Higher Du Bois and higher rational singularities

Introduction

The filtered de Rham complex is related to the complex of Kähler differentials via the canonical comparison Kähler-to-Du Bois map $\phi^{p}:\Omega_{Y}^{p}\to\underline{\Omega}^{p}_{Y}$. The maps $\phi^{p}$ are isomorphisms for all $p$ only when $Y$ is smooth, at least when $Y$ is a local complete intersection [MP22a, Theorem 3.39, Theorem F]. It is thus natural to consider the case when $\phi^{p}$ is a quasi-isomorphism in a certain range. Steenbrink [Ste83, §3] introduced the notion of Du Bois singularities, which play a role in the study of compactifications of moduli. By definition, $Y$ is Du Bois if $\phi^{0}$ is a quasi-isomorphism. Following [MOPW23] and [JKSY22a], we say that $Y$ is $k$-Du Bois if $\phi^{p}$ is a quasi-isomorphism for $0\leq p\leq k$. Thus -Du Bois singularities are exactly the Du Bois singularities in Steenbrink’s terminology. A key property satisfied by Du Bois singularities is the following:

Let $f:\mathcal{Y}\to S$ be a flat proper family of complex algebraic varieties. Assume that some fiber $Y_{s}$ has Du Bois singularities. Then, possibly after replacing $S$ by a neighborhood of $s$, for all $q\geq 0$, the sheaves $R^{q}f_{*}\mathcal{O}_{\mathcal{Y}}$ are locally free of finite type and compatible with base change.

The theorem can be interpreted (in particular) as giving a relation between the mixed Hodge structure $H^{*}(Y_{0})$ of a singular fiber $Y_{0}$ with the limit mixed Hodge $H^{*}_{\lim}$ associated to a one-parameter smoothing $\mathcal{X}/\Delta$. In this version, the theorem (for dimension $2$ slc hypersurface singularities) was independently established by Shah [Sha79], and plays a key role in the study of degenerations of $K3$ surfaces (e.g. [Sha80]) and related objects (e.g. [Laz10], [KLSV18]). Theorem 1.1 continues to have important consequences for the study of compact moduli of varieties of general type (see e.g. [Kol23], esp. §2.5 of loc. cit.).

Here, we prove the following generalization of Theorem 1.1 for the case of a local complete intersection (lci) morphism:

Let $f:\mathcal{Y}\to S$ be a flat proper family of complex algebraic varieties and let $s\in S$. Suppose that the fiber $Y_{s}$ has $k$-Du Bois lci singularities. Then, possibly after replacing $S$ by a neighborhood of $s$, the higher direct image sheaves $R^{q}f_{*}\Omega^{p}_{\mathcal{Y}/S}$ of the relative Kähler differentials are locally free and compatible with base change for $0\leq p\leq k$ and all $q\geq 0$.

We use the lci assumption to control the sheaves of Kähler and relative Kähler differentials in two ways. First, a result of [MP22a] gives an estimate on the codimension of the singular locus for $k$-Du Bois lci singularities. Using this and some results of Greuel, we prove a key technical point: under the lci assumption, the sheaves $\Omega^{p}_{\mathcal{X}/S}$ are flat over $S$ for $p\leq k$ (Theorem 2.5). In case $k=0$, both the codimension estimate and the flatness are automatic and one recovers Theorem 1.1 as a special case.

Let $f\colon\mathcal{Y}\to S$ be a flat proper family of complex algebraic varieties over an irreducible base. For $s\in S$, suppose that the fiber $Y_{s}$ has $k$-Du Bois lci singularities. Then, for every fiber $t$ such that $Y_{t}$ is smooth, $\dim\operatorname{Gr}_{F}^{p}H^{p+q}(Y_{t})=\dim\operatorname{Gr}_{F}^{p}H^{p+q}(Y_{s})$ for every $q$ and for $0\leq p\leq k$. Equivalently $\underline{h}^{p,q}(Y_{s})=h^{p,q}(Y_{t})$ for all $p\leq k$. ∎

In the case of hypersurface singularities, results similar to Corollary 1.4 were obtained by Kerr-Laza [KL20, KL23] with further clarifications given by Saito (personal communications) based on [Sai16]. These type of results are for example relevant to the study of the moduli of cubic fourfolds [Laz10].

Another application of Theorem 1.2 is the following generalization of the results of Kawamata [Kaw92], Ran [Ran92], and Tian [Tia92] on the unobstructedness of deformations for nodal Calabi-Yau varieties in any dimension, where the special case of isolated hypersurface singularities was established in §6 of the first version of [FL22a]:

Let $Y$ be a canonical Calabi-Yau variety (Definition 4.4) which is additionally a scheme with $1$-Du Bois lci (not necessarily isolated) singularities. Then the functor $\mathbf{Def}(Y)$ is unobstructed.

In this paper, we propose a more intrinsic definition of $k$-rational singularities in general (Definition 3.13) and show that it agrees with the usual definition of rational singularities for $k=0$ and with the definition of [FL22a, §3] (under mild assumptions; see Corollaries 3.16 and 3.18). Additionally, for hypersurface singularities, M. Saito proves that the new definition proposed here is indeed equivalent to the previous numerical definition mentioned above (Theorem A.1). The main advantage of the definition of higher rational singularities given here is that it naturally factors through the higher Du Bois condition. In analogy with the case $k=0$, we conjecture that $k$-rational implies $k$-Du Bois in general. In the first version of [FL22a, §3] (and expanded in [FL22b]), we verified this conjecture under the assumption of isolated lci singularities. Here, we generalize this in both directions, for arbitrary isolated or lci singularities:

Let $X$ have either lci singularities (not necessarily isolated) or isolated singularities (not necessarily lci). If $X$ is $k$-rational, then $X$ is $k$-Du Bois.

Mustaţă and Popa gave an independent proof of Theorem 1.6 for the case of lci singularities ([MP22b, Thm. B]).

The isolated complete intersection case (see [FL22b]) sheds light on the tight relationship between higher rational and higher Du Bois singularities. In the first version of this paper, we made the following conjecture:

If $X$ has lci singularities and $X$ is $k$-Du Bois, then $X$ is $(k-1)$-rational.

