Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

Overview of the paper

This paper is about the following “extension problem” for holomorphic differential forms on complex spaces. Let $X$ be a reduced complex space, and let $r\colon\widetilde{X}\textrightarrow X$ be a resolution of singularities. Which holomorphic $p$-forms on the regular locus $X_{\operatorname{reg}}$ extend to holomorphic $p$-forms on the complex manifold $\widetilde{X}$? Standard facts about resolution of singularities imply that the answer is independent of the choice of resolution. (If the exceptional locus of $r$ is a normal crossing divisor $E$, one can also ask for an extension with at worst logarithmic poles along $E$.)

The best existing result concerning this problem is due to Greb, Kebekus, Kovács, and Peternell [GKKP11, Thm. 1.4]. They show that if $X$ underlies a normal algebraic variety with Kawamata log terminal (=klt) singularities, then all $p$-forms on $X_{\operatorname{reg}}$ extend to $\widetilde{X}$, for every $0≤p≤\dim X$. Their theorem has many applications, including hyperbolicity of moduli, the structure of minimal varieties with trivial canonical class, the nonabelian Hodge correspondence for singular spaces, and quasi-étale uniformisation. Section 1.7 recalls some of these in more detail and gives references.

In this paper, we use the Decomposition Theorem and Saito’s theory of mixed Hodge modules to solve the extension problem in general. Our main result is a simple necessary and sufficient condition for a holomorphic $p$-form on $X_{\operatorname{reg}}$ to extend to a holomorphic (or logarithmic) $p$-form on $\widetilde{X}$. One surprising consequence is that the extension problem for forms of a given degree also controls what happens for forms of smaller degrees. Another consequence is that if $X$ is a complex space with rational singularities, then all $p$-forms on $X_{\operatorname{reg}}$ extend to $\widetilde{X}$, for every $0≤p≤\dim X$. This result is a crucial step in the recent work of Bakker and Lehn [BL18] on the global moduli theory of symplectic varieties.

2. Main result

Let $X$ be a reduced complex space of pure dimension $n$. It is well-known that a holomorphic $n$-form $α∈H⁰(X_{\operatorname{reg}},Ω^{n}_{X})$ extends to a holomorphic $n$-form on any resolution of singularities of $X$ if and only if $αΛ\overline{α}$ is locally integrable on $X$. Griffiths [Gri76, §IIa] gave a similar criterion for extension of $p$-forms in terms of integrals over $p$-dimensional analytic cycles in $X$, but his condition is not easy to verify in practice. Our first main result is the following intrinsic description of those holomorphic forms on $X_{\operatorname{reg}}$ that extend holomorphically to one (and hence any) resolution of singularities.

Let $X$ be a reduced complex space of pure dimension $n$, and $r\colon\widetilde{X}\textrightarrow X$ a resolution of singularities. A holomorphic $p$-form $α∈H⁰(X_{\operatorname{reg}},Ω^{p}_{X})$ extends to a holomorphic $p$-form on $\widetilde{X}$ if, and only if, for every open subset $U⊆X$, and for every pair of Kähler differentials $β∈H⁰(U,Ω^{n-p}_{X})$ and $γ∈H⁰(U,Ω^{n-p-1}_{X})$, the holomorphic $n$-forms $αΛβ$ and $dαΛγ$ on $U_{\operatorname{reg}}$ extend to holomorphic $n$-forms on $r^{-1}(U)$.

Our proof of this result is based on the Decomposition Theorem for Hodge modules. We think that it would also be very interesting to have an analytic proof, in terms of $L²$-Hodge theory for the $\bar{∂}$-operator. The following analogue of Theorem 1.1 for forms with logarithmic poles needs some additional results about mixed Hodge modules. Recall that a resolution of singularities $r\colon\widetilde{X}\textrightarrow X$ of a complex space is called a (strong) log resolution if the $r$-exceptional set is a divisor with (simple) normal crossings on $\widetilde{X}$.

Let $X$ be a reduced complex space of pure dimension $n$, and $r\colon\widetilde{X}\textrightarrow X$ a log resolution of singularities with exceptional divisor $E⊆X$. A holomorphic $p$-form $α∈H⁰(X_{\operatorname{reg}},Ω^{p}_{X})$ extends to a holomorphic section of the bundle $Ω_{\widetilde{X}}^{p}(\log E)$ on $\widetilde{X}$ if, and only if, for every open subset $U⊆X$, and for every pair of Kähler differentials $β∈H⁰(U,Ω^{n-p}_{X})$ and $γ∈H⁰(U,Ω^{n-p-1}_{X})$, the holomorphic $n$-forms $αΛβ$ and $dαΛγ$ on $U_{\operatorname{reg}}$ extend to holomorphic sections of the bundle $Ω_{\widetilde{X}}^{n}(\log E)$ on $r^{-1}(U)$.

3. Consequences

The extension problem for holomorphic (or logarithmic) forms on a complex space $X$ is of course closely related to the singularities of $X$. Since there might not be any global $p$-forms on $X_{\operatorname{reg}}$, the effect of the singularities is better captured by the following local version of the problem. Given a resolution of singularities $r\colon\widetilde{X}\textrightarrow X$ of a reduced complex space $X$, and an arbitrary open subset $U⊆X$, which holomorphic $p$-forms on $U_{\operatorname{reg}}$ extend to holomorphic $p$-forms on $r^{-1}(U)$? If $j\colon X_{\operatorname{reg}}↪X$ denotes the embedding of the regular locus, this amounts to asking for a description of the subsheaf $r_{*}Ω^{p}_{\widetilde{X}}↪j_{*}Ω^{p}_{X_{\operatorname{reg}}}$. This subsheaf is $𝒪_{X}$-coherent (by Grauert’s theorem) and independent of the choice of resolution (because any two resolutions are dominated by a common third). If the exceptional locus of $r$ is a normal crossing divisor $E$, one can also ask for a description of the subsheaf $r_{*}Ω^{p}_{\widetilde{X}}(\log E)↪j_{*}Ω^{p}_{X_{\operatorname{reg}}}$, which has similar properties.

