2. Preliminaries

2.1 The cdh differentials and the Du Bois complexes

Let X be a Noetherian scheme over a field F. The cdh differentials Footnote ¹ of X are defined as the sheafification of the Kähler differentials $\Omega_{X/F}^p$ in the cdh topology introduced by Suslin and Voevodsky [Reference Suslin and VoevodskySV00]. More precisely,

\begin{align*}\Omega_{{\rm cdh}, X/F}^p := {\bf R} a_*a^*\Omega_{X/F}^p,\end{align*}

where $a: X_{\rm cdh}\to X_{\rm zar}$ is the natural change-of-topology morphism from the cdh site to the Zariski site.

On the other hand, for a complex algebraic variety (i.e. when $F=\mathbb C$ ), Du Bois [Reference Du BoisDu81] introduced the notion of Du Bois complexes (or filtered de Rham complexes) following ideas of Deligne. The same construction applies to define the Du Bois complexes of any quasi-excellent Noetherian scheme X over a field F of characteristic zero. Indeed, by Hironaka’s theorems, the assumptions on X guarantee the existence of a hyperresolution $\epsilon_{\bullet}: X_{\bullet}\to X$ . We then define the pth Du Bois complex of X over F as

\begin{align*}\underline{\Omega}_{X/F}^p:= {\bf R} \epsilon_{\bullet*} \Omega^p_{X_{\bullet}/F}.\end{align*}

We refer readers to [Reference Guillén, Navarro Aznar, Pascual Gainza and PuertaGNP⁺82, Chapter V] and [Reference Peters and SteenbrinkPS08, Chapter 7.3] for the construction of hyperresolutions, and for the definition of derived pushforward from a hyperresolution. Being natural objects arising from Deligne’s mixed Hodge theory, the Du Bois complexes are mostly studied for complex algebraic varieties. In this case, the Du Bois complexes $\underline{\Omega}_{X/\mathbb C}^p$ in our notation are usually denoted simply by $\underline{\Omega}_X^p$ in the literature.

The relationship between the Du Bois complexes and the cdh differentials is observed in [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺11, Lemma 2.1].

Lemma 2.1. Let X be a finite-type F-scheme, and let $k\subset F$ be a subfield. Then for $p\ge 0$ , there is an isomorphism

\begin{align*}\underline{\Omega}_{X/k}^p\simeq \Omega_{{\rm cdh}, X/k}^p.\end{align*}

Proof. When $k=F$ , this is exactly [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺11, Lemma 2.1]. For $k\subset F$ , the result follows by the same proof, using [Reference Cortinãs, Haesemeyer and WeibelCHW08, Corollary 2.5] and [Reference Suslin and VoevodskySV00, Lemma 12.1].

For an F-scheme X, we denote by $\underline{\Omega}_{X/F}^{\le p}$ the (derived) cdh sheafification of the following complex of quasi-coherent sheaves:

\begin{align*}\mathcal O_X\to \Omega_{X/F}^1 \to \cdots \to \Omega_{X/F}^p\end{align*}

supported in cohomological degrees $0,1,\ldots, p$ . Explicitly, $\Omega_{{\rm cdh}, X/F}^{\le p} := {\bf R} a_*a^*\Omega_{X/F}^{\le p}$ , where $a: X_{\rm cdh}\to X_{\rm zar}$ is the natural change-of-topology morphism from the cdh site to the Zariski site.

When X is of finite type over F, this represents an object in the derived category of differential complexes on X. When $F=\mathbb C$ , it follows from Lemma 2.1 that for each $p\ge 0$ ,

\begin{align*}\underline{\Omega}_{X/\mathbb C}^{\le p} = \underline{\Omega}_{X/\mathbb C}^{\bullet}/F^{p+1}\end{align*}

agrees with the quotient of the filtered de Rham complex by the Hodge filtration. It is immediate from the definition that we have the following exact triangle for each $p\ge 0$ :

(1) \begin{align} \underline{\Omega}_{X/F}^p[-p]\to \underline{\Omega}_{X/F}^{\le p} \to \underline{\Omega}_{X/F}^{\le p-1}\xrightarrow{+1}.\end{align}

In the proof of [Reference Cortinãs, Haesemeyer, Schlichting and WeibelCHS⁺08, Theorem 6.1] and [Reference Cortinãs, Haesemeyer and WeibelCHW08, Proposition 2.6], the following useful properties of the objects $\underline{\Omega}_{X/F}^p$ and $\underline{\Omega}_{X/F}^{\le p}$ are established.

Proposition 2.2. Let X be a separated, finite-type F-scheme, and let $k \subset F$ be a subfield. Then for each $p\ge 0$ , the following assertions hold:

1. (1) $\dim \operatorname{supp} \mathcal H^q \underline{\Omega}_{X/k}^p \le \dim X - q - 1$ if $q\gt0$ ; in particular, $\mathcal H^i\underline{\Omega}_{X/k}^p=0$ for all p and all $i\ge d$ ;

2. (2) $\mathbb H^q(X,\underline{\Omega}_{X/k}^p)=0$ for all $q\gt\dim X$ ;

3. (3) $\mathbb H^q(X,\underline{\Omega}_{X/k}^{\le p})=0$ for all $q\gt\dim X+p$ .

Proof. Assertion (1) is stated in the proof of [Reference Cortinãs, Haesemeyer, Schlichting and WeibelCHS⁺08, Theorem 6.1] when $p=0$ (in which case $\underline{\Omega}_{X/k}^0=\underline{\Omega}_{X/F}^0$ ). The proof for $p\ge 1$ is identical when $k=F$ , and the case for general $k\subset F$ follows from Lemma 2.11(1). Assertion (2) follows from (1) and the spectral sequence

\begin{align*}E_1^{ij}=H^i(X, \mathcal H^j\underline{\Omega}_{X/F}^p)\implies \mathbb H^{i+j}(X,\underline{\Omega}_{X/F}^p).\end{align*}

Finally, (3) follows from (2) by induction using the exact triangle (1).

2.2 Derived de Rham complex and the cotangent complex

Following Bhatt [Reference BhattBha12, Construction 4.1], for an F-algebra A, we define the (Hodge-completed) derived de Rham complex of A over F as

\begin{align*}\widehat{\operatorname{dR}}_{A/F}:=\lim_{p\in \mathbb N^{op}} L\Omega_{A/F}^{\le p},\end{align*}

where $L\Omega_{A/F}^{\le p}:=\operatorname{Tot}(\Omega_{P/F}^{\le p}\otimes_P A)$ is the single complex associated to the double complex $\Omega_{P}^{\le p}\otimes_P A$ , in which $P\to A$ is a polynomial F-algebra resolution of A. It can be shown that the objects $L\Omega_{A/F}^{\le p}$ and $\widehat{\operatorname{dR}}_{A/F}$ are independent of the choice of the polynomial resolution $P\to A$ (up to homotopy), and that the construction globalizes to give definitions of $L\Omega_{X/F}^{\le p}$ and $\widehat{\operatorname{dR}}_{X/F}$ for any F-scheme X.

By construction, there is a natural derived filtration $\operatorname{Fil}_H$ on $\widehat{\operatorname{dR}}_{X/F}$ , with respect to which the graded pieces are given by

\begin{align*}\operatorname{gr}^p_H \widehat{\operatorname{dR}}_{X/F}= \mathbb L^p_{X/F}[-p]\end{align*}

where $\mathbb L^p_{X/F}$ is the pth derived wedge product of the cotangent complex, introduced by Illusie [Reference IllusieIll71].

As explained in [Reference BhattBha12, § 5], for a finite-type F-scheme X, there is a map

\begin{align*}\varphi:\widehat{\operatorname{dR}}_{X/F}\to \underline{\Omega}_{X/F}^{\bullet}\end{align*}

preserving the derived filtration on $\widehat{\operatorname{dR}}_{X/F}$ and the Hodge filtration on $\underline{\Omega}_{X/F}^{\bullet}$ . Taking associated graded pieces with respect to the two filtrations, we obtain a natural map

\begin{align*}\mathbb L_{X/F}^p\to \underline{\Omega}_{X/F}^p\end{align*}

for each p. It is a natural question when these maps are isomorphisms. For local complete intersections over $F=\mathbb C$ , this question is studied in the theory of higher Du Bois singularities. When X is not a local complete intersection, this map is rarely an isomorphism unless $p=0$ ; see Remark 2.5.

2.3 Background in higher Du Bois singularities

In the literature, higher Du Bois singularities are defined and studied for complex algebraic varieties. For the purpose of relating to K-regularity, we follow the notation in the previous subsections, and work with these singularities over more general base fields.

Let X be a local complete intersection over F, and $k\subset F$ a subfield. Following [Reference Mustaţă, Olano, Popa and WitaszekMOP⁺23, Reference Jung, Kim, Saito and YoonJKS⁺22] (with the obvious replacement of $\mathbb C$ by F), we say X has m-Du Bois singularities over k if the natural maps $\Omega_{X/k}^p\to\underline{\Omega}_{X/k}^p$ are isomorphisms for all $p\le m$ , where $\Omega_{X/k}^p$ is the (underived) pth wedge product of the Kähler differential $\Omega_{X/k}^1$ relative to k. One important property of m-Du Bois singularities is the following codimension bound on the singular locus, obtained by combining [Reference Mustaţă and PopaMP22, Theorem F] and [Reference Mustaţă and PopaMP22, Corollary 9.26]. See e.g. [Reference Friedman and LazaFL24, Theorem 3.8].

Theorem 2.3 (Mustaţă–Popa). Let X be a local complete intersection. If X has m-Du Bois singularities, then

\begin{align*}\operatorname{codim} X_{\operatorname{sing}}\ge 2m+1.\end{align*}

As a consequence, for local complete intersections one can replace $\Omega_{X/k}^p$ in the definition of m-Du Bois singularities by the more natural object $\mathbb L_{X/k}^p$ , the pth derived wedge product of the cotangent complex of X over k.

Lemma 2.4. Let X be a local complete intersection over F, and $k\subset F$ a subfield. Then the following assertions are equivalent:

1. (1) X has m-Du Bois singularities over k;

2. (2) X has m-Du Bois singularities over F;

3. (3) X has m-Du Bois singularities over Fʹ, for any field extension $F\subset F'$ ;

4. (4) the natural maps $\mathbb L_{X/k}^p\to \underline{\Omega}_{X/k}^p$ are isomorphisms for all $p\le m$ ;

5. (5) the natural maps $\mathbb L_{X/F}^p\to \underline{\Omega}_{X/F}^p$ are isomorphisms for all $p\le m$ ;

6. (6) the natural maps $\mathbb L_{X\times_F F'/F'}^p\to \underline{\Omega}_{X\times_F F'/F'}^p$ are isomorphisms for all $p\le m$ , and any field extension $F\subset F'$ .

Proof. The equivalences $(1)\iff (2)$ and $(4)\iff (5)$ follow from Lemma 2.11(1), and the equivalences $(2)\iff (3)$ and $(5)\iff (6)$ follows from Lemma 2.11(2).

We will show $(2)\iff (5)$ : using Lemma 2.11, we reduce to the case $k=\mathbb C$ .Footnote ² For ease of notation, we then write $\mathbb L_{X/\mathbb C}^p$ and $\underline{\Omega}_{X/\mathbb C}^p$ simply as $\mathbb L_{X}^p$ and $\underline{\Omega}_{X}^p$ .