For an isolated hypersurface singularity, Conjecture 1.8 is an immediate consequence of the following result:

Let $(X,x)$ be an isolated hypersurface singularity and let $\widetilde{\alpha}_{X,x}=\widetilde{\alpha}_{X}$ be the minimal exponent as defined by Saito [Sai93]. Then

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

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

Here (i) follows from [JKSY22a, Thm. 1] and [MOPW23, Thm. 1.1] and holds true for a general, not necessarily isolated, hypersurface singularity, and (ii) is proved in [FL22b, Corollary 6.6]. In Appendix A, M. Saito proves (ii) for the case of a general hypersurface singularity, based on the results of [JKSY22a]. Thus implies Conjecture 1.8 for the case of general hypersurface singularities, not necessarily isolated (Cor. A.2). Mustaţă and Popa have proved (ii) of Proposition 1.9, and hence Conjecture 1.8 in this case as well (cf. [MP22b, Thm. E, Cor. F]). Additionally, in [FL22b, Corollary 5.5], we established Conjecture 1.8 for isolated lci singularities. Recently, Chen-Dirks-Mustaţă [CDM22] have proved Conjecture 1.8 in general.

Since $k$-rational singularities are milder than $k$-Du Bois singularities, one expects that more of the Hodge diamond is preserved in families with $k$-rational singularities. Indeed, this is the case as shown in [KL20, Cor. 4.2] (isolated hypersurface $k$-rational singularities) and [KLS22, Thm. 1] (arbitrary rational singularities). Here, we extend these results to $k$-rational lci singularities, and clarify the difference between $k$-Du Bois and $k$-rational in this context. Essentially, for $k$-rational singularities, in addition to the preservation given by Corollary 1.4, one gains Hodge symmetry in a certain range. Informally, we can say that the frontier Hodge diamond up to coniveau $k$ is preserved for deformations of $k$-rational singularities.

Let $f\colon\mathcal{Y}\to S$ be a flat proper family of complex algebraic varieties over an irreducible base. For $s\in S$, suppose that the fiber $Y_{s}$ has $k$-rational lci singularities. Then, for every fiber $t$ such that $Y_{t}$ is smooth, and for all $p\leq k$,

where $\pi_{s}:\hat{Y}_{s}\to Y_{s}$ is an arbitrary projective resolution.

(4) Fourfolds $Y$ with ADE hypersurface singularities (such as those occurring in [Laz10]) are examples of $1$-rational singularities. For such fourfolds, Corollary 1.10 implies that only $\underline{h}^{2,2}(Y)$ can vary in small deformations. Thus, any smoothing of $Y$ will have finite monodromy (compare [KLS22, Cor. 1]).

A brief description of the contents of this paper is as follows. Section 2 deals with some basic results about Kähler differentials in the lci case. These include Theorem 2.5 regarding the flatness of the relative Kähler differentials and Proposition 2.7 on restricting to a generic hypersurface section. In Section 3, we give a quick review of the definition and the basic facts about higher Du Bois singularities (following [MOPW23], [JKSY22a], and [FL22a, §3]) and define higher rational singularities. After these preliminaries, we establish Theorem 1.2. Our argument is close to the original argument ([DBJ74, Lemma 1]) used to establish Theorem 1.1, following a suggestion of J. Kollár. Finally, in Section 5, we prove Theorem 1.6 following the strategy of [Kov99], and deduce a consequence about the Hodge numbers of a smoothing along the lines of Corollary 1.4. An appendix section by M. Saito discusses Conjecture 1.8 in the hypersurface case.

Finally, beyond the conjectures and speculation we have already made, we emphasize the importance of extending these results wherever possible to the non-lci case. Along these lines, Shen, Venkatesh and Vo have recently posted a preprint [SVV23] proposing different definitions of $k$-Du Bois and $k$-rational singularities which agree with the previous ones in the lci case.

Acknowledgement

We have benefited from discussions and correspondence with J. Kollár, M. Mustaţă, and M. Popa on higher du Bois singularities while preparing [FL22a]. The second author had several related discussions with M. Kerr and M. Saito while preparing previous joint work. We also thank J. de Jong, M. Saito and C. Schnell for some further comments related to this paper. After circulating an earlier version of this paper, M. Mustaţă, M. Popa informed us of some their recent work ([MP22b], [CDM22]) related to Theorem 1.6 and Conjecture 1.8. We are grateful to them for these communications. M. Saito kindly provided us with proofs of Proposition 1.9(ii) and Conjecture 1.8 in the hypersurface case, and agreed to include those as an appendix to our paper. Finally, we would like the referees for a very carefully reading of the first version of this paper and for many helpful comments.

Some results on Kähler and relative Kähler differentials

The Kähler differentials are coherent sheaves that are determined by certain universal properties, including compatibility with base change. For smooth families $\mathcal{Y}/S$, the proof of the constancy of the Hodge numbers uses in an essential way the semi-continuity of $h^{q}(\Omega^{p}_{Y_{t}})$, which in turn depends on the flatness of $\Omega^{p}_{\mathcal{Y}/S}$. Here, we generalize this key point, noting that, for an lci morphism, $\Omega^{p}_{\mathcal{X}/S}$ is flat over $S$ (Theorem 2.5) for $p$ satisfying a bound depending on the dimension $d$ of the singular locus. Our argument depends essentially on the lci assumption, and it is a consequence of some depth estimates for $\Omega_{X}^{p}$ for $X$ with lci singularities due to Greuel.

A second result (Proposition 2.7) that follows by related arguments are higher adjunction type results regarding restrictions of Kähler differentials to generic hypersurface sections.

We begin by recalling some basic notions concerning depth. Recall that, if $R$ is a local ring with maximal ideal $\mathfrak{m}$ and $M$ is an $R$-module, then $\operatorname{depth}M$ is the maximal length of a regular sequence for $M$ [Mat80, p. 120]. If $\mathcal{F}$ is a coherent sheaf on a complex space $X$ and $x\in X$, let $\operatorname{depth}_{x}\mathcal{F}=\operatorname{depth}\mathcal{F}_{x}$, viewed as an $\mathcal{O}_{X,x}$-module. A key technical result we will need is then the following theorem [Sch64], [Gre75, Satz 1.2]:

Suppose that $X$ is an analytic space, $A$ a closed analytic subspace, and $\mathcal{F}$ a coherent sheaf on $X$. Let $\rho\colon H^{0}(X;\mathcal{F})\to H^{0}(X-A;\mathcal{F}|X-A)$ be the restriction map.