When $X$ is reduced and irreducible, it is easy to see that $r_{*}𝒪_{\widetilde{X}}↪j_{*}𝒪_{X_{\operatorname{reg}}}$ is an isomorphism if and only if $\dim X_{\operatorname{sing}}≤\dim X-2$. (Use the normalisation of $X$.)

One consequence of Theorem 1.1 is that the extension problem for holomorphic forms of a given degree also controls what happens for all forms of smaller degree.

Let $X$ be a reduced and irreducible complex space. Let $r\colon\widetilde{X}\textrightarrow X$ be any resolution of singularities, and $j\colon X_{\operatorname{reg}}↪X$ the inclusion of the regular locus. If the morphism $r_{*}Ω^{k}_{\widetilde{X}}↪j_{*}Ω^{k}_{X_{\operatorname{reg}}}$ is an isomorphism for some $0≤k≤\dim X$, then $\dim X_{\operatorname{sing}}≤\dim X-2$, and $r_{*}Ω^{p}_{\widetilde{X}}↪j_{*}Ω^{p}_{X_{\operatorname{reg}}}$ is an isomorphism for every $0≤p≤k$.

An outline of the proof can be found in Section 2 below. The key idea is to use the Decomposition Theorem [BBD82, Sai88], in order to relate the coherent $𝒪_{X}$-module $r_{*}Ω^{p}_{\widetilde{X}}$ to the intersection complex of $X$, viewed as a polarisable Hodge module. In Appendix B, we look at the example of cones over smooth projective varieties; it gives a hint that the extension problem for all $p$-forms should be governed by what happens for $n$-forms. When $X$ is normal, an equivalent formulation of Theorem 1.4 is that, if the coherent $𝒪_{X}$-module $r_{*}Ω^{k}_{\widetilde{X}}$ is reflexive for some $k≤\dim X$, then $r_{*}Ω^{p}_{\widetilde{X}}$ is reflexive for every $p≤k$.

One can easily generalise Theorem 1.4 to arbitrary reduced complex spaces. The precise (but somewhat cumbersome) statement is that if the morphism $r_{*}Ω^{k}_{\widetilde{X}}↪j_{*}Ω^{k}_{X_{\operatorname{reg}}}$ is an isomorphism for some $k≥0$, and if $Z⊆X$ denotes the union of all the irreducible components of $X$ of dimension $≥k$, then $\dim Z_{\operatorname{sing}}≤k-2$, and the restriction to $Z$ of the morphism $r_{*}Ω^{p}_{\widetilde{X}}↪j_{*}Ω^{p}_{X_{\operatorname{reg}}}$ is an isomorphism for every $0≤p≤k$. The reason is that the irreducible components of $X$ are separated in any resolution of singularities, and so one can simply apply Theorem 1.4 one component at a time.

We also establish a version of Theorem 1.4 with log poles, by adapting the techniques in the proof to a certain class of mixed Hodge modules.

Let $X$ be a reduced and irreducible complex space. Let $r\colon\widetilde{X}\textrightarrow X$ be a log resolution with exceptional divisor $E⊆\widetilde{X}$, and $j\colon X_{\operatorname{reg}}↪X$ the inclusion of the regular locus. If the morphism $r_{*}Ω^{k}_{\widetilde{X}}(\log E)↪j_{*}Ω^{k}_{X_{\operatorname{reg}}}$ is an isomorphism for some $0≤k≤\dim X$, then $\dim X_{\operatorname{sing}}≤\dim X-2$, and $r_{*}Ω^{p}_{\widetilde{X}}(\log E)↪j_{*}Ω^{p}_{X_{\operatorname{reg}}}$ is an isomorphism for every $0≤p≤k$.

By a result of Kovács, Schwede, and Smith [KSS10, Thm. 1], a complex algebraic variety $X$ that is normal and Cohen-Macaulay has Du Bois singularities if and only if $r_{*}ω_{\widetilde{X}}(E)$ is a reflexive $𝒪_{X}$-module for some log resolution $r\colon\widetilde{X}\textrightarrow X$. We think that it would be interesting to know the precise relationship between Du Bois singularities and the extension problem for logarithmic $n$-forms. The tools we develop for the proof of Theorem 1.5 also lead to a slightly better answer in the case of holomorphic forms of degree $\dim X-1$.

Let $X$ be a reduced and irreducible complex space. Let $r\colon\widetilde{X}\textrightarrow X$ be a log resolution with exceptional divisor $E⊆\widetilde{X}$, and $j\colon X_{\operatorname{reg}}↪X$ the inclusion of the regular locus. If the natural morphism $r_{*}Ω^{n}_{\widetilde{X}}↪j_{*}Ω^{n}_{X_{\operatorname{reg}}}$ is an isomorphism, where $n=\dim X$, then the two morphisms

4. Rational and weakly rational singularities

An important class of singular spaces where Theorem 1.4 applies is normal complex spaces with rational singularities. Recall that $X$ has rational singularities if the following equivalent conditions hold. We refer to [KM98, §5.1] for details.

$X$ is normal, and if $r\colon\widetilde{X}\textrightarrow X$ is any resolution of singularities, then $Rⁱr_{*}𝒪_{\widetilde{X}}=0$ for every $i≥1$.

$X$ is Cohen-Macaulay and $ω_{X}^{\operatorname{GR}}=ω_{X}$.

$X$ is Cohen-Macaulay and $ω_{X}^{\operatorname{GR}}$ is reflexive.

Here $ω_{X}^{\operatorname{GR}}=r_{*}ω_{\widetilde{X}}$ is sometimes called the Grauert-Riemenschneider sheaf, because it appears in the Grauert-Riemenschneider vanishing theorem. In view of Condition (1.7.3), we say that a normal space $X$ has weakly rational singularities if the Grauert-Riemenschneider sheaf $ω_{X}^{\operatorname{GR}}$ is reflexive. With this notation, Theorem 1.4 has the following immediate corollary.