If the natural maps $\mathbb L_X^p\to \underline{\Omega}_X^p$ are isomorphisms for all $p\le m$ , taking cohomology in degree greater than or equal to 0 shows that X has m-Du Bois singularities.

For the converse, suppose X is a local complete intersection subvariety of a smooth variety Y, and that X has m-Du Bois singularities. Denote by $\mathcal I=\mathcal I_{X/Y}$ the ideal sheaf defining X in Y. Then the cotangent complex $\mathbb L^p_X$ can be represented by

\begin{align*}\mathcal I^p/\mathcal I^{p+1}\to \mathcal I^{p-1}/\mathcal I^p \otimes \Omega_Y^1|_X \to \cdots \to \mathcal I/\mathcal I^2 \otimes \Omega_Y^{p-1}|_X\to \Omega_Y^p|_X,\end{align*}

supported in cohomological degrees $-p, -p+1, \ldots, -1, 0$ .

This is the Eagon–Northcott complex associated to (the dual of) the map $\mathcal I/\mathcal I^2\to \Omega_Y^1|_X$ . As explained in [Reference Mustaţă and PopaMP22] before the proof of Theorem F,Footnote ³ by the depth sensitivity of the Eagon–Northcott complex, it is a resolution of $\Omega_X^p$ when $\operatorname{codim} X_{\operatorname{sing}} \ge p$ . This is satisfied when X has m-Du Bois singularities for $m\ge 1$ (see Theorem 2.3). Since the Lemma is clear when $m=0$ (in which case $\mathbb L_X^0 = \mathcal O_X = \Omega_X^0$ ), this completes the proof.

In view of the equivalence $(1)\iff (2)$ , we will sometimes simply say X has m-Du Bois singularities when the base field causes no confusion.

When X is not necessarily a local complete intersection, as explained in [Reference Shen, Venkatesh and VoSVV24], the conditions $\Omega_{X/F}^1\xrightarrow{\sim} \underline{\Omega}_{X/F}^1$ and $\mathbb L_{X/F}^1\xrightarrow{\sim} \underline{\Omega}_{X/F}^1$ are too restrictive due to the bad behaviors of the Kähler differentials.

Remark 2.5. For a variety X over F, the sheaf $\mathcal H^0\underline{\Omega}_{X/F}^1$ (even $\mathcal H^0\underline{\Omega}_{X/F}^p$ for all $p\ge 1$ ) is:

• – torsion-free by [Reference Huber and JörderHJ14, Remark 3.8];

• – reflexive when X has rational singularities, by [Reference Kebekus and SchnellKS21, Corollary 1.2] and the identification of $\mathcal H^0\underline{\Omega}_{X/F}^1$ with the h-differential in [Reference Huber and JörderHJ14].

On the other hand, when X is not a local complete intersection, the Kähler differentials typically have torsion, and are not reflexive even when X has mild singularities; see [Reference Greb and RollenskeGR11]. For such varieties, one does not have $\Omega_{X/F}^1\simeq \mathcal H^0\underline{\Omega}_{X/F}^1$ .

As a consequence, when X is not a local complete intersection, the maps $\Omega_{X/F}^1\to \underline{\Omega}_{X/F}^1$ and $\mathbb L_{X/F}^1\to \underline{\Omega}_{X/F}^1$ need not be isomorphisms even when X has mild singularities.

In fact, currently there is no known example of a variety satisfying $\Omega_X^1\simeq \mathcal H^0\underline{\Omega}_X^1$ that is not a local complete intersection. Therefore, for arbitrary varieties, we work with the following definition. The equivalences follow from Lemma 2.11.

Definition 2.6. Let X be a variety over F, and $k\subset F$ a subfield. We say that X has pre-m-Du Bois singularities if one of the following equivalent conditions holds:

1. (1) $\mathcal H^i\underline{\Omega}_{X/F}^p=0$ for all $p\le m$ and $i\gt0$ ;

2. (2) $\mathcal H^i\underline{\Omega}_{X/k}^p=0$ for all $p\le m$ and $i\gt0$ ;

3. (3) $\mathcal H^i\underline{\Omega}_{X\times_F F'/F'}^p=0$ for all $p\le m$ and $i\gt0$ , for some (any) field extension F’ of F.

Remark 2.7. Criteria for various classes of varieties (that are typically not local complete intersections) to have pre-m-Du Bois singularities have been worked out. See [Reference Guillén, Navarro Aznar, Pascual Gainza and PuertaGNP⁺82, V Theorem 4.6] for toric varieties, [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺13, Reference Popa and ShenPS] for cones over projective varieties, and [Reference Olano, Raychaudhury and SongORS24] for secant varieties.

For local complete intersections, one can characterize m-Du Bois singularities using the minimal exponent [Reference Jung, Kim, Saito and YoonJKS⁺22, Reference Mustaţă, Olano, Popa and WitaszekMOP⁺23, Reference Chen, Dirks, Mustaţă and OlanoCDM⁺24, Reference Chen, Dirks and MustaţăCDM24], which is an invariant that refines the log canonical threshold.

On the other hand, necessary and sufficient criteria for an arbitrary variety to have pre-m-Du Bois singularities are not known at this moment.

Following [Reference Shen, Venkatesh and VoSVV24], we say X has m-Du Bois singularities if it is seminormal, pre-m-Du Bois, the sheaves $\mathcal H^0\underline{\Omega}_{X/F}^p$ are reflexive for all $p\le m$ , and $\operatorname{codim} X_{\operatorname{sing}}\ge 2m+1$ . As explained in [Reference Shen, Venkatesh and VoSVV24, § 5], this new definition agrees with the well-studied notion in the case of local complete intersections, and includes more examples in general. Varieties with m-Du Bois singularities in this sense provide examples that are $K_{-d+m+1}$ -regular; see Example 4.2.

2.4 Background in K-regularity

Let X be a scheme. We say X is $K_m$ -regular if the natural maps

\begin{align*}K_m(X)\to K_m(X\times \mathbb A^r)\end{align*}

between algebraic K-groups are isomorphisms for all $r\ge 0$ . It is a fact that if X is regular, then it is $K_m$ -regular for all m. Also, $K_m$ -regularity implies $K_{m-1}$ -regularity for all m; see [Reference VorstVor79] and [Reference Dayton and WeibelDW80, Corollary 4.4].

In a sequence of foundational papers [Reference Cortinãs, Haesemeyer, Schlichting and WeibelCHS⁺08, Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10, Reference Cortinãs, Haesemeyer and WeibelCHW08] in this area, it was discovered that $K_m$ -regularity of X is controlled by the Kähler differentials, the cdh differentials and the Hochschild homology of X. Following these papers, we denote by

\begin{align*}N K_m(X) = \operatorname{coker}(K_m(X)\to K_m(X\times \mathbb A^1))\end{align*}

the NK-groups of Bass. For $r\ge 2$ , the groups $N^r K(X)$ are inductively defined by

\begin{align*}N^r K_m(X):= N(N^{r-1} K_m)(X) := \operatorname{coker}(N^{r-1} K_m(X)\to N^{r-1} K_m(X\times \mathbb A^1))\end{align*}

By [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10, Theorem 0.1], we have

\begin{align*} N^r K_m(X) = 0 &\iff NK_m(X) = NK_{m-1}(X) = \cdots = NK_{m-r+1}(X) = 0 \\ &\iff K_m(X) \xrightarrow{\sim} K_m(X\times \mathbb A^r) \text{ is an isomorphism}.\end{align*}

This reduces the study of K-regularity to that of the NK-groups. Another theorem in [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10] expresses NK, or more precisely, $NK^{(i)}$ , the ith eigenspace of the $\lambda$ -operation on NK, in terms of the Kähler differentials, the cdh differentials and the Hochschild homology of X. We rephrase the statement with the notation of Du Bois complexes, using Lemma 2.1.

Theorem 2.8 [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10, Theorem 0.4]. Let $X=\operatorname{Spec} R$ be an affine scheme over $\mathbb Q$ . Then

\begin{align*}NK_n^{(i)}(X) \cong TK_n^{(i)}(X)\otimes_{\mathbb Q} t\mathbb Q[t]\end{align*}

where t is the coordinate on $\mathbb A^1$ in the definition of NK, and

\begin{align*}TK_n^{(i)}(X) \simeq\begin{cases} \mathcal H^{i-n-1}\underline{\Omega}_{X/\mathbb Q}^{i-1} &n\le i-2, \\ \operatorname{coker}(\Omega_{X/\mathbb Q}^i\to \mathcal H^0\underline{\Omega}_{X/\mathbb Q}^i) &n = i-1, \\ \ker(\Omega_{X/\mathbb Q}^i\to \mathcal H^0\underline{\Omega}_{X/\mathbb Q}^i) &n=i, \\ HH^{(i-1)}_{n-1}(X) &n\ge i+1.\end{cases}\end{align*}

Remark 2.9. Since $HH_n^{(i)}(X)\simeq \mathcal H_{n-i} \mathbb L^i \simeq \mathcal H^{i-n} \mathbb L^i$ , this theorem can be rephrased in the following way. Let $C^i$ be the cone of the natural map $\mathbb L^i_{X/\mathbb Q} \to \underline{\Omega}^i_{X/\mathbb Q}$ . Then the $TK_n^{(i)}(X)$ are determined by the cohomologies of $C^i$ :

\begin{align*}TK^{(i+1)}_{i-j}(X) \simeq \mathcal H^j C^i.\end{align*}

As a consequence, X is $K_m$ -regular if and only if

\begin{align*}\mathcal H^j C^i =0 \quad\text{for all } j-i\ge -m.\end{align*}

Thus, in view of Lemma 2.4, the notion of K-regularity is naturally related to higher singularities, at least for local complete intersections.

To study the K-regularity for varieties that are not necessarily affine, it is convenient to work with a local version of Bass’s NK-groups, denoted by $\mathcal N \mathcal K_m$ , and defined as the sheafification of

\begin{align*}U\mapsto NK_m(\Gamma(U))\end{align*}

on the étale site of X in [Reference van der KallenvdK86]. As shown in loc. cit., we have

\begin{align*}H^i_{\unicode{x00E9}{\text{t}}}(X, \mathcal N \mathcal K_m) = H^i(X, \mathcal N \mathcal K_m),\end{align*}

where $H^i$ denotes the cohomology taken in the Zariski topology. When X is affine,

\begin{align*}H^i_{\unicode{x00E9}{\text{t}}}(X, \mathcal N \mathcal K_m) = \begin{cases}NK_m(X) &\text{if $i=0$}, \\0 &\text{if $i\gt0$}.\end{cases}\end{align*}

We have a local-to-global spectral sequence relating local and global NK-groups

\begin{align*}E_2^{pq} = H^p(X, \mathcal N \mathcal K_q(X))\implies NK_{q-p}(X).\end{align*}

In view of these results, Theorem 2.8 can be stated in this local setting as follows.

Theorem 2.10 [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10]. Let X be any scheme over $\mathbb Q$ . Then

\begin{align*}\mathcal N \mathcal K_m(X) \simeq \bigoplus_{j-i=-m} \mathcal H^j C^i \otimes t\mathbb Q[t],\end{align*}

where $C^i = \operatorname{Cone}(\mathbb L^i_{X/\mathbb Q}\to \underline{\Omega}_{X/\mathbb Q}^i)$ . In particular, X is locally $K_m$ -regular, that is, $\mathcal N \mathcal K_q(X)=0$ for all $q\le m$ , if and only if

\begin{align*}\mathcal H^j C^i = 0\quad\text{for } j-i \ge -m.\end{align*}

3. Vanishing theorems for the Du Bois complexes

In this section, we collect vanishing theorems for the Du Bois complexes, with a focus on the effect of the change of base fields on the statements. These will be applied later to study K-regularity.