If $\operatorname{depth}\mathcal{F}_{x}\geq\dim A+1$ for every $x\in X$, the homomorphism $\rho$ is injective.

If $\operatorname{depth}\mathcal{F}_{x}\geq\dim A+2$ for every $x\in X$, the homomorphism $\rho$ is an isomorphism. ∎

It is well known that Kähler $p$-differentials on singular spaces can have torsion. For instance this is already the case for $\Omega^{1}_{C}$ where $C$ is a nodal curve. However, we have the following [Gre75, Lemma 1.8]

Suppose that $X$ is an lci singularity of dimension $n$ and that $\dim\Sigma\leq d$. Then, for all $x\in X$, $\operatorname{depth}_{x}\Omega^{p}_{X}\geq n-p$ for $p\leq n-d$. More generally, let $f\colon\mathcal{X}\to S$ be a flat lci morphism of relative dimension $n$ over a smooth base $S$ and let $\mathcal{X}_{\text{\rm{crit}}}$ denote the critical locus of $f$, i.e. the points of $\mathcal{X}$ where $f$ is not a smooth morphism. If $\dim S=m$ and the relative dimension of $\mathcal{X}_{\text{\rm{crit}}}$ is at most $d$ then, for every $x\in\mathcal{X}$, then $\operatorname{depth}_{x}\Omega^{p}_{\mathcal{X}/S}\geq\dim\mathcal{X}-p=n+m-p$ for $p\leq n-d$. ∎

Before stating the next corollary, we fix the following notation which will be used for the rest of the section: If $f\colon\mathcal{X}\to S$ is a morphism and $s\in S$, we denote by $X_{s}$ the fiber $f^{-1}(s)$ and by $\Sigma_{s}$ the singular locus of $X_{s}$: $\Sigma_{s}=(X_{s})_{\text{\rm{sing}}}$.

Suppose that $f\colon\mathcal{X}\to S$ is a flat lci morphism of relative dimension $n$ over a smooth base $S$ of dimension $m$, and that $\dim\Sigma_{s}\leq d$, with $n-d\geq 2k+1$ for some integer $k\geq 1$ and every point $s\in S$. Let $\mathcal{X}_{\text{\rm{crit}}}$ denote the critical locus of $f$, i.e. the points of $\mathcal{X}$ where $f$ is not a smooth morphism. Then, for every $x\in\mathcal{X}$ and $p\leq n-d$,

and, for all $p\leq k$ and every open subset $\mathcal{U}$ of $\mathcal{X}$, the restriction map

By assumption, $\dim\mathcal{X}_{\text{\rm{crit}}}\leq d+m$. Note that $n-d\geq 2k+1\geq k$, and hence, if $p\leq k$, then $p\leq n-d$. Thus Theorem 2.2 implies that, for all $x\in\mathcal{X}$,

Then $H^{0}(\mathcal{U};\Omega^{p}_{\mathcal{X}/S}|\mathcal{U})\to H^{0}(\mathcal{U}-\mathcal{X}_{\text{\rm{crit}}};\Omega^{p}_{\mathcal{X}/S}|\mathcal{U}-\mathcal{X}_{\text{\rm{crit}}})$ is an isomorphism, by Theorem 2.1(ii). ∎

Suppose that $f\colon\mathcal{X}\to S$ is a flat lci morphism of relative dimension $n$ over a smooth base $S$ of dimension $m$, and that $\dim\Sigma_{s}\leq d$, with $n-d\geq 2k+1$ for some integer $k\geq 1$ and every point $s\in S$. Suppose that $\mathcal{X}\subseteq A\times S$, where $A$ is smooth, and let $I$ be the defining ideal of $\mathcal{X}$ in $A\times S$. Then there exists a filtration of $\bigwedge^{k}\pi_{1}^{*}\Omega_{A}^{1}|\mathcal{X}=\pi_{1}^{*}\Omega_{A}^{k}|\mathcal{X}$ by coherent subsheaves $\mathcal{K}^{a}$, $0\leq a\leq k$, such that $\mathcal{K}^{0}=\pi_{1}^{*}\Omega_{A}^{k}|\mathcal{X}$, $\operatorname{depth}_{x}\mathcal{K}^{a}\geq d+m+2$ for all $x\in\mathcal{X}$, and, for all $a\geq 0$,

The proof is by induction on $k$. The case $k=1$ is the conormal sequence

where we let $\mathcal{K}^{1}$ be the image of $I/I^{2}$. In general, for all $0\leq a\leq k$, let $\mathcal{K}^{a}$ be the image of $\bigwedge^{a}(I/I^{2})\otimes\pi_{1}^{*}\Omega_{A}^{k-a}|\mathcal{X}$ in $\pi_{1}^{*}\Omega_{A}^{k}|\mathcal{X}$. Thus $\{\mathcal{K}^{a}\}$ is the usual Koszul filtration, and the proposition is clear over all points of $\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$. Moreover, $\mathcal{K}^{k}$ is either or $\bigwedge^{k}(I/I^{2})$. By hypothesis $k<n=\operatorname{rank}\Omega_{\mathcal{X}/S}^{1}$. Let $r=\operatorname{rank}I/I^{2}$. We can clearly assume that $r\geq 1$. If $r\geq k$, then $\mathcal{K}^{k}=\bigwedge^{k}(I/I^{2})$. If $r<k$, then $\mathcal{K}^{k}=0$, and the largest $p$ such that $\mathcal{K}^{p}\neq 0$ is $\mathcal{K}^{r}$, the image of $\bigwedge^{r}(I/I^{2})\otimes\pi_{1}^{*}\Omega_{A}^{k-r}|\mathcal{X}$. In this case, the image of $\bigwedge^{r}(I/I^{2})\otimes(I/I^{2})\otimes\pi_{1}^{*}\Omega_{A}^{k-r-1}|\mathcal{X}$ in $\bigwedge^{r}(I/I^{2})\otimes\pi_{1}^{*}\Omega_{A}^{k-r}|\mathcal{X}$ maps to in $\pi_{1}^{*}\Omega_{A}^{k}|\mathcal{X}$. By the inductive hypothesis, the sequence

is exact, so there is an induced map $\varphi_{r}\colon\bigwedge^{r}(I/I^{2})\otimes\Omega_{\mathcal{X}/S}^{k-r}\to\pi_{1}^{*}\Omega_{A}^{k}|\mathcal{X}$. Then the image of $\varphi_{r}$ is contained in the torsion free sheaf $\mathcal{K}^{r}$ and is equal to $\mathcal{K}^{r}$ over $\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$. Moreover $\varphi_{r}$ is injective over $\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$. By the inductive hypothesis on $k$,

and hence $\operatorname{Im}\varphi_{r}=\mathcal{K}^{r}$, and $\varphi_{r}\colon\bigwedge^{r}(I/I^{2})\otimes\Omega_{\mathcal{X}/S}^{k-r}\cong\mathcal{K}^{r}=\mathcal{K}^{r}/\mathcal{K}^{r+1}$ is an isomorphism.