Let $X$ be a normal complex space with weakly rational singularities, and let $r\colon\widetilde{X}\textrightarrow X$ be a resolution of singularities. Then every holomorphic form defined on $X_{\operatorname{reg}}$ extends uniquely to a holomorphic form on $\widetilde{X}$. ∎

Rational singularities are weakly rational by definition. In particular, recall from [KM98, Thm. 5.22 and references there] that klt spaces have rational (and hence weakly rational) singularities. For algebraic klt varieties, the extension result was shown previously in [GKKP11, Thm. 1.4].

As we will see in Section 6.1, having weakly rational singularities turns out to be equivalent to the collection of inequalities

One can also describe the class of weakly rational singularities in more analytic terms: a normal complex space $X$ of dimension $n$ has weakly rational singularities if and only if, for every open subset $U⊆X$ and every holomorphic $n$-form $ω∈H⁰(U_{\operatorname{reg}},Ω^{n}_{U_{\operatorname{reg}}})$, the $(n,n)$-form $ωΛ\overline{ω}$ on $U_{\operatorname{reg}}$ is locally integrable on all of $U$. Appendix A discusses examples and establishes elementary properties of this class of singularities.

5. Local vanishing conjecture

The methods developed in this paper also settle the “local vanishing conjecture” proposed by Mustaţă, Olano, and Popa [MOP20, Conj. A]. The original conjecture contained the assumption that $X$ is a normal algebraic variety with rational singularities. In fact, the weaker assumption $R^{\dim X-1}r_{*}𝒪_{\widetilde{X}}=0$ is sufficient.

Let $X$ be a reduced and irreducible complex space of dimension $n$. Let $r\colon\widetilde{X}\textrightarrow X$ be a log resolution, with exceptional divisor $E⊆\widetilde{X}$. If $R^{n-1}r_{*}𝒪_{\widetilde{X}}=0$, then $R^{n-1}r_{*}Ω¹_{\widetilde{X}}(\log E)=0$.

As shown in [MOP20], this result has interesting consequences for the Hodge filtration on the complement of a hypersurface with at worst rational singularities.

6. Functorial pull-back

One can interpret Theorem 1.4 as saying that any differential form σ∈H⁰\bigl{(}X_{\operatorname{reg}},\,Ω¹_{X_{\operatorname{reg}}}\bigr{)}=H⁰\bigl{(}X,\,Ω^{}_{X}\bigr{)} induces a pull-back form \widetilde{σ}∈H⁰\bigl{(}\widetilde{X},\,Ω¹_{\widetilde{X}}\bigr{)}. More generally, we show that pull-back exists for reflexive differentials and arbitrary morphisms between varieties with rational singularities. The paper [Keb13b] discusses these matters in detail.

Let $f\colon X\textrightarrow Y$ be any morphism between normal complex spaces with rational singularities. Write Ω^{[p]}_{X}:=\bigl{(}Ω^{p}_{X}\bigr{)}^{**}, ditto for $Ω^{[p]}_{Y}$. Then there exists a pull-back morphism

uniquely determined by natural universal properties.

We refer to Theorem 14.1 and Section 14 for a precise formulation of the “natural universal properties” mentioned in Theorem 1.11. In essence, it is required that the pull-back morphisms agree with the pull-back of Kähler differentials wherever this makes sense, and that they satisfy the composition law.

Theorem 1.11 applies to morphisms $X\textrightarrow Y$ whose image is entirely contained in the singular locus of $Y$. Taking the inclusion of the singular set for a morphism, Theorem 1.11 implies that every differential form on $Y_{\operatorname{reg}}$ induces a differential form on every stratum on the singularity stratification.

One can also reformulate Theorem 1.11 in terms of $\operatorname{h}$-differentials; these are obtained as the sheafification of Kähler differentials with respect to the $\operatorname{h}$-topology on the category of complex spaces, as introduced by Voevodsky. We refer the reader to [HJ14] and to the survey [Hub16] for a gentle introduction to these matters. Using the description of $\operatorname{h}$-differentials found in [HJ14, Thm. 1], the following is an immediate consequence of Theorem 1.11.

Let $X$ be a normal complex space with rational singularities. Write Ω^{[p]}_{X}:=\bigl{(}Ω^{p}_{X}\bigr{)}^{**}. Then, $\operatorname{h}$-differentials and reflexive differentials agree: $Ω^{p}_{\operatorname{h}}(X)=Ω^{[p]}_{X}(X)$. ∎

The sheaf $Ω^{p}_{\operatorname{h}}$ of $\operatorname{h}$-differentials appears under a different name in the work of Barlet, [Bar18], who describes it in analytic terms (“integral dependence equations for differential forms”) as a subsheaf of $Ω^{[p]}_{X}$ and relates it to the normalised Nash transform.

7. Applications

The main results of this paper allow to study (sheaves of) reflexive differential forms on singular spaces, by pulling them back to a resolution of singularities. At times, this allows to prove results of Hodge-theoretic flavour in settings where classic Hodge-theory is not readily available. We give two immediate application of Theorem 1.4, which can be proven in just a few lines, following [GKKP11, Sect. 6 and 7] verbatim.

Let $X$ be a normal complex projective variety. If $ω_{X}^{\operatorname{GR}}$ is reflexive, then any holomorphic differential form on $X_{\operatorname{reg}}$ is closed. If $𝒜⊆Ω^{[p]}_{X}$ is an invertible subsheaf, then $κ(𝒜)≤p$. ∎

Let $X$ be a normal complex space where $ω_{X}^{\operatorname{GR}}$ is reflexive. If the tangent sheaf $𝒯_{X}$ is locally free, then $X$ is smooth. ∎

To illustrate the range of applicability, we mention some recent (and much more substantial) results that rely on the previously known extension theorem for klt spaces by Greb, Kebekus, Kovács, and Peternell.