The first general result is due to Steenbrink when $F=\mathbb C$ . The general case follows from Lemma 2.11.

Theorem 3.1 [Reference SteenbrinkSte85]. Let X be a variety over F. Then

\begin{align*}\mathcal H^q \underline{\Omega}_{X/F}^p = 0\end{align*}

for $p+q\gt\dim X$ .

In view of Lemma 2.11(1), Steenbrink’s vanishing theorem need not hold for Du Bois complexes over a subfield $k\subset F$ . Instead, we have the following weaker statement.

Proposition 3.2. Let X be a variety over F, and $k\subset F$ a subfield. If X has pre-m-Du Bois singularities, then

\begin{align*}\mathcal H^q \underline{\Omega}_{X/k}^p = 0\end{align*}

for all $q\ge \dim X-m-1$ and $q\gt0$ .

Complementary to Steenbrink’s theorem, we have vanishing results for $\mathcal H^q\underline{\Omega}_{X/k}^p$ when $p+q$ is small.

Theorem 3.3 [Reference Mustaţă and PopaMP25]. Let X be a local complete intersection over F, and $k\subset F$ a subfield. Denote by $X_{\operatorname{sing}}$ the singular locus of X. Then for all $p\ge 0$ , we have

\begin{align*}\mathcal H^q \underline{\Omega}_{X/k}^p = 0\end{align*}

for $1\le q\lt \operatorname{codim} X_{\operatorname{sing}}-p-1$ .

Beyond local complete intersections, we have the following result.

1. (1) $\mathcal H^q\underline{\Omega}_{X/F}^0 = \mathcal H^q\underline{\Omega}_{X/k}^0=0$ for $1\le q \lt \operatorname{depth} \mathcal O_X-\dim X_{\operatorname{sing}}-1$ ;

2. (2) if X is pre- $(p-1)$ -Du Bois with $\dim X_{\operatorname{sing}} = s$ , then

   \begin{align*}\mathcal H^q \underline{\Omega}_{X/F}^p = 0\end{align*}
   for $1\le q\lt \operatorname{depth} \mathcal H^0\underline{\Omega}_{X/F}^p-s-1$ ; in particular, if $\operatorname{depth} \mathcal H^0\underline{\Omega}_{X/F}^p\ge \dim X-p$ for all $p\ge 0$ , then
   \begin{align*}\mathcal H^q\underline{\Omega}_{X/k}^p = 0\end{align*}
   for $1\le q \lt \dim X - p -s - 1$ , and any subfield $k\subset F$ .

4. Higher Du Bois singularities and K-regularity

We start by establishing a general relationship between higher Du Bois singularities and K-regularity. Later, we will apply vanishing theorems for the Du Bois complexes to improve these results under an additional hypothesis.

1. (1) If X has m-Du Bois singularities, then it is $K_{-d+m+1}$ -regular.

2. (2) Suppose X either is affine or has isolated singularities. If X is $K_m$ -regular, then it is pre- $(m+1)$ -Du Bois.

• – X is seminormal;

• – X has pre-m-Du Bois singularities;

• – $\operatorname{codim} X_{\operatorname{sing}}\ge m+2$ if $m\ge 1$ .

To show X is $K_{-d+m+1}$ -regular, by the local-to-global spectral sequence

\begin{align*}E_2^{pq} = H^p(X, \mathcal N\mathcal K_q(X)) \implies NK_{q-p}(X),\end{align*}

it suffices to show

\begin{align*}H^p(X, \mathcal N\mathcal K_q(X))=0\quad\text{for all $q-p \le -d+m+1$}.\end{align*}

By Theorem 2.10, this is equivalent to showing

\begin{align*}H^p(X, \mathcal H^s C^t) =0\quad\text{for $s-t=-q$ and $q-p\le -d+m+1$},\end{align*}

where $C^t$ is the cone of the natural map $\mathbb L_{X/\mathbb Q}^t\to \underline{\Omega}_{X/\mathbb Q}^t$ .

We verify this through a case-by-case analysis.

• – If $s\gt0$ and $t\le m$ , we have $\mathcal H^s C^t = 0$ because X is pre-m-Du Bois.

• – If $s\gt0$ and $t\ge m+1$ , then $p\ge q+d-m-1 = t-s+d-m-1\ge d-s$ . By Proposition 2.2,

  \begin{align*}\dim \operatorname{supp} \mathcal H^s C^t = \dim \operatorname{supp} \mathcal H^s \underline{\Omega}_{X/\mathbb Q}^t\le d-s-1.\end{align*}
  Thus, we have $H^p(\mathcal H^s C^t)=0$ .

• – If $s\le 0$ , then $q\ge s+q = t\ge 0$ , and $p\ge q+d-m-1\ge d-m-1$ . By assumption, the sheaves $\mathcal H^s C^t$ are supported on $X_{\operatorname{sing}}$ , which has dimension $\le d-m-2$ when $m\ge 1$ , so we have $H^p(\mathcal H^s C^t)=0$ . For $m=0$ , some further work is needed.

  • * If $t=0$ , then $\mathcal H^s C^t = 0$ .

  • * If $t\ge 1$ , then $p\ge q+d-1 = d-1-s+t\ge d$ , so $H^p(\mathcal H^s C^t)=0$ , because $\mathcal H^s C^t$ is supported on $X_{\operatorname{sing}}$ , which has dimension $\le d-1$ .

This completes the proof that X is $K_{-d+m+1}$ -regular.

(2) Suppose X is affine and $K_m$ -regular. By Theorem 2.8, we have

\begin{align*}\mathcal H^s C^t = 0\quad\text{for $s-t\ge -m$}.\end{align*}

In particular, taking $s\gt0$ , this implies

\begin{align*}\mathcal H^s\underline{\Omega}_{X/\mathbb Q}^p=0\end{align*}

for all $p\le m+1$ and $s\gt0$ , that is, that X is pre- $(m+1)$ -Du Bois.

Now, suppose X has isolated singularities. Then the spectral sequence

\begin{align*}E_2^{pq} = H^p(X,\mathcal N \mathcal K_q(X))\implies NK_{q-p}(X)\end{align*}

degenerates, giving that the local NK-groups $\mathcal N \mathcal K_q(X)=0$ for $q\le m$ . It then follows from the affine case that X is pre- $(m+1)$ -Du Bois.

Proposition 4.1(1) allows us to construct many examples of $K_{-d+m+1}$ -regular varieties.

1. (1) A codimension r local complete intersection X has m-Du Bois singularities if and only if its minimal exponent $\tilde{\alpha}_X\ge m+r$ ; see [Reference SaitoSai94, Reference Jung, Kim, Saito and YoonJKS⁺22, Reference Mustaţă, Olano, Popa and WitaszekMOP⁺23, Reference Chen, Dirks and MustaţăCDM24]. Playing around with the minimal exponents, one can construct varieties with m-Du Bois singularities.

2. (2) For toric varieties and cones over smooth projective varieties, one can find criteria for them to have m-Du Bois singularities in [Reference Shen, Venkatesh and VoSVV24, §§ 6 and 7].

3. (3) Criteria for secant varieties to have m-Du Bois singularities are worked out in [Reference Olano, Raychaudhury and SongORS24].

By Proposition 4.1(1), these provide examples of $K_{-d+m+1}$ -regular varieties.

Next, applying vanishing theorems in § 3, we obtain the following strengthening of Proposition 4.1 in the local setting.

1. (1) The variety X is locally $K_{-t}$ -regular if $t\ge \max\{1, d-2m-2\}$ .

2. (2) If X is seminormal, then it is locally $K_0$ -regular if $m\ge ({d-2})/{2}$ .

3. (3) If X is seminormal, and the natural map $\Omega_{X/F}^1\to \mathcal H^0\underline{\Omega}_{X/F}^1$ is surjective, then it is locally $K_1$ -regular if $m\ge ({d-1})/{2}$ .

4. (4) Let $s=\dim X_{\operatorname{sing}}$ . Then X is $K_{-t}$ -regular if $t\ge \max\{s+1, d-2m-2+s\}$ ; if X is in addition seminormal, then the bound can be improved to $t\ge \max\{s, d-2m-2+s\}$ .

5. (5) If X is either affine, or has isolated singularities, then we can remove the adjective ‘locally’ in (1), (2) and (3).

As an application of this proposition, we give some examples that are $K_0$ -regular.

1. (1) $K_n$ -regular for all $n\lt0$ ;

2. (2) $K_0$ -regular if X is seminormal.

Examples include toric varieties [Reference Guillén, Navarro Aznar, Pascual Gainza and PuertaGNP⁺82], finite or more generally geometric quotients [Reference Du BoisDu81], and cones over varieties satisfying Bott vanishing ([Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺13], [Reference Shen, Venkatesh and VoSVV24, Corollary 7.6]). Since all these varieties are normal, they are locally $K_0$ -regular.

Next, we investigate the converse to the implication in Proposition 4.3.

1. (1) X is a local complete intersection;

2. (2) $\operatorname{depth} \mathcal H^0\underline{\Omega}_X^p\ge d-p$ for all $p\ge 0$ .

If X is $K_{-d+2m+1+s}$ -regular, then it is pre-m-Du Bois.

Proof. Since X is $K_{-d+2m+1+s}$ -regular, we know by Theorem 2.8 that

\begin{align*}\mathcal H^i \underline{\Omega}_{X/\mathbb Q}^j = 0\quad\text{for $i\ge d-2m-1-s+j$ and $i\gt0$}.\end{align*}

Applying Theorem 3.3 in case (1), and Theorem 3.4 in case (2) (with $k=\mathbb Q$ and F the field of definition of X), we have

\begin{align*}\mathcal H^i\underline{\Omega}_{X/\mathbb Q}^j = 0\quad\text{for $1\le i\le d-s-2-j$}.\end{align*}

For $j\le m$ , the two inequalities of i cover all indices $i\ge 1$ . It follows that X is pre-m-Du Bois.

Combining the two propositions above, we obtain the following corollary.

1. (1) X is a local complete intersection;

2. (2) $\operatorname{depth} \mathcal H^0\underline{\Omega}_X^p\ge d-p$ for all $p\ge 0$ .

Then

\begin{align*} \textit{X}\text{ has pre-}\textit{m}\text{-Du Bois singularities} &\iff \textit{X}\text{ is $K_{-d+2m+1}$-regular} \\ &\iff \textit{X}\text{ is $K_{-d+2m+2}$-regular} \end{align*}

when $-d+2m+2\lt0$ .

Proof. By Propositions 4.3 and 4.5 (applied with $s=0$ ), we have the implications

\begin{align*}\text{$K_{-d+2m+1}$-regular} \implies \text{pre-}\textit{m}\text{-Du Bois} \implies \text{$K_{-d+2m+2}$-regular}.\end{align*}

Since $K_{-d+2m+2}$ -regular implies $K_{-d+2m+1}$ -regular, all three conditions are in fact equivalent.

When $m=0$ , this gives the following result.