For a general $a$, there is a commutative diagram

where by definition $\mathcal{J}=\operatorname{Ker}\{\bigwedge^{a}(I/I^{2})\otimes\pi_{1}^{*}\Omega_{A}^{k-a}|\mathcal{X}\to\bigwedge^{a}(I/I^{2})\otimes\Omega_{\mathcal{X}/S}^{k-a}\}$ and $\varphi_{a}$, which exists and is an isomorphism over $\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$, is yet to be constructed over all of $\mathcal{X}$.

For a fixed $k$, arguing by descending induction on $a$, and starting either at $k$ or at $r$, we can assume that $\operatorname{depth}_{x}\mathcal{K}^{a+1}\geq d+m+2$. Moreover, $\mathcal{K}^{a+1}|\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$ is a subbundle of $\mathcal{K}^{a}|\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$, so the torsion subsheaf of $\mathcal{K}^{a}/\mathcal{K}^{a+1}$ is supported on $\mathcal{X}_{\text{\rm{crit}}}$. If $\sigma$ is a section of $\mathcal{K}^{a}$ over some open subset $\mathcal{U}$ and $\sigma|V\in\mathcal{K}^{a+1}$ for an open subset $\mathcal{V}=\mathcal{U}-W$ of $\mathcal{U}$, where $W$ is an analytic subset of $\mathcal{U}$, then we may assume that $W=\mathcal{X}_{\text{\rm{crit}}}$. Since $\mathcal{K}^{a}$ is torsion free and $\operatorname{depth}_{x}\mathcal{K}^{a+1}\geq d+m+2$, $\sigma\in\mathcal{K}^{a+1}(\mathcal{U})$. Thus $\mathcal{K}^{a}/\mathcal{K}^{a+1}$ is torsion free.

Let $\mathcal{I}$ be the kernel of the map

Since $\mathcal{K}^{a}/\mathcal{K}^{a+1}$ is torsion free, a standard argument shows that, for every open subset $\mathcal{U}$ of $\mathcal{X}$, the map

is an isomorphism. Over $\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$, there is an isomorphism $\mathcal{I}|\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}\cong\mathcal{J}|\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$. Since$\bigwedge^{a}(I/I^{2})$ is locally free, $\operatorname{depth}_{x}\mathcal{J}\geq\operatorname{depth}_{x}\bigwedge^{a}(I/I^{2})\otimes\Omega_{\mathcal{X}/S}^{k-a}\geq d+m+2$ by Corollary 2.3. In particular,

is also an isomorphism. Thus, $\mathcal{J}=\mathcal{I}$, and there is an induced injective homomorphism $\varphi_{a}$ as in the diagram. Since $\varphi_{a}$ is an isomorphism over $\mathcal{X}-\mathcal{X}_{\text{\rm{crit}}}$ and $\operatorname{depth}_{x}\bigwedge^{a}(I/I^{2})\otimes\Omega_{\mathcal{X}/S}^{k-a}\geq d+m+2$, $\varphi_{a}$ is an isomorphism. Finally, since both $\operatorname{depth}_{x}\mathcal{K}^{a+1}\geq d+m+2$ and $\operatorname{depth}_{x}\mathcal{K}^{a}/\mathcal{K}^{a+1}\geq d+m+2$, $\operatorname{depth}_{x}\mathcal{K}^{a}\geq d+m+2$ as well. ∎

A key technical step in establishing Theorem 1.2, where the lci assumption seems crucial, is the following:

Suppose that $f\colon\mathcal{X}\to S$ is a flat lci morphism, where $S$ is an arbitrary base space. Let $s\in S$ and suppose that $\operatorname{codim}_{X_{s}}\Sigma_{s}\geq 2k+1$. Then, possibly after replacing $S$ by a neighborhood of $s$, the sheaf of relative differentials $\Omega^{p}_{\mathcal{X}/S}$ is flat over $S$ for all $p\leq k$. For $p=0,1$, $\Omega^{p}_{\mathcal{X}/S}$ is flat over $S$ with no assumption on the dimension of the singular locus.

In the same spirit, with respect to Theorem 1.2, suppose that $\pi\colon\mathcal{C}\to\Delta$ is a family of smooth projective curves acquiring a single ordinary double point over , with the same local picture as above (so that $\mathcal{C}$ is also smooth). Then the sheaf $\Omega^{1}_{\mathcal{C}/\Delta}$ of relative Kähler differentials is flat over $\Delta$ either by the case $p=1$ of Theorem 2.5 or by the direct computation that $\Omega^{1}_{\mathcal{C}/\Delta}$ is isomorphic in a neighborhood of the singular point $p\in\mathcal{C}$ to the maximal ideal $\mathfrak{m}_{p}$, and in particular is torsion free over $\mathcal{O}_{\Delta}$. Let $C_{t}$ denote the fiber over $t$. If $C_{0}$ is reducible, the values of $h^{0}(\Omega^{1}_{C_{t}})$ and $h^{1}(\Omega^{1}_{C_{t}})$ jump up at $t=0$. Hence $R^{1}\pi_{*}\Omega^{1}_{\mathcal{C}/\Delta}$ has torsion at , and in particular is not locally free. This follows from very general results about cohomology and base change. In our situation, since $\Delta$ is smooth of dimension one, it follows directly by applying $R^{i}\pi_{*}$ to the exact sequence

to see that the dimension of the $\mathcal{O}_{\Delta}$-module $R^{1}\pi_{*}\Omega^{1}_{\mathcal{C}/\Delta}/tR^{1}\pi_{*}\Omega^{1}_{\mathcal{C}/\Delta}$ is larger than the rank of $R^{1}\pi_{*}\Omega^{1}_{\mathcal{C}/\Delta}$.