The standard conjectures of minimal model theory predict that the minimal model of any projective manifold $X$ of Kodaira dimension $κ(X)=0$ is a singular space with vanishing first Chern class. A series of papers [GKP16a, Dru18, GGK19, HP19] extended the classic Beauville-Bogomolov Decomposition Theorem to the singular setting. A recent paper [KLSV18] studies degenerations of hyper-Kähler manifolds, using the minimal model program to reduce any degeneration to (singular) “Kulikov type form”.

A series of papers [GKPT19b, GKPT19c, GKPT19a] extends the classic non-abelian Hodge correspondence from Kähler manifolds to singular spaces. For klt spaces, this relates representations of the fundamental group with (singular) Higgs sheaves and yields new results on quasi-étale uniformisation, [GKP16b, LT18, GKT18].

The extension results are used analysis in the study of (singular) Kähler-Einstein metrics, [BG14, LT19]

The extension result is used in holomorphic dynamics and foliations, for the classification of foliations [AD14, AD13], but also in the study of compactifications of Drinfeld half-spaces over a finite field, [Lan19].

Our generalisation of the extension theorem to (possibly non-algebraic) complex spaces with rational singularities (in Corollary 1.8 above) is used in a crucial way in the recent work of Bakker and Lehn [BL18] on global moduli for symplectic varieties, where a “symplectic variety” is a normal Kähler space $X$ with a nondegenerate holomorphic $2$-form on $X_{\operatorname{reg}}$ that extends holomorphically to any resolution of singularities.

8. Earlier results

As mentioned above, Theorem 1.4 was already known for spaces with Kawamata log terminal (=klt) singularities, where $r_{*}ω_{\widetilde{X}}$ is reflexive by definition [GKK10, GKKP11]. If one is only interested in $p$-forms of small degree (compared to $\dim X$), there are earlier results of Steenbrink-van Straten [vSS85] and Flenner [Fle88]. In the special case where $p=1$, Graf-Kovács relate the extension problem to the notion of Du Bois singularities [GK14]. For morphisms between varieties with klt singularities, the existence of a pull-back functor was shown in [Keb13b].

We refer the reader to the paper [GKKP11] or to the survey [Keb13a, §4] for a more detailed introduction, and for remarks on the history of the problem. The book [Kol13, §8.5] puts the results into perspective.

9. Acknowledgements

Both authors would like to thank Mark de Cataldo, Bradley Drew, Philippe Eyssidieux, Annette Huber-Klawitter, Florian Ivorra, Mihai Păun, and Mihnea Popa for helpful discussions, and the Freiburg Institute for Advanced Studies for support and for its stimulating atmosphere. We thank Fabio Bernasconi for showing us a counterexample to the extension theorem in positive characteristic, and two anonymous referees for careful reading and for suggestions that help to improve the exposition of this paper.

Stefan Kebekus gratefully acknowledges the support through a senior fellowship of the Freiburg Institute of Advanced Studies (FRIAS).

During the preparation of this paper, Christian Schnell was supported by a Mercator Fellowship from the Deutsche Forschungsgemeinschaft (DFG), by research grant DMS-1404947 from the National Science Foundation (NSF), and by the Kavli Institute for the Physics and Mathematics of the Universe (IPMU) through the World Premier International Research Center Initiative (WPI initiative), MEXT, Japan.

Techniques and main ideas

In this section, we sketch some of the ideas that go into the proof of Theorem 1.4. The one-line summary is that it is a consequence of the Decomposition Theorem [BBD82, Sai90]. Appendix B contains a short section on cones over projective manifolds that illustrates the extension problem in a particularly transparent case and explains why one might even expect a result such as Theorem 1.4 to hold true.

We actually give two proofs for Theorem 1.4. The first proof (in Section 11) relies on Theorem 1.1, which characterises those holomorphic forms on the regular locus of a complex space that extend holomorphically to any resolution of singularities. This proof is very short and, shows clearly why the extension problem for $k$-forms also controls the extension problem for $(k-1)$-forms (and hence for all forms of smaller degrees).

2. Second proof of Theorem 1.4

To illustrate the main ideas and techniques used in this paper, we are now going to describe a second, more systematic proof for Theorem 1.4. It is longer, and covers only the case where $k=n$, but it has the advantage of producing a stronger result that has other applications (such as the proof of the local vanishing conjecture). We hope that the description below will make it clear why the Decomposition Theorem is useful in studying the extension problem for holomorphic forms.

Setup

We fix a reduced and irreducible complex space $X$ of dimension $n$, and a resolution of singularities $r\colon\widetilde{X}\textrightarrow X$. We denote by $j\colon X_{\operatorname{reg}}↪X$ the embedding of the set of regular points, and assume that the natural morphism $r_{*}Ω_{\widetilde{X}}^{n}↪j_{*}Ω_{X_{\operatorname{reg}}}^{n}$ is an isomorphism. This means concretely that, locally on $X$, holomorphic $n$-forms extend from the regular locus to the resolution. Rather than using the given resolution $\widetilde{X}$ to show that $p$-forms extend, we are going to prove directly that the natural morphism $r_{*}Ω^{p}_{\widetilde{X}}↪j_{*}Ω^{p}_{X_{\operatorname{reg}}}$ is an isomorphism for every $p∈\{0,1,…,n\}$. This is a statement about $X$ itself, because the subsheaf $r_{*}Ω^{p}_{\widetilde{X}}$ does not depend on the choice of resolution, as we have seen in the introduction.

Using independence of the resolution, we may assume without loss of generality that the resolution $r\colon\widetilde{X}\textrightarrow X$ is projective, and an isomorphism over $X_{\operatorname{reg}}$. Such resolutions exist for every reduced complex space by [BM97, Thm. 10.7].