1. (1) X has Du Bois singularities;

2. (2) X has pre-0-Du Bois singularities;

3. (3) X is $K_{-d+1}$ -regular;

4. (4) X is $K_{-d+2}$ -regular.

Since X is normal, hence seminormal, the equivalence $(1)\iff (2)$ follows from the well-known fact that $\mathcal H^0\underline{\Omega}_{X/\mathbb Q}^0 \simeq \mathcal H^0\underline{\Omega}_{X/\mathbb C}^0 \simeq \mathcal O_{X^{sn}}$ , the structure sheaf of the seminormalization of X.

1. (1) X is $K_{-1}$ -regular;

2. (2) X is $K_0$ -regular;

3. (3) X has Du Bois singularities;

4. (4) X has pre-0-Du Bois singularities;

5. (5) X has pre-1-Du Bois singularities.

The equivalence of (1)–(3) with (4) is remarked in [Reference Popa, Shen and VoPSV, Example 3.5]. See also Lemma 6.2.

In [Reference WeibelWei01, Theorem 5.9], characterizations for $K_{-1}$ -regularity that are more geometric in nature are obtained. For example, given a resolution of singularities $\tilde{X}\to X$ with exceptional divisor E, it is shown in loc. cit. that X is $K_{-1}$ -regular if and only if

\begin{align*}H^1(E, \mathcal I/\mathcal I^n)=0\quad\text{for all }n\ge 2,\end{align*}

where $\mathcal I$ is the ideal of $\mathcal O_{\tilde{X}}$ defining E.

One can directly see that this condition characterizes Du Bois singularities for normal surfaces, as follows: since X is seminormal, we have $\mathcal H^0\underline{\Omega}_X^0=\mathcal O_X$ ; moreover, by dimension considerations, $\mathcal H^2\underline{\Omega}_X^0=0$ . It follows that X has Du Bois singularities if and only if $\mathcal H^1 \underline{\Omega}_X^0=0$ .

By Steenbrink’s triangle for the Du Bois complexes,

\begin{align*}\underline{\Omega}_X^0 \to \mathcal R f_*\mathcal O_{\tilde{X}}(-E)\to \mathcal O_x \xrightarrow{+1},\end{align*}

where x is the isolated singularity of X. We thus see that

\begin{align*}\mathcal H^1\underline{\Omega}_X^0 = R^1 f_*\mathcal O_{\tilde{X}}(-E),\end{align*}

which by the formal function theorem is

\begin{align*}R^1f_*\mathcal O_{\tilde{X}}(-E) = \varprojlim_n H^1(E_n, \mathcal O_{\tilde{X}}(-E)|_{E_n}) = \varprojlim_n H^1(E, I/I^n).\end{align*}

Since $H^2(E,{-})=0$ , from the short exact sequence

\begin{align*}0\to I^n/I^{n+1} \to I/I^{n+1} \to I/I^{n}\to 0\end{align*}

we see that for all $n\ge 1$ ,

\begin{align*}H^1(E,I/I^n)=0 \implies H^1(E,I/I^{n+1})=0.\end{align*}

It follows that

\begin{align*}X \text{ is Du Bois}&\iff R^1 f_*\mathcal O_{\tilde{X}}(-E) =\varprojlim_n H^1(E, I/I^n)= 0 \\&\iff H^1(E, I/I^n) = 0\quad\text{for sufficiently large }\textit{n} \\&\iff H^1(E, I/I^n) = 0\quad\text{for all } \textit{n}.\end{align*}

I am grateful to Sung Gi Park for telling me about this argument.

• – C is always $K_{-1}$ -regular;

• – C is $K_0$ -regular $\iff$ C is $K_1$ -regular $\iff$ C is $\operatorname{Pic}$ -regular $\iff$ C is seminormal. This fits into our Hodge theory picture, as a curve is seminormal if and only if it has Du Bois singularities;

• – if C is $K_2$ -regular, then it is smooth.

With a bit more work, one can extend Corollary 4.6 beyond the bound $m\le ({d-3})/{2}$ for local complete intersections of dimension greater than or equal to 2. We will need the smoothness criterion given by Theorem 2.3, as well as the following result about the torsion and cotorsion of the Kähler differentials. See [Reference GrafGra15, Theorem 1.11] for the case of hypersurfaces, and [Reference Miller and VassiliadouMV19, Corollary 3.1] for local complete intersections.

1. (1) torsion-free if and only if $p\le \operatorname{codim} X_{\operatorname{sing}}-1$ ;

2. (2) reflexive, if and only if $p\le \operatorname{codim} X_{\operatorname{sing}}-2$ .

If it is reflexive, then we have $\Omega_{X/F}^p\simeq \mathcal H^0\underline{\Omega}_{X/F}^p$ .

Corollary 4.11. Let X be a local complete intersection of dimension d with isolated singularities. Then

\begin{align*} \textit{X}\text{ has }\textit{m}\text{-Du Bois singularities} &\iff \textit{X}\text{ is $K_{-d+2m+1}$-regular} \\ &\iff \textit{X}\text{ is $K_{-d+2m+2}$-regular}. \end{align*}

Proof. First, we show that

\begin{align*}\textit{m}\text{-Du Bois} \implies \text{$K_{-d+2m+2}$-regular}.\end{align*}

We check this case by case.

• – If $-d+2m+2\lt0$ , this follows by Proposition 4.3(1).

• – If $-d+2m+2=0$ , this follows by Proposition 4.3(2), because local complete intersections of dimension greater than or equal to 2 with isolated singularities are normal.

• – If $-d+2m+2=1$ , this follows by Proposition 4.3(3), because the surjectivity of $\Omega_{X/F}^1\to \mathcal H^0\underline{\Omega}_{X/F}^1$ (where F is the field of definition of X) holds by Theorem 4.10, as long as $d\ge 3$ .

• – If $-d+2m+2\ge 2$ , then $d\le 2m$ . Since X is m-Du Bois, it is smooth by Theorem 2.3, hence $K_{-d+2m+2}$ -regular.

Next, we show that

\begin{align*} \text{$K_{-d+2m+1}$-regular}\implies\textit{m}\text{-Du Bois}.\end{align*}

By Proposition 4.5 applied with $s=0$ , we know X is pre-m-Du Bois. If $m\le d-2$ , then X is m-Du Bois by Theorem 4.10. Thus, we may assume $m\ge d-1$ . We consider several cases.

• – If $d\ge 4$ , then one can check that

  \begin{align*}m\ge d-1 \ge \bigg\lfloor \frac{d+1}{2} \bigg\rfloor\quad \text{and}\quad \bigg\lfloor \frac{d+1}{2} \bigg\rfloor\le d-2,\end{align*}
  so X is $\lfloor ({d+1})/{2} \rfloor$ -Du Bois, hence smooth by Theorem 2.3.

• – If $d=3$ , then $m\ge 2$ . By assumption X is $K_2$ -regular, with the following consequences.

  • * The map $\Omega_{X/\mathbb Q}^2\to \mathcal H^0\underline{\Omega}_{X/\mathbb Q}^2$ is surjective by Theorem 2.10. Since X is 1-Du Bois, Lemma 2.11(1) implies the map $\Omega_{X/F}^2\to \mathcal H^0\underline{\Omega}_{X/F}^2$ is surjective,

  • * The map $\Omega_{X/F}^2\to \mathcal H^0\underline{\Omega}_{X/F}^2$ is injective by Theorem 4.10.

    It follows that X is 2-Du Bois, hence smooth by Theorem 2.3.

• – If $d=2$ , then $m\ge 1$ . By assumption X is $K_1$ -regular. Similar to the case $d=3$ , one checks that X is 1-Du Bois, hence smooth.

• – If $d=1$ , then $m\ge 0$ . If $m=0$ , then X is $K_0$ -regular, hence $\operatorname{Pic}$ -regular because $K_0(X)=\mathbb{Z}\oplus \operatorname{Pic}(X)$ for curves. By Traverso’s theorem [Reference TraversoTra70, Theorem 3.5], X is seminormal, hence has Du Bois singularities; when $m\ge 1$ , X is $K_2$ -regular, hence smooth by Vorst’s Theorem [Reference VorstVor79, Theorem 3.4].

Corollary 4.12 ( $\!\implies$ Theorem A). Let X be a complex local complete intersection with isolated singularities of dimension d. Let r be the codimension of X in its embedding, and $\tilde{\alpha}_X$ be the minimal exponent of X (which depends on the embedding). Then

\begin{align*}X \text{ is $K_m$-regular} \iff \tilde{\alpha}_X \ge \bigg\lceil \frac{m+d}{2}\bigg\rceil+r-1.\end{align*}

Proof. Combining Corollary 4.11 with the main theorems of [Reference Jung, Kim, Saito and YoonJKS⁺22, Reference Mustaţă, Olano, Popa and WitaszekMOP⁺23, Reference Chen, Dirks and MustaţăCDM24], stating that

\begin{align*}\textit{X}\text{ has }\textit{m}\text{-Du Bois singularities} \iff \tilde{\alpha}_X\ge m+r,\end{align*}

we obtain the result.

Remark 4.13. Similar arguments to Corollary 4.11 show that for an affine local complete intersection X with $\dim X_{\operatorname{sing}}=1$ , one has the following.

In particular,

\begin{align*}\tilde{\alpha}_X\ge m+r \iff X \text{ is } K_{-d+2m+2}\text{-regular}.\end{align*}

More generally, suppose X is an affine local complete intersection $\dim X_{\operatorname{sing}}=s$ . Then

\begin{align*} \textit{X}\text{ is $K_{-d+2m+1+s}$-regular}\implies &\textit{X}\text{ has }\textit{m}\text{-Du Bois singularities} \\ \implies &\textit{X}\text{ is $K_{-d+2m+2}$-regular}. \end{align*}

Thus, $K_m$ -regularity can be read off, at least partially, from the minimal exponent $\tilde{\alpha}_X$ and the codimension r of X in its embedding:

1. (1) X is $K_m$ -regular for $m\le -d+2\tilde{\alpha}_X-2r+2$ ;

2. (2) X is not $K_m$ -regular for $m=-d+2\lceil \tilde{\alpha}_X\rceil-2r+s$ .

5. Application to regularity criteria

In this section, we prove the following regularity criterion. Recall that $K_m$ -regularity implies $K_{m-1}$ -regularity for all m (see [Reference VorstVor79, Reference Dayton and WeibelDW80]), hence it immediately implies Theorem 2.

1. (1) If $\operatorname{codim} X_{\operatorname{sing}}$ is odd, and X is $K_2$ -regular, then it is regular.

2. (2) If $\operatorname{codim} X_{\operatorname{sing}}$ is even, and X is $K_1$ -regular, then it is regular.

The proof is similar to that of Corollary 4.11, and relies on three main ingredients:

1. (1) a regularity criterion due to Mustaţă–Popa (Theorem 2.3) – the relevance of this result in relation to extensions of Vorst’s conjecture is observed in [Reference Haesemeyer and WeibelHW25];

2. (2) results due to Graf and to Miller and Vassiliadou about the torsion and cotorsion of the Kähler differentials of local complete intersections (Theorem 4.10);

3. (3) the relationship between $K_m$ -regularity and pre-k-Du Bois singularities (Proposition 4.5), which in turn relies on a vanishing theorem of Mustaţă–Popa (Theorem 3.3).

