Theorem 1.12 and Theorem 1.15. The main point will be to relate the $k$-Du Bois property to the vanishing of certain of the $s_{p}$ (Theorem 1.12(i)):

Let $(X,x)$ be an isolated lci singularity with Milnor fiber $M$. Then $X$ is $k$-Du Bois $\iff$ $s_{p}=0$ for all $p\leq k$.

First suppose that $X$ is $k$-Du Bois. By a result of Scherk [Sch80, Corollary 3.11] for hypersurfaces and Steenbrink [Ste95, Theorem 1] in general, we can embed a smoothing $X_{t}$ of $X$ in a family of smoothings $\mathcal{Y}\to\Delta$ such that $Y_{0}$ has just one isolated singularity analytically isomorphic to $X$ and the restriction map $i^{*}\colon H^{n}(Y_{t})\to H^{n}(M)$ is surjective. This gives an exact sequence of mixed Hodge structures

Since $X$, and hence $Y_{0}$, are $k$-Du Bois, by [FL22b, Corollary 1.4] and counting dimensions, for all $p\leq k$, the map $\operatorname{Gr}_{F}^{p}H^{n}(Y_{0})\to\operatorname{Gr}_{F}^{p}H^{n}(Y_{t})$ is an isomorphism and hence $\operatorname{Gr}_{F}^{p}H^{n}(M)=0$. Thus $s_{p}=0$ for all $p\leq k$.

Given Theorem 5.1, it is now straightforward to give various characterizations of $k$-Du Bois and $k$-rational singularities and to prove extra vanishing statements. We begin with the $k$-Du Bois case:

Let $(X,x)$ be an isolated lci of dimension $n$ and suppose $k\leq\frac{1}{2}(n-1)$. Then the following are equivalent:

$\Omega_{X}^{p}\xrightarrow{\sim}\mathcal{H}^{0}(\underline{\Omega}^{p}_{X})$ is a quasi-isomorphism and $\mathcal{H}^{q}(\underline{\Omega}^{p}_{X})=0$ for $p\leq k$ and $q>0$.

$b^{p,q}=\dim H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E)(-E))=0$ for $0\leq p\leq k$ and $q>0$.

$X$ is $(k-1)$-Du Bois and $b^{k,n-k-1}=0$.

$b^{p,q}=\dim H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E)(-E))=0$ in the following cases: $0\leq p\leq k$ and $q>0$, or $q\geq n-k-1$, or $p+q\neq n-1,n$.

The equivalence of (i), (ii), (iii) is Theorem 1.3 of the introduction. The equivalence (iii) $\iff$ (iv) follows immediately from Lemma 2.5. The equivalence (i) $\iff$ (vi) is Theorem 5.1. Thus, the only remaining statement to prove is the equivalence of (v) with the others. Clearly, (v) $\implies$ (iii). Conversely, assuming (iii), the condition that $b^{p,q}=0$ for $p+q\neq n-1,n$ is automatic from Theorem 2.9, so we only have to show that $b^{p,q}=0$ for $q\geq n-k-1$. Also, the only relevant case (where $p>k$) is $p=k+1,q=n-k-1$. Then $b^{k+1,n-k-1}=0$ by Lemma 2.5. ∎

The following is the analogue of Theorem 5.2 in the $k$-rational case, and includes Theorem 1.12(ii).

Let $(X,x)$ be an isolated lci of dimension $n$, and suppose $k\leq\frac{1}{2}(n-1)$. Then the following are equivalent:

$\Omega_{X}^{p}\xrightarrow{\sim}R^{0}\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)$ is an isomorphism and $H^{q}(\widehat{X};\pi_{*}\Omega^{p}_{\widehat{X}}(\log E))=0$ for $p\leq k$ and $q>0$.

$H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E))=0$ for $0\leq p\leq k$ and $q>0$.