Criteria for extension

The first idea in the proof of Theorem 1.4 is to use duality.For the sake of exposition, we work directly on $X$ in this section. In the actual proof, we only use duality for coherent sheaves on complex manifolds, after locally embedding $X$ into a complex manifold. Let $ω_{X}^{\textbullet}∈\operatorname{D}_{\mathit{coh}}^{\mathit{b}}(𝒪_{X})$ denote the dualizing complex of $X$; on the $n$-dimensional complex manifold $\widetilde{X}$, one has $ω_{\widetilde{X}}^{\textbullet}≅ω_{\widetilde{X}}[n]$. The dualizing complex gives rise to a simple numerical criterion for whether sections of a coherent $𝒪_{X}$-module extend uniquely over $X_{\operatorname{sing}}$. Indeed, Proposition 6.1 – or rather its generalisation to singular spaces – says that sections of a coherent $𝒪_{X}$-module $ℱ$ extend uniquely over $X_{\operatorname{sing}}$ if and only if

When the support of $ℱ$ has pure dimension $n$, as is the case for the $𝒪_{X}$-module $r_{*}Ω^{p}_{\widetilde{X}}$ that we are interested in, this amounts to the following two conditions:

$\dim\operatorname{Supp}R^{k}\scr{H}\negthinspace om_{𝒪_{X}}(ℱ,ω_{X}^{\textbullet})≤-(k+2)$ for every $k≥-n+1$

Unfortunately, there is no good way to compute the dual complex of $r_{*}Ω^{p}_{\widetilde{X}}$. But if we work instead with the entire complex $\mathbf{R}r_{*}Ω^{p}_{\widetilde{X}}$, things get better: Grothendieck duality [RRV71], applied to the proper holomorphic mapping $r\colon\widetilde{X}\textrightarrow X$, yields

In Proposition 6.4, we prove the following variant of the criterion for section extension: if $K∈\operatorname{D}_{\mathit{coh}}^{\mathit{b}}(𝒪_{X})$ is a complex with $ℋ^{j}K=0$ for $j<0$, and if

then sections of the coherent $𝒪_{X}$-module $ℋ⁰K$ extend uniquely over $X_{\operatorname{sing}}$. This observation transforms the problem of showing that sections of $r_{*}Ω^{p}_{\widetilde{X}}$ extend uniquely over $X_{\operatorname{sing}}$ into the problem of showing that

In summary, we see that a good upper bound for the dimension of the support of $R^{k}r_{*}Ω^{n-p}_{\widetilde{X}}$ would be enough to conclude that $p$-forms extend. Or, to put it more simply, “vanishing implies extension”.

Hodge modules and the Decomposition Theorem

The problem with the approach outlined above is that the complex $\mathbf{R}r_{*}Ω^{n-p}_{\widetilde{X}}$ has too many potentially nonzero cohomology sheaves, which makes it hard to prove the required vanishing. For example, if the preimage of a singular point $x∈X_{\operatorname{sing}}$ is a divisor in the resolution $\widetilde{X}$, then $R^{n-1}r_{*}Ω^{n-p}_{\widetilde{X}}$ might be supported at $x$, violating the inequality in (2.0.2). Since we are not assuming that the singularities of $X$ are klt, we also do not have enough information about the fibres of $r\colon\widetilde{X}\textrightarrow X$ to prove vanishing by restricting to fibres as in [GKKP11, §18].

The second idea in the proof, which completely circumvents this problem, is to relate the $𝒪_{X}$-module $r_{*}Ω^{p}_{\widetilde{X}}$ to the intersection complex of $X$, viewed as a polarisable Hodge moduleSince the intersection complex is intrinsic to $X$, this also serves to explain once again why the $𝒪_{X}$-module $r_{*}Ω^{p}_{\widetilde{X}}$ does not depend on the choice of resolution.. In the process, we make use of the Decomposition Theorem. Roughly speaking, the Decomposition Theorem decomposes the push-forward of the constant sheaf into a “generic part” (that only depends on $X$) and a “special part” (that is affected by the positive-dimensional fibres of $r$). The upshot is that the generic part carries all the relevant information, and that the positive-dimensional fibres of $r$ are completely irrelevant for the extension problem. To be more precise, the Decomposition Theorem for the projective morphism $r$, together with Saito’s formalism of Hodge modules, leads to a (non-canonical) decomposition

into two complexes $K_{p},R_{p}∈\operatorname{D}_{\mathit{coh}}^{\mathit{b}}(𝒪_{X})$ with the following properties:

The support of $R_{p}$ is contained in the singular locus $X_{\operatorname{sing}}$.

The complexes $K_{p}$ and $K_{n-p}$ are related by Grothendieck duality in the same way that the complexes $\mathbf{R}r_{*}Ω^{p}_{\widetilde{X}}$ and $\mathbf{R}r_{*}Ω^{n-p}_{\widetilde{X}}$ are related in (2.0.1). More precisely, one has $\mathbf{R}\scr{H}\negthinspace om_{𝒪_{X}}(K_{p},ω_{X}^{\textbullet})≅K_{n-p}[n]$.

An improved criterion

As an immediate consequence of the decomposition in (2.0.3), we obtain a decomposition of the -th cohomology sheaves

Because $ℋ⁰R_{p}$ is supported inside $X_{\operatorname{sing}}$, whereas $Ω^{p}_{\widetilde{X}}$ is torsion free, we deduce that $ℋ⁰R_{p}=0$, and hence that $r_{*}Ω^{p}_{\widetilde{X}}≅ℋ⁰K_{p}$. According to the criterion for section extension in Proposition 6.4, now applied to the complex $K_{p}$, all we therefore need for sections of $r_{*}Ω^{p}_{\widetilde{X}}$ to extend uniquely over $X_{\operatorname{sing}}$ is to establish the collection of inequalities

Property (2.0.2) makes this a much more manageable task, compared to the analogous problem for the original complex $\mathbf{R}r_{*}Ω^{n-p}_{\widetilde{X}}$. We stress that, except in the case $p=n$, these inequalities are stronger than asking that sections of $r_{*}Ω^{p}_{\widetilde{X}}$ extend uniquely over $X_{\operatorname{sing}}$.