Proof of Theorem B. Let F be the field of definition of X. By Theorem 2.3, regularity of X is implied by having m-Du Bois singularities for $m\gt ({d-1})/{2}$ .

(1) Suppose $\operatorname{codim} X_{\operatorname{sing}}=2m+1$ is odd. To show X is smooth, it suffices to show X has $(m+1)$ -Du Bois singularities, that is, the following two conditions hold.

• – The intersection X has pre- $(m+1)$ -Du Bois singularities. By Proposition 4.5, this is implied by $K_{-d+2(m+1)+1+s} = K_2$ -regularity.

• – The maps $\Omega_{X/F}^p\to \mathcal H^0\underline{\Omega}_{X/F}^p$ are isomorphisms for $p\le m+1$ . By Theorem 4.10, this is the case when $p\le \operatorname{codim} X_{\operatorname{sing}} - 2$ . Thus, we have the desired isomorphism when $m+1\le \operatorname{codim} X_{\operatorname{sing}}-2 = (2m+1)-2$ , that is, when $m\ge 2$ .

When $m=1$ , we need to show $\Omega_{X/F}^2\to \mathcal H^0\underline{\Omega}_{X/F}^2$ is an isomorphism. By Theorem 2.8, $K_2$ -regularity implies that this map is surjective with F replaced by $\mathbb Q$ , and Lemma 2.11(2) implies the surjectivity over F; on the other hand, Theorem 4.10 implies that $\Omega_{X/F}^2$ is torsion-free, so the map is an isomorphism.

Finally, it is impossible that $m=0$ , as $K_2$ -regularity implies X has 1-Du Bois singularities, so $\operatorname{codim} X_{\operatorname{sing}} \ge 3$ .

Thus, X has $(m+1)$ -Du Bois singularities, hence is smooth.

(2) Suppose $\operatorname{codim} X_{\operatorname{sing}}=2m$ is even. To show X is smooth, it suffices to show X has m-Du Bois singularities, that is, the following two conditions hold.

• – The intersection X has pre-m-Du Bois singularities. By Proposition 4.5, this is implied by $K_{-d+2m+1+s} = K_1$ -regularity.

• – The maps $\Omega_{X/F}^p\to \mathcal H^0\underline{\Omega}_{X/F}^p$ are isomorphisms for $p\le m$ . By Theorem 4.10, this is the case when $p\le \operatorname{codim} X_{\operatorname{sing}} - 2$ . Thus, we have the desired isomorphism when $m\le \operatorname{codim} X_{\operatorname{sing}}-2 = 2m-2$ , that is, when $m\ge 2$ .

When $m=1$ , we need to show $\Omega_{X/F}^1\to \mathcal H^0\underline{\Omega}_{X/F}^1$ is an isomorphism. By Theorem 2.8 and Lemma 2.11(1), $K_1$ -regularity implies that this map is surjective; on the other hand, the results [Reference GrafGra15, Reference Miller and VassiliadouMV19] show that $\Omega_X^1$ is torsion-free, so the map is injective.

Thus, X has m-Du Bois singularities, hence is smooth.

The result above can be improved for local complete intersections of minimal embedding codimension $r\ge 2$ .

Definition 5.3. The minimal embedding codimension of a local complete intersection X is the minimal integer r such that for every $x\in X$ , there exist a regular local ring R, and a regular sequence $(f_1, \ldots, f_r)$ in R, such that

\begin{align*}\mathcal O_{X,x}\simeq R/(f_1, \ldots, f_r).\end{align*}

The following theorem, which I learned from B. Dirks, shows that the codimension bound in Theorem 2.3 can be improved when X is a local complete intersection of minimal embedding codimension r. The proof follows as in [Reference Chen, Dirks and MustaţăCDM24, Corollary 1.3]. See [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺13, Theorem 3.5] for a detailed argument.

Theorem 5.4 (Bradley Dirks). Let X be a local complete intersection of minimal embedding codimension r. Suppose X has m-Du Bois singularities. Then

\begin{align*}\operatorname{codim} X_{\operatorname{sing}}\ge 2m+r.\end{align*}

As a consequence, we have the following corollary.

1. (1) If $\operatorname{codim} X_{\operatorname{sing}} - r\ge 6-2r$ is even, and X is $K_{3-r}$ -regular, then X is smooth.

2. (2) If $\operatorname{codim} X_{\operatorname{sing}} - r\ge 5-2r$ is odd, and X is $K_{2-r}$ -regular, then X is smooth.

• – X is pre- $(m+1)$ -Du Bois, implied by $K_{-d+2(m+1)+1+s}=K_{3-r}$ -regularity;

• – $\Omega_X^p$ is reflexive for $p\le m+1$ ; this is the case if $m+1\le \operatorname{codim} X_{\operatorname{sing}} - 2 = 2m+r-2$ , that is, when $m\ge 3-r$ .

The proof of (2) is similar.

Remark 5.6. It is desirable to know about the implications of $K_2$ -regularity beyond affine local complete intersections. For local complete intersections with isolated singularities (that are not necessarily affine), Theorem B and Corollary 5.5 continue to hold, thanks to the degeneration of the local-to-global spectral sequence

\begin{align*}E_2^{pq} = H^p(X, \mathcal N\mathcal K_q(X)) \implies NK_{q-p}(X).\end{align*}

When the singular locus has higher dimension, it is not known whether $K_2$ -regular implies smooth; see Question 8.2.

Remark 5.7. By Theorem 2.8, if an affine variety X is $K_2$ -regular, then the natural map

\begin{align*}\Omega_X^1\to \mathcal H^0\underline{\Omega}_X^1\end{align*}

is an isomorphism. In the study of higher Du Bois singularities beyond the local complete intersection case, one difficulty is that there is no known variety satisfying this condition; see Remark 2.5. This suggests also that $K_2$ -regular varieties that are not local complete intersections (if there are any) are tricky to find.

Note that the normality assumption is necessary: the surface $(xyz=0)\subset \mathbb A^3$ is $K_1$ -regular. In fact, any affine simple normal crossings divisor is $K_1$ -regular.

6. Bass’s question and the Du Bois invariants

Let X be a variety over F. The Du Bois invariants, defined as

\begin{align*}b^{p,q} = \dim_F \mathcal H^q \underline{\Omega}_{X/F}^p \end{align*}

for $q\gt0$ , are introduced by Steenbrink [Reference SteenbrinkSte97] when X has isolated singularities. It follows from Proposition 2.2 that $b^{p,q}=0$ for all $q\ge d:=\dim X$ . We arrange the invariants $b^{p,q}$ for $0\le p \le d$ and $1\le q\le d-1$ into a $(d-1)\times (d+1)$ table, and refer to it as the Du Bois table.

For surfaces over $\mathbb Q$ , we have a $1\times 3$ table of the following form.

In fact, it is well known that for surfaces, the only interesting invariants are $b^{0,1}$ and $b^{1,1}$ .

Lemma 6.1 [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺11, Proposition 3.2]. Let X be a surface over F. For $q{\kern-1pt}\gt{\kern-1pt}0$ and $p{\kern-1pt}\ge{\kern-1pt} 0$ , we have

\begin{align*}\mathcal H^q\underline{\Omega}_{X/F}^p = 0\end{align*}

unless $(p,q)=(0,1)$ or (1,1).

Applications of the Du Bois invariants to K-theory started in [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺11], where the authors used an example of a surface with $b^{0,1}=1$ and $b^{1,1}=0$ to give negative answers to Bass’s question [Reference BassBas73], which asks whether

\begin{align*}K_m(X)\simeq K_m(X\times \mathbb A^1) \implies K_m(X)\simeq K_m(X\times \mathbb A^2).\end{align*}

Indeed, for surfaces, the main theorem of [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10] implies that Bass’s question for the negative K-groups has an affirmative answer if and only if $b^{0,1}$ and $b^{1,1}$ are both zero, or both non-zero. In view of the potential applications of these invariants to K-theory, it is desirable to have a better understanding of them. In this section, we give a complete classification of these invariants for surfaces.

First, along the lines of [Reference Popa, Shen and VoPSV, Proposition C], we show that $b^{0,1}$ and $b^{1,1}$ are constrained by the inequality $b^{0,1}\ge b^{1,1}$ .

Lemma 6.2. Let X be a surface over F. Then the map

\begin{align*}d: \mathcal H^1\underline{\Omega}_{X/F}^0\to \mathcal H^1\underline{\Omega}_{X/F}^1\end{align*}

is surjective. In particular, we have $b^{0,1}\ge b^{1,1}$ when X has isolated singularities.

Proof. By Lemma 2.11(2), it suffices to prove the result when $F=\mathbb C$ . In this case, we have by the Poincaré lemma that

\begin{align*}\mathbb C\xrightarrow{\sim} (\underline{\Omega}_{X/\mathbb C}^{\bullet})^{\operatorname{an}}\end{align*}

as sheaves in the analytic topology on $X(\mathbb C)$ .

By construction of the Du Bois complex (in the analytic category, see [Reference Du BoisDu81]), we have a spectral sequence

\begin{align*}E_1^{pq} = \mathcal H^q(\underline{\Omega}_{X/\mathbb C}^p)^{\operatorname{an}} \mapsto \mathcal H^{p+q}(\underline{\Omega}_{X/\mathbb C}^{\bullet})^{\operatorname{an}}\end{align*}

converging to $(\underline{\Omega}_{X/\mathbb C}^{\bullet})^{\operatorname{an}}\simeq \mathbb C[0]$ supported in cohomological degree zero. Analyzing the maps in the spectral sequence shows that there is a surjection

\begin{align*}d^{\operatorname{an}}: \mathcal H^1(\underline{\Omega}_{X/\mathbb C}^0)^{\operatorname{an}}\to \mathcal H^1(\underline{\Omega}_{X/\mathbb C}^1)^{\operatorname{an}}.\end{align*}

Using Serre’s GAGA theorem, we then conclude that the map d is surjective.

In fact, the invariants $b^{0,1}$ and $b^{1,1}$ can take on arbitrary values.

1. (1) If X is an affine cone over a projective curve, or a quasi-homogeneous hypersurface, then $b^{0,1}=b^{1,1}$ ; in these cases, Bass’s question for $K_m$ has positive answer when $m\le 0$ .

2. (2) Let $m\ge n\ge 0$ be any pair of non-negative integers. The hypersurface $(x^3+y^3+z^{3m}+xyz^{m+n}=0)$ has Du Bois invariants

   \begin{align*}b^{0,1}= m \quad \text{and}\quad b^{1,1}=n.\end{align*}

Proof. (1) When X is an affine cone over a projective curve C associated to a line bundle L on C, explicit calculation as in [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺13] and [Reference Popa and ShenPS] shows that

\begin{align*}\mathcal H^1\underline{\Omega}_{X/\mathbb Q}^0=\mathcal H^1\underline{\Omega}_{X/\mathbb Q}^1\simeq \bigoplus_{m\ge 0} \mathbb H^1(C,\underline{\Omega}_{C/\mathbb Q}^0\otimes L^m).\end{align*}

For quasi-homogeneous hypersurfaces, we use van Straten and Steenbrink’s computation of the geometric genus $p_g\le b^{0,1}$ and the alpha invariant $\alpha:=b^{0,1}-b^{1,1}$ : by [Reference van Straten and SteenbrinkvSS85, Theorem 3.1, Corollary 6.3], for a surface $X=(f=0)\subset \mathbb A^3=Y$ with isolated singularities at 0, we have

\begin{align*}p_g = \dim Q^f/V_{\gt0}\end{align*}

and

\begin{align*}p_g+\alpha = \dim Q^f/\overline{K}\cap V_{\gt0},\end{align*}

where $Q^f = \Omega_{Y,0}^3/df\wedge \Omega_{Y,0}^2$ and $\overline{K}$ is the kernel of the multiplication by f on $Q^f$ . If f is quasi-homogeneous, multiplication by f is zero on $Q^f$ , so $\overline{K}=Q^f$ , and it follows that $\alpha = b^{0,1}-b^{1,1}=0$ , as claimed.