$H^{q}(\widehat{X};\Omega^{p}_{\widehat{X}}(\log E))=0$ in the following cases: $0\leq p\leq k$ and $q>0$, or $q\geq n-k-1$ and $(p,q)\neq(n,n-1)$, or $p+q\neq n-1,n$.

The equivalence of (i), (ii), and (iii) is Theorem 1.4.

Let $(X,x)$ be an isolated lci of dimension $n$.

Thus, we have established (iii) $\iff$ (v), completing the implications in Theorem 5.3. ∎

We summarize the connection between $k$-rational and $k$-Du Bois in the following statement (Theorem 1.15 from the introduction):

Let $(X,x)$ be an isolated lci of dimension $n$.

$X$ is $k$-Du Bois $\iff$ $X$ is $(k-1)$-rational and $b^{k,n-k-1}=0$.

(i) was proved as Theorem 5.3(iv). As for (ii), if $X$ is $k$-Du Bois, then $X$ is $(k-1)$-rational by Corollary 4.3 and Theorem 5.3(vi), and $b^{k,n-k-1}=0$. Conversely, if $X$ is $(k-1)$-rational and $b^{k,n-k-1}=0$, then $X$ is $(k-1)$-Du Bois by (i), and Theorem 5.2(iv) then implies that $X$ is $k$-Du Bois. ∎

2. Inversion of adjunction

Methods similar to the proof of Theorem 4.5 also show the following (due to Steenbrink [Ste83, Theorem 3.11] in case $k=0$):

If $\mathcal{X}\to\Delta$ is a smoothing of the isolated lci singularity $X$ which admits a semistable model, then the following are equivalent:

$\mathcal{X}$ is $k$-Du Bois and $s_{k}=0$.

The proof is by a careful examination of the exact sequences (4.1)–(4.3) in the proof of Theorem 4.5.

(i) $\implies$ (iii): Theorem 5.1 implies that $s_{k}=0$. For the remaining statement, we argue by induction on $k$, starting with the case $k=-1$. By the $k$-Du Bois assumption on $X$, using (4.2),

Since $H^{n-k-1}(\widehat{\mathcal{X}};\Omega^{k}_{\widehat{\mathcal{X}}}(\log\mathcal{E})|\mathcal{E})=0$, the map

is injective, using (4.1). Finally, again by the $k$-Du Bois assumption and (4.3),

is injective, and its image is clearly $tH^{n-k}(\widehat{\mathcal{X}};\Omega^{k}_{\widehat{\mathcal{X}}}(\log(\widehat{X}+\mathcal{E})))$, where $t$ is the coordinate on $\Delta$. This implies that $tH^{n-k}(\widehat{\mathcal{X}};\Omega^{k}_{\widehat{\mathcal{X}}}(\log(\widehat{X}+\mathcal{E})))=H^{n-k}(\widehat{\mathcal{X}};\Omega^{k}_{\widehat{\mathcal{X}}}(\log(\widehat{X}+\mathcal{E})))$, and thus that $H^{n-k}(\widehat{\mathcal{X}};\Omega^{k}_{\widehat{\mathcal{X}}}(\log(\widehat{X}+\mathcal{E})))=0$. Hence

Thus $b^{k,n-k}(\mathcal{X})=0$, and by the inductive hypotheses $\mathcal{X}$ is $(k-1)$-Du Bois. Then $\mathcal{X}$ is $k$-Du Bois by Theorem 5.2(iv).