It suffices to consider the case $p=k$. Since every deformation is (locally) pulled back from the versal deformation, by standard properties of flatness and wedge product under base change we can assume that $S$ is smooth. First, with no assumption on the singular locus, $\mathcal{O}_{\mathcal{X}}$ is flat over $S$ by assumption. To see that $\Omega^{1}_{\mathcal{X}/S}$ is flat over $S$, again with no assumption on the singular locus, we have the conormal sequence

Let $I_{s}$ be the ideal of $X_{s}$ in $A$. By the lci assumption, $I_{s}=I\otimes\mathcal{O}_{S,s}/\mathfrak{m}_{s}$. Then the conormal sequence for $\mathcal{X}/S$ becomes the corresponding sequence for $\Omega^{1}_{X_{s}}$ after tensoring with $\mathcal{O}_{S,s}/\mathfrak{m}_{s}$, namely

In particular, $\bar{u}$ is injective. Hence $\Omega_{\mathcal{X}/S}^{1}=\operatorname{Coker}u$ is flat over $S$ by the local criterion of flatness [Mat80, (20.E) pp. 150–151].

For $k\geq 2$, by induction on $k$ and descending induction on $a$, $\mathcal{K}^{a+1}$ and $\mathcal{K}^{a}/\mathcal{K}^{a+1}$ are flat over $S$ for all $a\geq 1$, and hence so is $\mathcal{K}^{a}$. Let $\mathcal{K}^{a}_{s}$ be the image of $\bigwedge^{a}(I_{s}/I_{s}^{2})\otimes\Omega_{A}^{k-a}|X_{s}$ in $\Omega_{A}^{k}|X_{s}$. Proposition 2.4 applied to the case $S=$ pt implies that $\mathcal{K}_{s}^{a}/\mathcal{K}_{s}^{a+1}\cong\bigwedge^{a}(I_{s}/I_{s}^{2})\otimes\Omega_{X_{s}}^{k-a}$. We have a commutative diagram

Here, the top row is exact for $a\geq 1$ by the flatness of $\mathcal{K}^{a}/\mathcal{K}^{a+1}$ and induction on $k$. By descending induction on $a$, the above diagram implies that the natural map $\mathcal{K}^{a}\otimes(\mathcal{O}_{S,s}/\mathfrak{m}_{s})\to\mathcal{K}_{s}^{a}$ is an isomorphism for all $a\geq 1$. In particular, for $a=1$, we have the map $u\colon\mathcal{K}^{1}\to\pi_{1}^{*}\Omega_{A}^{k}|\mathcal{X}$ with cokernel $\Omega_{\mathcal{X}/S}^{k}$, and the reduction $\bar{u}$ of $u$ mod $\mathfrak{m}_{s}$ is the natural inclusion $\mathcal{K}_{s}^{1}\to\Omega_{A}^{k}|X_{s}$. Since $\bar{u}$ is injective, $\Omega_{\mathcal{X}/S}^{k}=\operatorname{Coker}u$ is flat over $S$ by the local criterion of flatness as above. This completes the proof of Theorem 2.5. ∎

2. Kähler differentials and generic hyperplane sections

Let $X$ be a reduced local complete intersection with $X_{\text{\rm{sing}}}=\Sigma$ and let $H$ be a Cartier divisor on $X$ which is a generic element of a base point free linear system, with ideal sheaf $\mathcal{O}_{X}(-H)$. If $\operatorname{codim}\Sigma\geq 2k-1$, then, setting $\mathcal{O}_{H}(-H)=\mathcal{O}_{X}(-H)|H$, we have

with $\mathcal{O}_{H}(-H)\cong\mathcal{O}_{H}$ locally free and $H$ reduced (as it is generically reduced and lci).

By induction on $k$, it suffices to consider the case $p=k$. By the remarks above, we can assume $k\geq 2$. By the de Rham lemma [Gre75, Lemma 1.6], we have the following:

Let $d=\dim\Sigma$. For $0\leq p<n-d$, the following sequence is exact:

Since $f$ is generic, the critical set $C(f)$ of $f$ (in the notation of [Gre75]) is just $\Sigma$. Then the result follows immediately from [Gre75, Lemma 1.6] (with $g=h$, $k=1$, and $C(f,g)\cap N(h)=\Sigma$). ∎

From the lemma, since $n-d=\operatorname{codim}\Sigma\geq 2k-1>k$, because $k>1$, there is an exact sequence

Tensoring with $\mathcal{O}_{H}$ gives an exact sequence

So we have to show that, for a general choice of $f$, $\mathit{Tor}_{1}^{\mathcal{O}_{X}}(\Omega^{k}_{X}/df\wedge\Omega^{k-1}_{X},\mathcal{O}_{H})=0$. Since $\mathcal{O}_{H}$ has the free resolution

it suffices to show that $\Omega^{k}_{X}/df\wedge\Omega^{k-1}_{X}$ has no $f$-torsion, i.e. that multiplication by $f$ is injective. Since there is an inclusion $\Omega^{k}_{X}/df\wedge\Omega^{k-1}_{X}\to\Omega^{k+1}_{X}$, it suffices to prove that $\Omega^{k+1}_{X}$ has no $f$-torsion. Let $\tau$ be an $f$-torsion local section of $\Omega^{k+1}_{X}$. Note that the support of $\tau$ must be contained in $\Sigma\cap V(f)$, since $\Omega^{k+1}_{X}|X-\Sigma$ is torsion free and $f$ is invertible on $X-V(f)$. By the assumption that $H$ is general, every component of $\Sigma\cap V(f)$ has dimension $\leq d-1$. By [Gre75, Lemma 1.8], since $k+1\leq 2k-1\leq\operatorname{codim}\Sigma$ as long as $k\geq 2$, we have, for every $x\in X$,

Using $n-d\geq 2k-1$ gives $n-k-1\geq d+k-2\geq d$, since we are assuming $k\geq 2$. Thus

It follows from Theorem 2.1(i) that, for every open subset $U$ of $X$, the restriction map

is injective. In particular, $\tau=0$. Hence $(*)_{p}$ is exact for $p\leq k$. ∎

Note that we showed the following in the course of proving Proposition 2.7:

With $X$ and $H$ as in the statement of Proposition 2.7, $\mathit{Tor}_{1}^{\mathcal{O}_{X}}(\Omega^{p}_{X},\mathcal{O}_{H})=0$ for all $p\leq k+1$. ∎

k𝑘k-Du Bois and k𝑘k-rational singularities

We now review the notion of Du Bois and higher Du Bois singularities, and propose a general definition for higher rational singularities. After verifying that our definition agrees with the standard definition of rational singularities, as well as with previous definitions of higher rational singularities (under mild assumptions, expected to hold in general), we discuss the connection between higher rational and higher Du Bois singularities.