The case of isolated singularities

We conclude this outline with a brief sketch how (2.0.4) is proved in the case of isolated singularities. In Section 6.2, we more or less reduce the general case to this special case by locally cutting with hypersurfaces; note that this works because we are proving a stronger statement than just extension of $p$-forms.

Because of Property (2.0.2), we have $ℋ^{k}K_{n-p}=0$ for $k≥p+1$. Since $\dim X_{\operatorname{sing}}=0$, the inequality in (2.0.4) is therefore true by default as long as $p≤n-2$. In this way, we recover the result of Steenbrink and van Straten [vSS85, Thm. 1.3] mentioned in the introduction: on an $n$-dimensional complex space with isolated singularities, $p$-forms extend for every $p≤n-2$. This only leaves two cases, namely $p=n-1$ and $p=n$.

The case $p=n$ is covered by the assumption that $n$-forms extend. We have $K_{n}≅ℋ⁰K_{n}≅r_{*}Ω_{\widetilde{X}}^{n}$, and sections of $r_{*}Ω_{\widetilde{X}}^{n}$ extend uniquely over $X_{\operatorname{sing}}$. Because of the isomorphism \mathbf{R}\scr{H}\negthinspace om_{𝒪_{X}}\bigl{(}K_{n},ω_{X}^{\textbullet}\bigr{)}≅K_{0}[n], Proposition 6.1 gives us the desired inequalities

In the other case $p=n-1$, the inequalities in (2.0.4) are easily seen to be equivalent to the single vanishing $ℋ^{n-1}K_{1}=0$. Using the fact that $ℋ^{k}K_{0}=0$ for $k≥n-1$, one shows that the $𝒪_{X}$-module $ℋ^{n-1}K_{1}$ is a quotient of the (constructible) -th cohomology sheaf of the intersection complex of $X$. But the intersection complex is known to be concentrated in strictly negative degrees, and therefore $ℋ^{n-1}K_{1}=0$.

Conventions

Throughout this paper, all complex spaces are assumed to be countable at infinity. All schemes and algebraic varieties are assumed to be defined over the field of complex numbers. We follow the notation used in the standard reference books [Har77, GR84]. In particular, varieties are assumed to be irreducible, and the support of a coherent sheaf $ℱ$ on $X$ is a closed subset of $X$, with the induced reduced structure. For clarity, we will always say explicitly when a complex space needs to be reduced, irreducible, or of pure dimension.

2. 𝒟𝒟𝒟-modules

Unless otherwise noted, we use left $𝒟$-modules throughout this paper. This choice agrees with the notation of the paper [Sch16], which we will frequently cite. It is, however, incompatible with the conventions of the reference papers [Sai88, Sai90] and of the survey [Sch14] that use right $𝒟$-modules throughout. We refer the reader to [Sch16, §A.5], where the conversion rules for left and right $𝒟$-modules are recalled.

3. Complexes

Let $K$ be a complex of sheaves of Abelian groups on a topological space, for example a complex of sheaves of $𝒪_{X}$-modules (or $𝒟_{X}$-modules) on a complex manifold $X$. We use the notation $ℋ^{j}K$ for the $j$-th cohomology sheaf of the complex. We use the notation $K[n]$ for the shift of $K$. We have $ℋ^{j}K[n]=ℋ^{j+n}K$, and all differentials in the shifted complex are multiplied by $(-1)^{n}$.

4. The dualizing complex

If $X$ is any complex space, we write $ω_{X}^{\textbullet}∈\operatorname{D}_{\mathit{coh}}^{\mathit{b}}(𝒪_{X})$ for the dualizing complex as introduced by Ramis and Ruget, see [BS76, VII Thm. 2.6] and the original reference [RR70]. Given a complex of $𝒪_{X}$-modules $K∈\operatorname{D}_{\mathit{coh}}^{\mathit{b}}(𝒪_{X})$ with bounded coherent cohomology, we call the complex $\mathbf{R}\scr{H}\negthinspace om_{𝒪_{X}}(K,ω_{X}^{\textbullet})∈\operatorname{D}_{\mathit{coh}}^{\mathit{b}}(𝒪_{X})$ the dual complex of $K$.

When $X$ is a complex manifold of pure dimension, one has $ω_{X}^{\textbullet}≅ω_{X}[\dim X]$.

5. Reflexive sheaves on normal spaces

Let $X$ be a normal complex space, and $ℱ$ a coherent $𝒪_{X}$-module. Recall that $ℱ$ is called reflexive if the natural morphism from $ℱ$ to its double dual ℱ^{**}:=\scr{H}\negthinspace om_{𝒪_{X}}\bigl{(}\scr{H}\negthinspace om_{𝒪_{X}}(ℱ,𝒪_{X}),𝒪_{X}\bigr{)} is an isomorphism. The following notation will be used.

Given a normal complex space $X$ and a coherent sheaf $ℱ$ on $X$, write Ω^{[p]}_{X}:=\bigl{(}Ω^{p}_{X}\bigr{)}^{**}, ℱ^{[m]}:=\bigl{(}ℰ^{⊗m}\bigr{)}^{**} and \det ℱ:=\bigl{(}Λ^{\operatorname{rank}ℰ}ℱ\bigr{)}^{**}. Given any morphism $f\colon Y\textrightarrow X$ of normal complex spaces, write $f^{[*]}ℱ:=(f^{*}ℱ)^{**}$, etc. Ditto for quasi-projective varieties.