(ii) $\implies$ (i): Again by induction, we can assume that $X$ is $(k-1)$-Du Bois, hence, via the exact sequence (4.3), $H^{q}(\widehat{\mathcal{X}};\Omega^{k}_{\widehat{\mathcal{X}}}(\log(\widehat{X}+\mathcal{E})))=0$ for all $q>0$. Then $H^{q}(\widehat{\mathcal{X}};\Omega^{k}_{\widehat{\mathcal{X}}}(\log(\widehat{X}+\mathcal{E}))(-\widehat{X}-\mathcal{E}))=0$ for $q>0$ as well, and $H^{q}(\widehat{\mathcal{X}};\Omega^{k}_{\widehat{\mathcal{X}}}(\log\mathcal{E})(-\mathcal{E}))=0$ since $\mathcal{X}$ is $k$-Du Bois. Via the exact sequence (4.2), $H^{q}(\widehat{X};\Omega^{k}_{\widehat{X}}(\log E)(-E))=0$ for all $q>0$, hence $X$ is $k$-Du Bois. ∎

As a corollary, we get an easy proof of the following inverse of adjunction statement (which has been proved more generally and precisely in the hypersurface case by Dirks-Mustaţă [DM22, Theorem 1.1]):

Let $\mathcal{X}$ be an isolated lci singularity such that there exists a hypersurface section $X$ of $\mathcal{X}$ passing through the singular point with an isolated $k$-Du Bois singularity. Then $\mathcal{X}$ is $k$-rational.

Suppose that $X=V(f)$. Then $f$ defines a $1$-parameter smoothing $\mathcal{X}\to\Delta$. After a base change, we find a finite cyclic cover $\mathcal{X}^{\prime}$ of $\mathcal{X}$ and a smoothing $\mathcal{X}^{\prime}\to\Delta$ of $X$ which admits a semistable model. By Theorem 5.6, $\mathcal{X}^{\prime}$ is $k$-rational. So we have to prove that the same holds for $\mathcal{X}$. More generally, let $G$ be a finite cyclic group and let $\ (\widehat{\mathcal{X}}^{\prime},\mathcal{E}^{\prime})\to(\widehat{\mathcal{X}},\mathcal{E})$ be a $G$-equivariant log resolution. Then, as $G$ is finite cyclic, $(\Omega^{p}_{\widehat{\mathcal{X}}^{\prime}}(\log\mathcal{E}^{\prime}))^{G}=\Omega^{p}_{\widehat{\mathcal{X}}}(\log\mathcal{E})$, and thus

Since $\mathcal{X}^{\prime}$ is $k$-rational, $H^{q}(\widehat{\mathcal{X}};\Omega^{p}_{\widehat{\mathcal{X}}}(\log\mathcal{E}))=0$ for $p\leq k$ and $q>0$. Hence $\mathcal{X}$ is $k$-rational. ∎

The hypersurface case

For the remainder of this paper, we assume unless otherwise stated that $X$ is an isolated hypersurface singularity. In this case, we can relate the previous invariants $s_{p}$ to the so-called spectrum of $X$. We briefly review this connection and then analyze the case where $X$ is $k$-Du Bois but not $k$-rational.

where $b=n-p+a$, and hence $p$ is the unique integer for which $-1<a\leq 0$. ∎

In the above notation, for all $d\geq 0$,

The first two equalities are immediate from the definitions and the isomorphism $\operatorname{Gr}_{V}^{b}Q_{f}\cong\operatorname{Gr}_{F}^{p}H^{n}(M)(a)$. The second two equalities follow from the fact that $\alpha\in(n-p,n-p+1]$ $\iff$ $n+1-\alpha\in[p,p+1)$ and the symmetry $m_{n+1-\alpha}=m_{\alpha}$. The last statement follows from the formulas for $s_{p}$ and $s_{n-p}$. ∎

Following Saito [Sai93], we define the minimal exponent $\widetilde{\alpha}_{X,x}=\widetilde{\alpha}_{X}$ to be the minimal spectral number $\alpha_{\min}=\min\{\alpha:\alpha\in\operatorname{Sp}(X,x)\}$. Equivalently, if $b<\widetilde{\alpha}_{X}-1$, then $\operatorname{Gr}_{V}^{b}Q_{f}=0$, and $\operatorname{Gr}_{V}^{b}Q_{f}\neq 0$ for $b=\widetilde{\alpha}_{X}-1$.