If $\pi\colon\widehat{X}\to X$ is a log resolution with reduced exceptional divisor $E$, and we define $\underline{\Omega}^{\bullet}_{X,\Sigma}=R\pi_{*}\Omega^{\bullet}_{\widehat{X}}(\log E)(-E)$, where $\Sigma$ is the singular locus of $X$, then there is the distinguished triangle of relative cohomology in the filtered derived category

Thus, in the derived category, there is a corresponding distinguished triangle

In the proper case, there is the following fundamental result of Du Bois, based on Deligne’s construction of the mixed Hodge structure on $Y$:

The result above allows us to define Hodge numbers in the singular case:

Let $Y$ be a compact complex algebraic variety. We define the Hodge-Du Bois numbers

for $0\leq p,q\leq n$. In particular, if $Y$ is smooth, $\underline{h}^{p,q}(Y)=h^{p,q}(Y)$. In general, by Theorem 3.2,

where $h^{p,r}_{p+q}(Y):=\dim\operatorname{Gr}^{p}_{F}\operatorname{Gr}^{W}_{p+r}H^{p+q}(Y)$ are the Hodge-Deligne numbers associated to the mixed Hodge structure on $Y$.

As the example of nodal curves shows (Remark 1.11) the Hodge-Du Bois diamond will not satisfy either of the Hodge symmetries: $\underline{h}^{p,q}=\underline{h}^{q,p}$ or $\underline{h}^{p,q}=\underline{h}^{n-p,n-q}$. Nonetheless, some vestige of these symmetries remains (in the form of inequalities) as those given by Lemma 3.23 below.

Another key consequence for us is the following:

This is a consequence of the following general fact: If $(\mathcal{C}^{\bullet},d,\mathcal{F}^{\bullet}\mathcal{C}^{\bullet})$ is a filtered complex, then the associated spectral sequence degenerates at the $E_{1}$ page $\iff$ the differential $d$ is strict with respect to the filtration $\mathcal{F}^{\bullet}\mathcal{C}^{\bullet}$. Then, assuming $E_{1}$-degeneration, an easy argument shows that $H^{i}(\mathcal{C}^{\bullet},d)\to H^{i}(\mathcal{C}^{\bullet}/\mathcal{F}^{k+1}\mathcal{C}^{\bullet},d)$ is surjective for every $i$ and $k$. Since the induced filtration on $\mathcal{C}^{\bullet}/\mathcal{F}^{k+1}\mathcal{C}^{\bullet}$ is also strict with respect to $d$, the corresponding spectral sequence for the complex $\mathcal{C}^{\bullet}/\mathcal{F}^{k+1}\mathcal{C}^{\bullet}$ also degenerates at $E_{1}$. ∎

2. Du Bois and higher Du Bois singularities

[PS08, p.175], where $\sigma^{\bullet}$ is the trivial or naive filtration on $\Omega_{X}^{\bullet}$. Following [MOPW23] and [JKSY22a], one defines:

Let $X$ be a complex algebraic variety. Then $X$ is $k$-Du Bois if the natural maps

are quasi-isomorphisms for $0\leq p\leq k$. Note that the case $k=0$ coincides with the usual definition of Du Bois singularities [PS08, Def. 7.34].

Let $X$ be a complex algebraic variety with lci singularities. Assume $X$ is $k$-Du Bois with $k\geq 1$. Then $X$ is normal and regular in codimension $2k$, i.e. $\operatorname{codim}\Sigma\geq 2k+1$.

The normality claim is [MP22a, Cor. 5.6]. For hypersurface singularities, various dimension bounds (covering the claim of the theorem) were obtained in both [MOPW23] and [JKSY22a]. The general lci case follows from [MP22a, Thm. F] (numerical characterization of $k$-Du Bois) and [MP22a, Cor. 3.40] (bounds on $\dim\Sigma$ in terms of the relevant numerical invariant). ∎

Suppose that $X$ has lci $k$-Du Bois singularities and that $f\colon\mathcal{X}\to S$ is a flat morphism, where $S$ is arbitrary, with $X=X_{s}=f^{-1}(s)$ for some $s\in S$. Then possibly after replacing $S$ by a neighborhood of $s$, the sheaf of relative differentials $\Omega^{p}_{\mathcal{X}/S}$ is flat over $S$ for all $p\leq k$.

This is immediate from Theorem 2.5 and Theorem 3.8. ∎

(i) Theorem 3.8 and Theorem 2.2 imply the following previously known results:

If $X$ has hypersurface $k$-Du Bois singularities, then $\Omega_{X}^{p}$ is torsion free for $0\leq p\leq k$ ([JKSY22a, Prop. 2.2]).

If $X$ has lci $k$-Du Bois singularities, then $\Omega_{X}^{p}$ is reflexive for $0\leq p\leq k$ ([MP22a, Cor. 5.6]).

(ii) For $p=1$ and lci singularities, the situation is well understood by a result of Kunz [Kun86, Prop. 9.7]: $\Omega_{X}^{1}$ satisfies Serre’s condition $S_{a}$ $\iff$ $X$ satisfies $R_{a}$ (i.e. is regular in codimension $a$).

For other examples of $k$-Du Bois and $k$-rational singularities, one can consult [KL20] (cf. Theorem 5.3 and §6.1).

3. Higher rational singularities

The standard definition of a rational singularity involves the choice of a resolution (e.g. [KM98, Def. 5.8]). As we will explain below, it is possible to give an equivalent definition for rational singularities without reference to resolutions, but using instead the dualizing complex. In addition to being more intrinsic, it generalizes to higher rational singularities, and it factors naturally through the higher Du Bois condition.