(2) For semi-quasi-homogeneous hypersurfaces of weight $(w_x, w_y,w_z)$ with isolated singularities, the V-filtration on $Q^f$ is computed in [Reference SaitoSai88]. One has that $V_{\gt l} Q^f$ is generated by monomials $x^a y^b z^c$ satisfying

\begin{align*}w_x (a+1)+w_y(b+1)+w_z(c+1) - 1 \gt l.\end{align*}

In particular, for $f=x^3+y^3+z^{3m}+xyz^{m+n}$ , we have

\begin{align*}V_{\gt0} = (x, y, z^m),\end{align*}

and a Gröbner basis computation shows that

\begin{align*}\overline{K} = (x, y, z^{2m-n}).\end{align*}

This gives that

\begin{align*}p_g = \dim Q^f/V_{\gt0} = m \ge b^{0,1}\end{align*}

and

\begin{align*}p_g+\alpha = \dim Q^f/\overline{K}\cap V_{\gt0} = 2m-n.\end{align*}

Next, using the deformation argument in [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺11] (see Theorem 2.14 and the proof of Proposition 4.3 in loc. cit.), we see that $b^{0,1}$ is independent of n. When $n=0$ , we have $\alpha=m$ , and since $0\le b^{1,1}=b^{0,1}-\alpha=b^{0,1}-m\le 0$ , we have $b^{0,1}=m$ and $b^{1,1}=0$ . It follows that for $n\ge 1$ , we have

\begin{align*}b^{0,1}= m\quad \text{and} \quad b^{1,1} = n,\end{align*}

as required.

Moving to higher dimensions, we show that for homogeneous hypersurfaces with isolated singularities, Bass’s question has positive answer for the non-positive K-groups. The proof relies on the computation of the cohomology groups $H^q(S, \Omega_S^p(l))=0$ for smooth hypersurfaces in projective spaces. Complete computations can be found in [Reference VoVo24]. We state part of the theorem below in the form of what we need.

1. (1) For $l\ge 0$ , we have $H^q(S, \Omega_S^p(l)) = 0$ if $p+q\neq \dim S$ .

2. (2) For $p+q=\dim S$ ,

   \begin{align*}H^q(S, \Omega_S^p(l)) = 0 \text{ for all $l\ge 1$ }\iff (p+1)d \le n+1.\end{align*}

When $d=1$ , the result follows from Bott vanishing for projective spaces; when $n=1$ , the statement is vacuous; when $n=2,3$ , one can compute $H^q(S, \Omega_S^p(l))$ using the long exact sequence associated to the following short exact sequences (see [Reference VoVo24, Proposition 1] for one implication in (2) when $n=3$ ):

\begin{align*}0&\to \mathcal O_{\mathbb P^n}(-d+l)\to \mathcal O_{\mathbb P^n(l)}\to \mathcal O_S(l)\to 0, \quad n=2,3,\\0&\to \Omega_{\mathbb P^3}^1|_S(l)\to \mathcal O_S(l-1)^{\oplus 4}\to\mathcal O_S(l)\to 0,\\0&\to \mathcal O_S(l-d)\to \Omega_{\mathbb P^3}^1|_S(l)\to \Omega_S^1(l)\to 0,\end{align*}

and verify that the results hold true.

Proposition 6.5. Let X be a homogeneous hypersurface with at worst isolated singularities. Then for every m,

\begin{align*}K_m(X)\xrightarrow{\sim} K_m(X\times \mathbb A^1) \implies K_m(X)\xrightarrow{\sim} K_m(X\times \mathbb A^r)\end{align*}

for all $r\ge 1$ . In particular, if $K_2(X)\simeq K_2(X\times \mathbb A^1)$ , then X is smooth.

By [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10, Theorem 0.1], the implication we need to show is equivalent to

\begin{align*}NK_m(X)=0\implies NK_i(X)=0\quad\text{for $i\lt m$}. \end{align*}

Applying [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10, Theorem 0.4], this is in turn equivalent to

\begin{align*}\mathcal H^qC^p=0 \text{ for $q-p=-m$} \implies \mathcal H^qC^p=0 \text{ for $q-p=-m-1$},\end{align*}

where $C^p = \operatorname{Cone}(\mathbb L_{X/\mathbb Q}^p\to \underline{\Omega}_{X/\mathbb Q}^p)$ . Since F is algebraic over $\mathbb Q$ , we have $\Omega_{F/\mathbb Q}^p=0$ for all $p\ge 1$ , so the spectral sequence in Lemma 2.11(1) degenerates, giving that

\begin{align*}\mathcal H^q\underline{\Omega}_{X/F}^p \simeq \mathcal H^q\underline{\Omega}_{X/\mathbb Q}^p\end{align*}

and

\begin{align*}\mathcal H^q\mathbb L_{X/F}^p \simeq \mathcal H^q\mathbb L_{X/\mathbb Q}^p.\end{align*}

Note that since X is a hypersurface with isolated singularities, as in the proof of Lemma 2.4, we have $\mathbb L_{X/F}^p \simeq \Omega_{X/F}^p$ for $p\le \dim X$ . Moreover, by Theorem 4.10, we have $\mathcal H^0 C^p=0$ for $p\le \dim X-2$ and $\mathcal H^{-1} C^p = 0$ for $p\le \dim X-1$ . Consequently, the exact sequences in [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺13, Proposition 1.9] give the isomorphisms

\begin{align*}\operatorname{tors} \Omega_{X/F}^{n} \simeq \operatorname{tors} \Omega_{X/F}^{n + 1}\end{align*}

and

\begin{align*}\mathcal H^0\underline{\Omega}_{X/F}^{n-1}/\Omega_{X/F}^{n-1} \simeq \mathcal H^0\underline{\Omega}_{X/F}^{n}/\Omega_{X/F}^{n},\end{align*}

where $n=\dim X$ . Therefore, it suffices for us to check

\begin{align*}\mathcal H^q\underline{\Omega}_{X/F}^p=0 \text{ for $q-p=-m$} \implies \mathcal H^q\underline{\Omega}_{X/F}^p=0 \text{ for $q-p=-m-1$}.\end{align*}

Since X is a homogeneous hypersurface with isolated singularities, we can view it as an affine cone over a smooth projective hypersurface S. Using the computation of $\mathcal H^q\underline{\Omega}_{X/F}^p$ for cones [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺13], as well as Theorem 6.4, we see that the Du Bois table of X takes the following form.

Thus, we only need to check

\begin{align*}H^q(\Omega^p_S(m))=0 \text{ for all $m\ge 1$} \implies H^{q+1}(\Omega^{p-1}_S(m))=0\text{ for all $m\ge 1$}\end{align*}

for $p+q = \dim S$ . This follows from Theorem 6.4(2).

1. (1) X has (pre-)p-Du Bois singularities;

2. (2) X is $K_{-d+2+2p}$ -regular;

3. (3) $NK_{-d+2+2p}(X)=0$ ;

4. (4) $\deg f\le ({d+1})/({p+1})$ .

For X to be singular, one needs $2\le \deg f\le ({d+1})/({p+1})$ , in which case $-d+2+2p\le 1$ , so X can be at most $K_1$ -regular, as predicted by Theorem B.

7. K-regularity for projective varieties

In this section, we apply the main theorem of [Reference Cortinãs, Haesemeyer, Schlichting and WeibelCHS⁺08] to characterize K-regularity for projective varieties. I thank Dori Bejleri, Anh Duc Vo, and especially Elden Elmanto for discussions that shaped the contents presented here.

Our main result is the following characterization of $K_m$ -regularity for projective varieties over a field F of characteristic zero.

Theorem 7.1 ( $\!\implies$ Theorem D). Let X be a projective variety over F. Then X is $K_m$ -regular if and only if

\begin{align*}\mathbb H^i(X,\mathbb L_{X/F}^p)\xrightarrow{\sim} \mathbb H^i(X, \underline{\Omega}_{X/F}^p)\end{align*}

are isomorphisms for $i-p\ge -m+1$ .

Remark 7.2. Let X be a projective variety of dimension d. It follows from Theorem 7.1 that

\begin{align*}\textit{X}\text{ is $K_{-d+1}$-regular} \iff \mathbb H^d(X,\mathcal O_X)\to \mathbb H^d(X,\underline{\Omega}_X^0)\text{ is an isomorphism}.\end{align*}

In particular, if X has Du Bois singularities then it is $K_{-d+1}$ -regular, which is consistent with Proposition 4.1(1).

By [Reference Cortinãs, Haesemeyer, Schlichting and WeibelCHS⁺08, Theorem 5.5], there is a homotopy fibration sequence

\begin{align*}\tilde{C_j}{\bf HC}\to \tilde{C}_jL_{\rm cdh}{\bf HC}\to \tilde{C_j} \mathcal K,\end{align*}

where $\mathcal K$ denotes the non-connective K-theory spectrum of perfect complexes on $\operatorname{Sch/F}$ , $\tilde{C}_j \mathcal E$ denotes the cofiber of the map $\mathcal E\to \mathcal E(-\times \mathbb A^j)$ for a presheaf of spectra on $\operatorname{Sch/F}$ , and HC is the presheaf of complexes computing cyclic cohomologies over $\mathbb Q$ . Following loc. cit., we write $H^i_{\rm cdh}$ for the cohomology taken in the cdh topology, and $H^i$ for the cohomology taken in the Zariski topology. By definition, a scheme X is $K_m$ -regular if and only if $\pi_m\tilde{C_j}\mathcal K(X)=\ker(K_m(X)\to K_m(X\times \mathbb A^j))=0$ for all $j\ge 0$ .

Taking cohomology, we obtain a long exact sequence

(2) \begin{align}\begin{split}\cdots \to &H^{d-1}(\tilde{C}_j{\bf HC}(X))\to H^{d-1}_{\rm cdh}(\tilde{C}_j {\bf HC}(X))\to \pi_{-d+1}\tilde{C}_j \mathcal K(X) \\ \to & H^{d}(\tilde{C}_j{\bf HC}(X))\to H^{d}_{\rm cdh}(\tilde{C}_j{\bf HC}(X))\to \pi_{-d}\tilde{C}_j \mathcal K(X) \to H^{d+1}(\tilde{C}_j{\bf HC}(X)).\end{split}\end{align}

Thus, Theorem 7.1 can be proved by analyzing the surjectivity of the maps

\begin{align*}H^i(\tilde{C}_j{\bf HC}(X))\to H^i_{\rm cdh}(\tilde{C}_j {\bf HC}(X)).\end{align*}

The main Hodge-theoretic input for the proof is the following result, whose statement goes back to at least Du Bois [Reference Du BoisDu81] when $F=\mathbb C$ and $p=0$ .