By [Sai93], $\widetilde{\alpha}_{X}>1$ $\iff$ $X$ is a rational singularity, and $\widetilde{\alpha}_{X}\geq 1$ $\iff$ $X$ is a Du Bois singularity. We can generalize this as follows:

Let $(X,x)$ be an isolated hypersurface singularity. Then

$\widetilde{\alpha}_{X}\geq k+1$ $\iff$ $s_{p}=0$ for $0\leq p\leq k$.

$\widetilde{\alpha}_{X}>k+1$ $\iff$ $s_{n-p}=0$ for $0\leq p\leq k$.

Let $(X,x)$ be an isolated hypersurface singularity. Then

$X$ is $k$-Du Bois $\iff$ $\widetilde{\alpha}_{X}\geq k+1$.

$X$ is $k$-rational $\iff$ $\widetilde{\alpha}_{X}>k+1$. ∎

As noted in the introduction (i) holds for a general, not necessarily isolated, hypersurface singularity by [JKSY22, Thm. 1] and [MOPW23, Thm. 1.1.], and (ii) holds under the same assumption by [FL22b, Appendix] and [MP22b].

(Saito) $X$ is $k$-Du Bois $\iff$ $\sum_{i=1}^{n+1}w_{i}\geq k+1$.

$X$ is $k$-rational $\iff$ $\sum_{i=1}^{n+1}w_{i}>k+1$.

It is straightforward to extend the results of Corollary 6.8 to the case of a positive weight deformation of an isolated weighted homogeneous hypersurface singularity, using the result of Varchenko [Var82] that such a deformation is a $\mu=$ constant deformation and the theorem of Steenbrink [Ste85] that the spectrum is semicontinuous in an appropriate sense.

2. k𝑘k-liminal singularities

Let $X$ be the germ of an isolated singularity. The space $X$ is $k$-liminal if $X$ is $k$-Du Bois but not $k$-rational. Thus, for a nonnegative integer $k$, if $X$ is an isolated hypersurface singularity, it follows from Theorem 5.2 and Theorem 5.3 that $X$ is $k$-liminal $\iff$ $\widetilde{\alpha}_{X}=k+1$.

We analyze the property that $X$ is $k$-liminal in more detail in the hypersurface case. First, we recall some general results: We note the following theorem of Dimca-Saito [DS12, §4.11] (cf. also [Sai18, Remark 3.7]):

For an isolated hypersurface singularity $X$, if $b=\widetilde{\alpha}_{X}-1$, then $\dim\operatorname{Gr}_{V}^{b}Q_{f}=\dim\operatorname{Gr}_{V}^{\widetilde{\alpha}_{X}-1}Q_{f}=1$. Equivalently, $m_{\widetilde{\alpha}_{X}}=1$. Moreover, $V_{>b}Q_{f}=\mathfrak{m}_{x}Q_{f}$. ∎

If $X$ is an isolated hypersurface singularity of dimension $n$, then $\widetilde{\alpha}_{X}\leq\displaystyle\frac{n+1}{2}$, and $\widetilde{\alpha}_{X}=\displaystyle\frac{n+1}{2}$ $\iff$ $X$ is an ordinary double point. In particular, if $n=2k+1$ is odd, then there is no isolated hypersurface singularity $X$ of dimension $n$ which is $k$-rational, and $X$ is $k$-Du Bois $\iff$ $X$ is $k$-liminal $\iff$ $X$ is an ordinary double point.

(i) This statement has been generalized to the case where $X$ is not assumed a priori to have isolated singularities by Dirks-Mustaţă [DM22, Corollary 6.3]. There is also a sharper bound due to Mustaţă-Popa [MP20, Theorem E(3)].

(ii) The case $\dim X=3$ and $k=1$ in the final statement of Corollary 6.12 is due to Namikawa-Steenbrink [NS95, Theorem 2.2].

The above results then imply the following corollary:

References