For all $p$, there exists a natural sequence of maps in the derived category

Since $\pi$ is proper, Grothendieck duality gives

as claimed. The map $\psi^{p}$ is easily seen to be independent of the choice of a resolution, by the usual factorization arguments. ∎

The variety $X$ has $k$-rational singularities if the maps

are quasi-isomorphisms for all $0\leq p\leq k$.

The following lemma connects our definition to more standard ones:

Suppose that $\dim\Sigma\leq d$. Then, for all $p<n-d$,

By Theorem 3.1, there is the distinguished triangle of relative cohomology

Since $n-p>d$, $\underline{\Omega}^{n-p}_{\Sigma}=0$ and hence $\underline{\Omega}^{n-p}_{X}\cong\underline{\Omega}^{n-p}_{X,\Sigma}=R\pi_{*}\Omega^{n-p}_{\widehat{X}}(\log E)(-E)$. Applying Grothendieck duality, it follows as in [MOPW23, §2.2] that

The final statement is clear since $n-d\geq 2k+1$ $\implies$ $n-p\geq n-k\geq d+k+1>d$. ∎

$X$ is -rational $\iff$ $X$ has rational singularities.

Since $X$ is reduced, $\dim\Sigma\leq n-1$. Thus $X$ is -rational $\iff$ the natural map $\mathcal{O}_{X}\to R\pi_{*}\mathcal{O}_{\widehat{X}}$ is an isomorphism $\iff$ $X$ has rational singularities in the usual sense. ∎

is a quasi-isomorphism. This formulation occurs for instance in [KK20], and it is equivalent to that given by [KM98, Thm. 5.10(3)] (note that the quasi-isomorphism $\omega_{X}^{GR}\cong\omega_{X}^{\bullet}$ forces $X$ to be Cohen-Macaulay).

In [FL22a, §3], we defined $k$-rational singularities for an isolated singularity by the condition that $R\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)\cong\Omega_{X}^{p}$. This is equivalent to Definition 3.13 (under a mild assumption):

In the above notation, suppose that $\operatorname{codim}\Sigma\geq 2k+1$.

$X$ is $k$-rational $\iff$ the natural map $\Omega_{X}^{p}\to R\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)$ is an isomorphism for all $p\leq k$.

$X$ is $k$-rational $\iff$ $X$ is $(k-1)$-rational and $\Omega_{X}^{k}\to R\pi_{*}\Omega^{k}_{\widehat{X}}(\log E)$ is an isomorphism. ∎

In the lci case, the assumption on $R^{0}\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)$ is automatic:

Suppose that $X$ has lci singularities and $\operatorname{codim}\Sigma\geq 2k+1$. Then, for all $p\leq k$, $\psi^{p}\circ\phi^{p}\colon\Omega_{X}^{p}\to R^{0}\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)$ is an isomorphism. Hence $X$ is $k$-rational $\iff$ for all $p\leq k$ and all $q>0$, $R^{q}\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)=0$.

This follows easily from Theorem 2.2 and Corollary 2.3, as $R^{0}\pi_{*}\Omega^{p}_{\widehat{X}}(\log E)$ is torsion free for all $p$. ∎

Suppose that $X$ has an isolated singularity $x$, so that $\Sigma=\{x\}$. Then by Theorem 3.1, there is the distinguished triangle of relative cohomology

4. k𝑘k-rational vs. k𝑘k-Du Bois singularities

Steenbrink [Ste83] proved that, if $X$ is an isolated rational singularity, then $X$ is Du Bois, and Kovács [Kov99] generalized this, showing that that any rational singularity is Du Bois. Saito gave a different proof in [Sai00, Thm. 5.4]. The method of [Ste83] generalizes to prove that isolated $k$-rational singularities are $k$-Du Bois (see [FL22b, Theorem 3.2]), and the method of [Kov99] can be generalized to handle both isolated and lci singularities. Using the ideas of [Kov99], we shall show the following in Section 5:

Suppose either that $X$ has isolated singularities or $X$ is lci. If $X$ is $k$-rational, then $X$ is $k$-Du Bois.

If $Y$ is a compact complex algebraic variety of dimension $n$ with lci $k$-rational singularities, then, for $0\leq p\leq k$,

In particular, taking $p+q=n$ gives $\underline{h}^{p,n-p}=\underline{h}^{n-p,p}$ for $p\leq k$.

Using Theorem 3.2, we get the following identifications

where the middle isomorphism is given by the quasi-isomorphism $\psi^{p}$ (for $p\leq k$). ∎

It remains to discuss the Hodge symmetry $h^{p,q}=h^{q,p}$, which is induced by complex conjugation in the smooth case. We recall that the cohomology of a compact singular algebraic variety $Y$ carries a mixed Hodge structure $(H^{*}(Y),F^{\bullet},W_{\bullet})$. The Hodge-Deligne numbers $h^{p,r}_{i}=\operatorname{Gr}^{p}_{F}\operatorname{Gr}_{p+r}^{W}H^{i}(Y)$ satisfy the symmetry given by conjugation: $h^{p,r}_{i}=h^{r,p}_{i}$ (as $\operatorname{Gr}_{p+r}^{W}H^{i}(Y)$ is a pure Hodge structure). Since $Y$ is compact, the weights on $H^{i}(Y)$ are at most $i$, and in fact between $2i-2n$ and $i$ if $i\geq n$. It follows that, for $i\leq n$, $\underline{h}^{p,i-p}=\sum_{r=0}^{i-p}h^{p,r}_{i}$. However, the Hodge-Du Bois numbers do not satisfy the same kind of symmetry as they reflect only the Hodge filtration $F^{\bullet}$. In fact, we note the following:

Let $Y$ be a compact complex algebraic variety of dimension $n$. For $0\leq p\leq i\leq n$,

Furthermore, equality holds above for all $p\leq k$ $\iff$ $\underline{h}^{p,i-p}=\underline{h}^{i-p,p}$ for all $p\leq k$ $\iff$ $\operatorname{Gr}_{F}^{p}W_{i-1}H^{i}(Y)=0$ for all $p\leq k$.

Clearly $\sum_{a=0}^{p}\underline{h}^{i-a,a}=\sum_{a=0}^{p}\underline{h}^{a,i-a}$ for all $p\leq k$ $\iff$ $\underline{h}^{p,i-p}=\underline{h}^{i-p,p}$ for all $p\leq k$.