Mixed Hodge modules

For the convenience of the reader, we briefly recall a number of facts concerning mixed Hodge modules, and lay down the notation that will be used throughout. In a nutshell, a (mixed) Hodge module is something like a variation of (mixed) Hodge structure with singularities, in the sense that the vector bundles with connection (respectively locally constant sheaves) in the definition of a variation of Hodge structure are replaced by regular holonomic $𝒟$-modules (respectively perverse sheaves). The standard references for mixed Hodge modules are the original papers by Saito [Sai88, Sai90]. The survey articles [Sai89, Sai94, Sch14] review some aspects of the theory in a smaller number of pages. A good reference for $𝒟$-modules is the book [HTT08]. We consider the following setting throughout the present section.

Assume that a complex manifold $Y$ of pure dimension $d$ and a graded-polarisable mixed Hodge module $M$ on $Y$ are given.

More precisely, a polarisable Hodge module $M$ of weight $w$ on $Y$ has three components: a regular holonomic left $𝒟_{Y}$-module $\mathcal{M}$, called the underlying $𝒟$-module; an increasing good filtration $F_{\textbullet}\mathcal{M}$ by coherent $𝒪_{Y}$-modules, called the Hodge filtration; and a perverse sheaf of $ℚ$-vector spaces $\operatorname{rat}M$, called the underlying perverse sheaf. These are subject to a number of conditions, including the existence of a polarisation, which together ensure that $M$ is determined by finitely many polarisable variations of Hodge structure of weight $w-\dim Z_{j}$ on locally closed submanifolds $Z_{j}⊆Y$. A graded-polarisable mixed Hodge module $M$ on $Y$ is an object of the same kind, but with an additional increasing filtration $W_{\textbullet}M$, called the weight filtration, such that each subquotient

is a polarisable Hodge module of weight $ℓ$. The support of $M$, denoted by $\operatorname{Supp}M$, is by definition the support of the $𝒟_{Y}$-module $\mathcal{M}$ (or, equivalently, of the perverse sheaf $\operatorname{rat}M$).

In Setting 4.1, we denote by $\operatorname{MHM}(Y)$ the Abelian category of graded-polarisable mixed Hodge modules on $Y$, and by $\operatorname{HM}(Y,w)$ the Abelian category of polarisable Hodge modules of weight $w$.

Conversely, every polarisable Hodge module $N∈\operatorname{HM}(Y,w)$ may be viewed as a graded-polarisable mixed Hodge module $N$ with $W_{w-1}N=0$ and $W_{w}N=N$.

Maintain Setting 4.1. Given any integer $k∈ℤ$, define $ℚ(k)=(2πi)^{k}ℚ⊆ℂ$. The Tate twist $M(k)$ is the mixed Hodge module whose underlying perverse sheaf is $ℚ(k)⊗\operatorname{rat}M$, whose underlying filtered $𝒟_{Y}$-module is $(\mathcal{M},F_{\textbullet-k}\mathcal{M})$, and whose weight filtration is given by $W_{ℓ}\,M(k)=W_{ℓ+2k}M$. When $M$ is pure of weight $w$, it follows that $M(k)$ is again pure of weight $w-2k$.

1.2. Decomposition by strict support

In Setting 4.1, one says that the mixed Hodge module $M$ has strict support if the support of every nontrivial subquotient of $M$ is equal to $\operatorname{Supp}M$. Ditto for perverse sheaves and regular holonomic $𝒟_{Y}$-modules. Note that the strict support property is generally not preserved by restriction to open subsets; for example, $\operatorname{Supp}M$ may be globally irreducible, but locally reducible. We use the symbol $\operatorname{HM}_{X}(Y,w)$ to denote the Abelian category of polarisable Hodge modules on $Y$ of weight $w$ with strict support $X$; this is a full subcategory of $\operatorname{HM}(Y,w)$.

If $M$ is a polarisable Hodge module, then $M$ has strict support if and only if the support of every nontrivial subobject (or quotient object) is equal to $\operatorname{Supp}M$; the reason is that polarisable Hodge modules are semisimple [Sai88, Cor. 5.2.13]. By definition, every polarisable Hodge module admits, on every open subset of $X$, a decomposition by strict support as a (locally finite) direct sum of polarisable Hodge modules with strict support [Sai88, §5.1.6].

1.3. Weight filtration and dual module

In Setting 4.1, we write $M^{\prime}=𝔻M∈\operatorname{MHM}(Y)$ for the dual mixed Hodge module. This is again a graded-polarisable mixed Hodge module [Sai90, Prop. 2.6], with the property that

In particular, if $M$ is pure of weight $ℓ$, then $𝔻M$ is again pure of weight $-ℓ$. The underlying perverse sheaf $\operatorname{rat}M^{\prime}$ is isomorphic to the Verdier dual [HTT08, Def. 4.5.2] of $\operatorname{rat}M$. The regular holonomic left $𝒟_{Y}$-module $(\mathcal{M}^{\prime},F_{\textbullet}\mathcal{M}^{\prime})$ underlying $M^{\prime}=𝔻M$ is isomorphic to the holonomic dual [HTT08, Def. 2.6.1]

of the regular holonomic left $𝒟_{Y}$-module $\mathcal{M}$.

1.4. The de Rham complex

In Setting 4.1, the complex of sheaves of $ℂ$-vector spaces

concentrated in degrees $-d,…,0$ is called the de Rham complex of $\mathcal{M}$. Since $\mathcal{M}$ is a regular holonomic $𝒟_{Y}$-module, the de Rham complex $\operatorname{DR}(\mathcal{M})$ has constructible cohomology sheaves, and is in fact a perverse sheaf on $Y$ by a theorem of Kashiwara [HTT08, Thm. 4.6.6]. In particular, it is always semiperverse, which means concretely that

The perverse sheaf $\operatorname{rat}M$ and the de Rham complex of $\mathcal{M}$ are related through an isomorphism $ℂ⊗_{ℚ}\operatorname{rat}M≅\operatorname{DR}(\mathcal{M})$ that is part of the data of a mixed Hodge module.