Proposition 7.3. Let X be a projective variety over F. Then there is a surjection

\begin{align*}\mathbb H^i(X, L\Omega_{X/F}^{\le p})\to \mathbb H^i(X, \underline{\Omega}_{X/F}^{\le p})\end{align*}

for all integers i and $p\ge 0$ .

Proof. By standard base change arguments (see the footnote to the proof of Lemma 2.11), we reduce to the case when $F=\mathbb C$ . When X is a complex projective variety, the Hodge-to-de-Rham spectral sequence

\begin{align*}E_1^{p,q}=H^q(X,\underline{\Omega}_{X/\mathbb C}^p)\implies \mathbb H^{p+q}(X,\mathbb C)\end{align*}

degenerates at $E_1$ , giving a surjection

\begin{align*}H^i(X,\mathbb C)\to H^i(X,\mathbb C)/F^{p+1}H^i(X,\mathbb C) = \mathbb H^i(X,\underline{\Omega}_{X/\mathbb C}^{\bullet}/F^{p+1}\underline{\Omega}_{X/\mathbb C}^{\bullet})=:\mathbb H^i(X,\underline{\Omega}_{X/\mathbb C}^{\le p})\end{align*}

for all integers i and $p\ge 0$ .

By [Reference BhattBha12], the derived de Rham complex $\widehat{\operatorname{dR}}_{X/\mathbb C}$ computes the Betti cohomology of $X(\mathbb C)$ , so we have the following commutative diagram.

Since the top horizontal row is a surjection, we have that the maps

\begin{align*}H^i(X,L\Omega_{X/\mathbb C}^{\le p})\to H^i_{\rm cdh}(X,\Omega_{X/\mathbb C}^{\le p})\simeq \mathbb H^i(X,\underline{\Omega}_{X/\mathbb C}^{\le p})\end{align*}

are surjective, for all integers i and $p\ge 0$ .

We are now ready to prove the main theorem of this section.

Proof of Theorem 7.1. First assume X is $K_m$ -regular. We want to show

\begin{align*}\mathbb H^i(X,\mathbb L_{X/F}^p)\xrightarrow{\sim} \mathbb H^i(X, \underline{\Omega}_{X/F}^p)\end{align*}

are isomorphisms when $i-p\ge -m+1$ . By (2), we have that

\begin{align*}H^i(X, \tilde{C}_j{\bf HC}(X))\xrightarrow{\sim} H^i_{\rm cdh}(X,\tilde{C}_j{\bf HC}(X))\end{align*}

are isomorphisms for $i\ge -m+1$ . Using the decomposition of cyclic homology

\begin{align*}{\bf HC}(X) = \bigoplus_{p\ge 0} L\Omega_{X/\mathbb Q}^{\le p}[2p],\end{align*}

and the fact that the cdh sheafification of $L\Omega_{X/\mathbb Q}^{\le p}$ is $\underline{\Omega}_{X/\mathbb Q}^{\le p}$ , we see that

\begin{align*}\mathbb H^{i+2p}(X, \tilde{C}_jL\Omega_{X/\mathbb Q}^{\le p})\xrightarrow{\sim} \mathbb H^{i+2p}_{\rm cdh}(X,\tilde{C}_j\underline{\Omega}_{X/\mathbb Q}^{\le p})\end{align*}

are isomorphisms for all $i\ge -m+1$ and $p\ge 0$ .Footnote ⁴

Proof of Claim 1. Applying $\tilde{C}_j$ in Lemma 2.11(1), we obtain spectral sequences

\begin{align*}E_1^{ij}&=\Omega^i_{F/\mathbb Q}\otimes \mathbb H^j(X,L\Omega_{X/F}^{\le p-i})\implies \mathbb H^{i+j}(X, L\Omega_{X/\mathbb Q}^{\le p}),\\E_1^{ij}&=\Omega^i_{F/\mathbb Q}\otimes \mathbb H^j(X, \underline{\Omega}_{X/F}^{\le p-i})\implies \mathbb H^{i+j}(X, \underline{\Omega}_{X/\mathbb Q}^{\le p}).\end{align*}

Analyzing the spectral sequence, we see that it is enough to show the maps

\begin{align*}\mathbb H^{-m+2p-1-i}(X,\tilde{C}_j L\Omega_{X/F}^{\le p-i})\to \mathbb H^{-m+2p-1-i}(X,\tilde{C}_j \underline{\Omega}_{X/F}^{\le p-i})\end{align*}

are surjective for all $i\ge 0$ . This is implied by the surjectivity of the maps

\begin{align*}\mathbb H^{-m+2p-1-i}(X,L\Omega_{X\times \mathbb A^j/F}^{\le p-i})\to \mathbb H^{-m+2p-1-i}(X,\underline{\Omega}_{X\times \mathbb A^j/F}^{\le p-i})\end{align*}

for $i\ge 0$ . To prove the latter surjectivity, we analyze the spectral sequences, proved along the same lines as Lemma 2.11:

\begin{align*}E_1^{ij}&= \mathbb H^j(p^*L\Omega^i_{X/F}\otimes \Omega_{X\times\mathbb A^j/X}^{\le p-i})\implies \mathbb H^{i+j}(L\Omega_{X\times \mathbb A^j/k}^{\le p}),\\E_1^{ij}&= \mathbb H^j(p^*\underline{\Omega}^i_{X/F}\otimes\Omega_{X\times \mathbb A^j/X}^{\le p-i})\implies \mathbb H^{i+j}(\underline{\Omega}_{X\times \mathbb A^j/k}^{\le p}).\end{align*}

We need to check surjectivity of the maps $E_1^{ij}\to \tilde{E}_1^{ij}$ for $i+j=-m+2p-1-i$ . That is, we need to show the maps

\begin{align*}\mathbb H^{-m+2p-1-i-k}(p^*L\Omega_{X/F}^k\otimes \Omega_{X\times \mathbb A^j/X}^{\le p-i-k})\to \mathbb H^{-m+2p-1-i-k}(p^*\underline{\Omega}_{X/F}^k\otimes \Omega_{X\times \mathbb A^j/X}^{\le p-i-k})\end{align*}

are surjective for $0\le k\le p-i$ . By induction using long exact sequences associated to

\begin{align*}\Omega_{X\times \mathbb A^j/X}^{p-i-k}[-p+i+k]\to \Omega_{X\times \mathbb A^j/X}^{\le p-i-k}\to \Omega_{X\times \mathbb A^j/X}^{\le p-i-k-1}\xrightarrow{+1},\end{align*}

it suffices to show the maps

\begin{align*}\mathbb H^{-m+p-1}(L\Omega_{X/F}^k)\to \mathbb H^{-m+p-1}(\underline{\Omega}_{X/F}^k)\end{align*}

are surjective, for all $p\ge 0$ and $k\le p$ . This follows by induction using

\begin{align*}\mathbb L_{X/F}^k[-k]&\to L\Omega_{X/F}^{\le k}\to L\Omega_{X/F}^{\le k-1}\xrightarrow{+1},\\ \underline{\Omega}_{X/F}^k[-k]&\to \underline{\Omega}_{X/F}^{\le k}\to \underline{\Omega}_{X/F}^{\le k-1}\xrightarrow{+1}\end{align*}

and Proposition 7.3. This finishes the proof of Claim 1.

Proof of Claim 2. Indeed, this follows by induction using

\begin{align*}\tilde{C}_j\mathbb L_{X/F}^p[-p]&\to \tilde{C}_jL\Omega_{X/F}^{\le p}\to \tilde{C}_jL\Omega_{X/F}^{\le p-1}\xrightarrow{+1},\\ \tilde{C}_j\underline{\Omega}_{X/F}^p[-p]&\to \tilde{C}_j\underline{\Omega}_{X/F}^{\le p}\to \tilde{C}_j\underline{\Omega}_{X/F}^{\le p-1}\xrightarrow{+1}\end{align*}

and Proposition 7.3.

Proof of Claim 3. We have a split exact triangle (the splitting is induced by the zero section $s: X\to X\times \mathbb A^j$ )

\begin{align*}p^*\Omega^1_{X/F} \to \Omega^1_{X\times \mathbb A^j/F}\to p^*\mathcal O_X\otimes \Omega_{X\times \mathbb A^j/X}^1\xrightarrow{+1}.\end{align*}

Taking the wedge product gives

\begin{align*}\Omega^p_{X\times \mathbb A^j/F}\simeq \bigoplus_{i=0}^p p^*\Omega_X^i\otimes \Omega^{p-i}_{X\times \mathbb A^j/X}\end{align*}

(with the convention $\Omega_{X\times \mathbb A^j/X}^p = \mathcal O_{X\times \mathbb A^j}$ when $p=0$ ).

Taking derived functor and cdh sheafification gives

\begin{align*}\mathbb L^p_{X\times \mathbb A^j/F}&\simeq \bigoplus_{i=0}^pp^*\mathbb L_X^i\otimes \Omega^{p-i}_{X\times \mathbb A^j/X},\\ \underline{\Omega}^p_{X\times \mathbb A^j/F}&\simeq \bigoplus_{i=0}^p p^*\underline{\Omega}_X^i\otimes \Omega^{p-i}_{X\times \mathbb A^j/X}.\end{align*}

Thus,

\begin{align*}\mathbb H^{i+p}(\tilde{C}_j\mathbb L^p_{X\times \mathbb A^j/F})&\simeq \mathbb H^{i+p}(\mathbb L_X^p\otimes V_j)\oplus \bigoplus_{i=0}^{p-1} \mathbb H^{i+p}(p^*\mathbb L_X^i\otimes \Omega^{p-i}_{X\times \mathbb A^j/X}),\\ \mathbb H^{i+p}(\tilde{C}_j\underline{\Omega}^p_{X\times \mathbb A^j/F})&\simeq \mathbb H^{i+p}(\underline{\Omega}_X^p\otimes V_j)\oplus \bigoplus_{i=0}^{p-1} \mathbb H^{i+p}(p^*\underline{\Omega}_X^i\otimes \Omega^{p-i}_{X\times \mathbb A^j/X}),\end{align*}

where $V_j = F[t_1,\ldots, t_j]/F$ .

Since the maps in Claim 2 are compatible with this direct sum decomposition, the maps $\mathbb H^{i+p}(\mathbb L_{X/F}^{p})\xrightarrow{\sim} \mathbb H^{i+p}_{\rm cdh}(\underline{\Omega}_{X/F}^{p})$ are isomorphisms for all $i\ge -m+1$ and $p\ge 0$ .

This completes the proof in one direction. For the converse, assume that the maps

\begin{align*}\mathbb H^i(X,\mathbb L_{X/F}^p)\xrightarrow{\sim} \mathbb H^i(X, \underline{\Omega}_{X/F}^p)\end{align*}

are isomorphisms for $i-p\ge -m+1$ . We will show X is $K_m$ -regular. By (2) and the decomposition of cyclic homology, we need to show:

1. (1) $\mathbb H^{-m+1+2p}(X, \tilde{C}_jL\Omega_{X/\mathbb Q}^{\le p})\xrightarrow{\sim} \mathbb H^{-m+1+2p}_{\rm cdh}(X,\tilde{C}_j\underline{\Omega}_{X/\mathbb Q}^{\le p})$ are isomorphisms for all $p\ge 0$ ;

2. (2) $\mathbb H^{-m+2p}(X, \tilde{C}_jL\Omega_{X/\mathbb Q}^{\le p})\xrightarrow{\sim} \mathbb H^{-m+2p}_{\rm cdh}(X,\tilde{C}_j\underline{\Omega}_{X/\mathbb Q}^{\le p})$ are surjective for all $p\ge 0$ .