Note that, if $i-p\leq s\leq i$ and $r+s\leq i$, then $r\leq i-s\leq p$, so the second sum is greater that the first, giving the inequality. For a given $p$, equality holds $\iff$ $h^{r,s}_{i}=0$ for $r\leq p$ and $s\leq i-p-1$. Moreover, $h^{r,s}_{i}=0$ for $r\leq p$ and $s\leq i-p-1$ for some $p\leq k$ $\iff$ $h^{r,s}_{i}=0$ for $r\leq k$, $r+s\leq i-1$. This is equivalent to: $\operatorname{Gr}_{F}^{p}W_{i-1}H^{i}(Y)=0$ for $p\leq k$. ∎

Recall that, for any resolution $\pi:\hat{Y}\to Y$,

(e.g. [PS08, Cor. 5.42]). Using this, we obtain:

If $Y$ is a compact complex algebraic variety of dimension $n$ with either isolated or lci $k$-rational singularities, then

In view of the discussion above, we define the discrepancy

By Lemma 3.23, the equality $\underline{h}^{p,i-p}=\underline{h}^{i-p,p}$ holds for $p\leq k$ $\iff$ $\delta_{i}^{p}=0$ for $p\leq k$. The map $\psi^{p}$ occurring in the definition of higher rationality (Definition 3.13 and Lemma 3.12) factors through the resolution $\pi:\hat{Y}\to Y$, and at the level of cohomology corresponds to the $\operatorname{Gr}^{p}_{F}$ piece (see also Corollary 3.22) of the natural map

where all spaces are endowed with the natural Hodge structures. On the graded piece $\operatorname{Gr}_{F}^{p}H^{i}(Y)=H^{i-p}(Y;\underline{\Omega}^{p}_{Y})$, $\operatorname{Gr}_{F}^{p}\Psi^{i}$ is the map $\psi^{p}\colon H^{i-p}(Y;\underline{\Omega}^{p}_{Y})\to H^{n-i+p}(Y;\underline{\Omega}^{n-p}_{Y})^{\vee}$, which is an isomorphism if $Y$ is both $k$-rational and $k$-Du Bois, and in particular if $Y$ is $k$-rational and has either isolated singularities or lci singularities. By the strictness of morphisms of Hodge filtrations, if $\operatorname{Gr}_{F}^{p}\Psi^{i}$ is an isomorphism then $\pi^{*}$ is injective on $\operatorname{Gr}_{F}^{p}$ and hence $\delta^{p}_{i}=0$. Thus, $k$-rationality implies $\delta^{p}_{i}=0$ for $p\leq k$ and all $i$, which in turn means $\underline{h}^{p,i-p}=\underline{h}^{i-p,i}$ in this range. ∎

As noted by one of the referees, in the above proof as well as in the proof of Lemma 3.12, the main point is to apply relative duality to a resolution of singularities morphism. Thus the proof does not take into account the full information of a hyperresolution.

Proof of Theorem 1.2 and Corollary 1.5

We turn now to the global setting of a deformation of a compact analytic space or proper scheme $Y$, and to the question of the local freeness of $\Omega^{p}_{\mathcal{Y}/S}$.

where the top row is exact since tensor product is right exact and the second is exact since $\mathcal{F}^{\bullet}$ is $A$-flat.

With notation as above, suppose that $Y$ is $k$-Du Bois. Then:

By the $k$-Du Bois condition, the natural map $\Omega^{\bullet}_{Y}/\sigma^{\geq k+1}\Omega^{\bullet}_{Y}\to\underline{\Omega}^{\bullet}_{Y}/F^{k+1}\underline{\Omega}^{\bullet}_{Y}$ is a quasi-isomorphism of filtered complexes. Thus there are isomorphisms

To prove (i), arguing as in [DBJ74], there is a commutative diagram

By Theorem 4.1 and Lemma 4.2, for every $i$,

Thus all of the inequalities must have been equalities, and in particular $H^{q}(\mathcal{Y};\Omega^{p}_{\mathcal{Y}/\operatorname{Spec}A})\to H^{q}(\overline{\mathcal{Y}};\Omega^{p}_{\overline{\mathcal{Y}}/\operatorname{Spec}\overline{A}})$ is surjective for every $q$ and for every quotient $A\to\overline{A}$. This completes the proof, by the remarks in the first paragraph of the proof. ∎

It seems likely that there is a more general result, in the non-isolated case, assuming sufficiently strong conditions on every subvariety of $Y_{\text{\rm{sing}}}$.

We turn now to a proof of Corollary 1.5. First, we define canonical Calabi-Yau varieties following [FL22a, Definition 6.1]:

is surjective. Arguing as in [Kaw92, Lemma 3], we claim that $\omega_{\mathcal{Y}/\operatorname{Spec}A}\cong\mathcal{O}_{\mathcal{Y}}$. To see this, note that, as $Y$ is $1$-Du Bois it is -Du Bois, and hence by Theorem 1.1 $R^{i}\pi_{*}\mathcal{O}_{\mathcal{Y}}=H^{i}(\mathcal{Y};\mathcal{O}_{\mathcal{Y}})$ is a free $A$-module for all $i$, compatible with base change. In particular, $R^{n}\pi_{*}\mathcal{O}_{\mathcal{Y}}$ is a free rank one $A$-module. An application of relative duality then shows that $(R^{i}\pi_{*}\mathcal{O}_{\mathcal{Y}})^{\vee}\cong R^{n-i}\omega_{\mathcal{Y}/\operatorname{Spec}A}$, and hence $R^{n-i}\omega_{\mathcal{Y}/\operatorname{Spec}A}$ is a free $A$-module for all $i$. Thus $R^{0}\pi_{*}\omega_{\mathcal{Y}/\operatorname{Spec}A}$ is a free rank one $A$-module and the natural map $R^{0}\pi_{*}\omega_{\mathcal{Y}/\operatorname{Spec}A}\otimes_{A}\mathcal{O}_{\mathcal{Y}}\to\omega_{\mathcal{Y}/\operatorname{Spec}A}$ is an isomorphism. Hence $\omega_{\mathcal{Y}/\operatorname{Spec}A}\cong\mathcal{O}_{\mathcal{Y}}$.