Analyzing the spectral sequences

\begin{align*}E_1^{ij}&=\Omega^i_{F/k}\otimes \tilde{C}_j\mathbb H^j(X,L\Omega_{X/F}^{\le p-i})\implies \tilde{C}_j\mathbb H^{i+j}(X,L\Omega_{X/\mathbb Q}^{\le p}),\\[4pt]E_1^{ij}&=\Omega^i_{F/k}\otimes \tilde{C}_j\mathbb H^j(X, \underline{\Omega}_{X/F}^{\le p-i})\implies \tilde{C}_j\mathbb H^{i+j}(X, \underline{\Omega}_{X/\mathbb Q}^{\le p})\end{align*}

shows that it is enough to show:

1. (1) $\mathbb H^{-m+1+2p}(X, \tilde{C}_jL\Omega_{X/F}^{\le p})\xrightarrow{\sim} \mathbb H^{-m+1+2p}_{\rm cdh}(X,\tilde{C}_j\underline{\Omega}_{X/F}^{\le p})$ are isomorphisms for all $p\ge 0$ ;

2. (2) $\mathbb H^{-m+2p}(X, \tilde{C}_jL\Omega_{X/F}^{\le p})\xrightarrow{\sim} \mathbb H^{-m+2p}_{\rm cdh}(X,\tilde{C}_j\underline{\Omega}_{X/F}^{\le p})$ are surjective for all $p\ge 0$ .

By definition of $\tilde{C}_j$ , it suffices to show:

1. (1) $\mathbb H^{-m+1+2p}(X, L\Omega_{X/F}^{\le p})\xrightarrow{\sim} \mathbb H^{-m+1+2p}(X,\underline{\Omega}_{X/F}^{\le p})$ are isomorphisms for all $p\ge 0$ ;

2. (2) $\mathbb H^{-m+1+2p}(X\times \mathbb A^j, L\Omega_{X\times \mathbb A^j/F}^{\le p})\xrightarrow{\sim} \mathbb H^{-m+1+2p}(X\times \mathbb A^j,\underline{\Omega}_{X\times \mathbb A^j/F}^{\le p})$ are isomorphisms for all $p,j\ge 0$ ;

3. (3) $\mathbb H^{-m+2p}(X\times \mathbb A^j, L\Omega_{X\times \mathbb A^j/F}^{\le p})\xrightarrow{\sim} \mathbb H^{-m+2p}(X\times \mathbb A^j,\underline{\Omega}_{X\times \mathbb A^j/F}^{\le p})$ are surjective for all $p,j\ge 0$ .

Part (1) follows by assumptions. For (2) and (3), analyzing the spectral sequences

\begin{align*}E_1^{ij}&= \mathbb H^j(p^*L\Omega^i_{X/F}\otimes \Omega_{X\times\mathbb A^j/X}^{\le p-i})\implies \mathbb H^{i+j}(L\Omega_{X\times\mathbb A^j/F}^{\le p}),\\[4pt]E_1^{ij}&= \mathbb H^j(p^*\underline{\Omega}^i_{X/F}\otimes\Omega_{X\times \mathbb A^j/X}^{\le p-i})\implies \mathbb H^{i+j}(\underline{\Omega}_{X\times \mathbb A^j/F}^{\le p}),\end{align*}

we see that it suffices to show:

1. (1) $\mathbb H^{-m+1+2p-k}(p^*L\Omega_{X/F}^k\otimes \Omega_{X\times \mathbb A^j/X}^{\le p-k})\to \mathbb H^{-m+1+2p-k}(p^*\underline{\Omega}_{X/F}^k\otimes \Omega_{X\times \mathbb A^j/X}^{\le p-k})$ are isomorphisms for all $p\ge 0$ and $k\le p$ ;

2. (2) $\mathbb H^{-m+2p-k}(p^*L\Omega_{X/F}^k\otimes \Omega_{X\times \mathbb A^j/X}^{\le p-k})\to \mathbb H^{-m+2p-k}(p^*\underline{\Omega}_{X/F}^k\otimes \Omega_{X\times \mathbb A^j/X}^{\le p-k})$ are surjective for all $p\ge 0$ and $k\le p$ .

By induction using long exact sequences associated to

\begin{align*}\Omega_{X\times \mathbb A^j/X}^{p-k}[-p+k]\to \Omega_{X\times \mathbb A^j/X}^{\le p-k}\to \Omega_{X\times \mathbb A^j/X}^{\le p-k-1}\xrightarrow{+1},\end{align*}

it suffices to show:

1. (1) $\mathbb H^{-m+1+p}(L\Omega_{X/F}^k)\to \mathbb H^{-m+1+p}(\underline{\Omega}_{X/F}^k)$ are isomorphisms for all $p\ge 0$ and $k\le p$ ;

2. (2) $\mathbb H^{-m+p}(L\Omega_{X/F}^k)\to \mathbb H^{-m+p}(\underline{\Omega}_{X/F}^k)$ are surjective for all $p\ge 0$ and $k\le p$ .

We have (1) by assumption, and (2) follows by Proposition 7.3.

As an application, we construct an example that shows local $K_m$ -regularity does not imply $K_m$ -regularity in general.

Let X be any projective surface with $\mathcal H^1\underline{\Omega}_{X}^0\simeq \mathbb C$ (e.g., the projective cone over a curve of genus 2), and Y be a curve of genus $g\ge 1$ , so that $H^1(Y,\mathcal O_Y)\cong \mathbb C$ . Take $Z=X\times Y$ . Then there is an exact triangle

\begin{align*}\mathcal O_Z\to \underline{\Omega}_Z^0\to \mathcal O_{Y}[-1]\xrightarrow{+1}.\end{align*}

Since $\mathcal H^2\underline{\Omega}_Z^0=0$ , we see that Z is locally $K_{-2}$ -regular. However Z is not $K_{-2}$ -regular by Theorem 7.1, because

\begin{align*}\mathbb H^2(\mathcal O_Y[-1])\simeq H^1(\mathcal O_Y)\neq 0.\end{align*}

For this reason, we do not expect pre-m-Du Bois varieties to be $K_{-d+m+2}$ -regular in general. We summarize the relationships between these concepts when $m=0$ as follows.

We conclude with the following example, which shows that Proposition 4.1(2) need not hold if X is not affine, and has higher-dimensional singular locus.

Let C be a genus 2 curve in $\mathbb P^2$ , and let $S=C\times \mathbb P^1$ . Denote by $\pi: S\to \mathbb P^1$ the projection map. The line bundle

\begin{align*} L &= \omega_S\otimes (\pi^*\omega_{\mathbb P^1}(1))^{\vee} \\ &= \omega_{S/\mathbb P^1}\otimes (\pi^*\mathcal O_{\mathbb P^1}(1))^{\vee} \end{align*}

is relatively ample over $\mathbb P^1$ , so we can consider the relative affine cone

\begin{align*}Y = {\bf Spec}_{\mathbb P^1} \bigoplus_{m\ge 0} \pi_* L^m\end{align*}

over S defined by L. One checks that the arguments in [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺13] computing the Du Bois complexes of cones carry over to the relative case, giving that

\begin{align*}\mathcal H^1\underline{\Omega}_Y^0 \simeq \bigoplus_{m\ge 1} R^1 \pi_* L^m\end{align*}

is a sheaf supported on $Y_{\operatorname{sing}}\simeq \mathbb P^1$ . We have $R^1\pi_* L \simeq \mathcal O_{\mathbb P^1}(-1)$ by Grothendieck duality, and $R^1\pi_* L^m =0$ by Kodaira vanishing. It follows that

\begin{align*}\mathcal H^1\underline{\Omega}_Y^0 \simeq \mathcal O_{\mathbb P^1}(-1).\end{align*}

Similarly, one checks that

\begin{align*}\mathcal H^1\underline{\Omega}_Y^1 \simeq \mathcal O_{\mathbb P^1}(-1).\end{align*}

Now, let X be a projectivization of Y that is smooth away from the singular points of Y. Then we have

\begin{align*}\mathcal H^1\underline{\Omega}_X^0 \simeq \mathcal H^1\underline{\Omega}_X^1 \simeq \mathcal O_{\mathbb P^1}(-1).\end{align*}

In particular, X is not pre-0-Du Bois. We claim that X is $K_{-1}$ -regular. Indeed, by Theorem 7.1, this is equivalent to the following maps being isomorphisms:

\begin{align*}\mathbb H^3(\mathcal O)&\xrightarrow{\sim} \mathbb H^3(\underline{\Omega}_X^0),\\ \mathbb H^2(\mathcal O)&\xrightarrow{\sim} \mathbb H^2(\underline{\Omega}_X^0),\\ \mathbb H^3(\mathbb L_X)&\xrightarrow{\sim} \mathbb H^3(\underline{\Omega}_X^1).\end{align*}

Since the cone of the map $\mathcal O_X\to \underline{\Omega}_X^0$ is $\mathcal O_{\mathbb P^1}(-1)[-1]$ , which has no cohomology, we have the two isomorphisms. The cone of the map $\mathbb L_X\to \underline{\Omega}_X^1$ is $\mathcal O_{\mathbb P^1}(-1)[-1]\oplus C$ , where the complex C is supported on the one-dimensional subvariety $X_{\operatorname{sing}}$ , and in cohomological degrees less than or equal to 0. Analyzing the spectral sequence

\begin{align*}E_2^{ij} = H^i(\mathcal H^j C)\implies \mathbb H^{i+j}(C)\end{align*}

gives the last isomorphism.

8. Further questions

(1) As noted in [Reference Haesemeyer and WeibelHW25], being $K_m$ -regular also implies strict $(m-1)$ -Du Bois, that is, the maps

\begin{align*}\Omega_{X/\mathbb C}^p\to \underline{\Omega}_{X/\mathbb C}^p\end{align*}

are isomorphisms for $p\le m-1$ . Since there is no known example of a variety with strict-1-Du Bois singularities that is not a local complete intersection, this leads to the following question.

(2) We showed in Theorem B that $K_2$ -regular affine local complete intersections are smooth. We do not know if this is the case when the variety is not necessarily affine.

One approach could be to construct projective varieties that are $K_2$ -regular using Theorem 7.1.

(3) The notion of higher-F-injectivity is introduced in [Reference Kawakami and WitaszekKW24] as an analog of higher Du Bois singularities in positive characteristic, extending the well-known notion of F-injectivity (analogous to the Du Bois singularities for complex algebraic varieties). One can ask the following question.

(4) In § 6, we classified all possible Du Bois tables in dimension 2. We ask if one can extend this classification to higher dimensions.

For 3-folds over $\mathbb Q$ , the Du Bois table takes the following form.

Using the spectral sequence argument as in Lemma 6.2, one can show that

\begin{align*}b^{0,2}\ge b^{1,2}, \quad b^{0,2}-b^{1,2}\ge b^{2,1}-b^{1,1}.\end{align*}

The question amounts to asking whether there are further relations among the entries in the Du Bois table. By the main theorem of [Reference Cortinãs, Haesemeyer, Walker and WeibelCHW⁺10], answers to this question will in particular give answers to Bass’s question.