1.5. Subquotients of the de Rham complex

Assume Setting 4.1. The filtration $F_{\textbullet}\mathcal{M}$ induces an increasing filtration on the de Rham complex by

The $p$-th subquotient of this filtration is the complex of $𝒪_{Y}$-modules

For a more detailed discussion of these complexes, see for example [Sch16, §7]. The following simple lemma will be useful later.

In Setting 4.1, if $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M})$ is acyclic for every $p≤m$, then $F_{m+\dim Y}\mathcal{M}=0$.

Since $F_{\textbullet}\mathcal{M}$ is a good filtration, there is, at least locally on $Y$, an integer $p_{0}$ such that $F_{p_{0}}\mathcal{M}=0$ and, hence, $\operatorname{gr}_{p}^{F}\mathcal{M}=0$ for every $p≤p_{0}$. To show that $F_{m+d}\mathcal{M}=0$, it is therefore enough to prove that $\operatorname{gr}_{p}^{F}\mathcal{M}=0$ for every $p≤m+d$. Because $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M})$ is acyclic for $p≤m$, this follows from (4.3.2) by induction on $p≥p_{0}$. ∎

2. Duality

Next, we review how the duality functor for mixed Hodge modules affects the subquotients of the de Rham complex. The following nontrivial result by Saito shows that the dual complex of $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M})$ is nothing but $\operatorname{gr}_{-p}^{F}\operatorname{DR}(\mathcal{M}^{\prime})$.

where $(\mathcal{M}^{\prime},F_{\textbullet}\mathcal{M}^{\prime})$ is the filtered $𝒟_{Y}$-module underlying $M^{\prime}=𝔻M$.

This is proved in [Sai88, §2.4.3]; see also [Sch16, Lem. 7.4]. The crucial point in the proof is that $\operatorname{gr}_{\textbullet}^{F}\mathcal{M}$ is a Cohen-Macaulay module over $\operatorname{gr}_{\textbullet}^{F}𝒟_{Y}$, due to the fact that $(\mathcal{M},F_{\textbullet}\mathcal{M})$ underlies a mixed Hodge module. ∎

In the special case where $M$ is a polarisable Hodge module, the de Rham complex is self-dual, up to a shift in the filtration. Duality therefore relates different subquotients of $\operatorname{DR}(\mathcal{M})$, in a way that will be very useful for the proof of Theorem 1.4.

Let $M∈\operatorname{HM}(Y,w)$ be a polarisable Hodge module of weight $w$ on a complex manifold $Y$. Any polarisation on $M$ induces an isomorphism

A polarisation on $M$ induces an isomorphism $𝔻M≅M(w)$ [Sai88, §5.2.10], and therefore an isomorphism $(\mathcal{M}^{\prime},F_{\textbullet}\mathcal{M}^{\prime})≅(\mathcal{M},F_{\textbullet-w}\mathcal{M})$. Now apply Proposition 4.5. ∎

The following proposition contains an acyclicity criterion for subquotients of the de Rham complex, involving both the weight filtration $W_{\textbullet}M$ and the Hodge filtration $F_{\textbullet}\mathcal{M}$.

Assume Setting 4.1. If $w$, $c∈ℤ$ are such that $W_{w-1}M=0$ and $F_{c-1}\mathcal{M}=0$, then $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M})$ is acyclic unless $c-d≤p≤d-w-c$.

Since $F_{c-1}\mathcal{M}=0$ and $d=\dim Y$, a look at the formula (4.3.2) for the $p$-th subquotient of $\operatorname{DR}(\mathcal{M})$ reveals that $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M})=0$ for $p≤c-1-d$. The other inequality is going to follow by duality. Let us first consider the pure case, meaning that $M∈\operatorname{HM}(Y,w^{\prime})$ is a polarisable Hodge module of weight $w^{\prime}$. By Corollary 4.6, we have

and since the complex on the right-hand side is acyclic for $-p-w^{\prime}≤c-1-d$, we get the result when $M$ is pure. The general case follows from this by considering the subquotients of the weight filtration $W_{\textbullet}M$. ∎

Proposition 4.7 is especially useful when combined with the following general fact, which an easy consequence of the filtration $F_{\textbullet}\mathcal{M}$ being exhaustive.

Let $Y$ be a complex manifold. Let $(\mathcal{M},F_{\textbullet}\mathcal{M})$ be a coherent $𝒟_{Y}$-module with a good filtration. If $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M})$ is acyclic for all $p≥p_{0}+1$, then the inclusion $F_{p_{0}}\operatorname{DR}(\mathcal{M})↪\operatorname{DR}(\mathcal{M})$ is a quasi-isomorphism. ∎

3. Direct images and the Decomposition Theorem

Let $f\colon X\textrightarrow Y$ be a projective holomorphic mapping between two complex manifolds, and let $M∈\operatorname{MHM}(X)$ be a graded-polarisable mixed Hodge module on $X$. One of the most important results in Saito’s theory is that, in this setting, one can define a direct image functor, compatible with the direct image functor for perverse sheaves and filtered $𝒟$-modules, and that the $i$-th higher direct image $Hⁱf_{*}M$ is again a graded-polarisable mixed Hodge module on $Y$. In this section, we briefly review this result and its implications for the underlying filtered $𝒟_{X}$-module $(\mathcal{M},F_{\textbullet}\mathcal{M})$ and the de Rham complex $\operatorname{DR}(\mathcal{M})$.

Let $X$ be a complex manifold. Following Saito, we denote by $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}F(𝒟_{X})$ the derived category of (certain cohomologically bounded and coherent complexes of) filtered $𝒟_{X}$-modules, as defined in [Sai88, §2.1.15]. The category of filtered $𝒟_{X}$-modules is only an exact category, but it embeds into the larger Abelian category of graded $R_{F}𝒟_{X}$-modules, where

is the Rees algebra of $𝒟_{X}$ with respect to the order filtration. The embedding takes a coherent filtered $𝒟_{X}$-module $(\mathcal{M},F_{\textbullet}\mathcal{M})$ to the associated Rees module

which is coherent over $R_{F}𝒟_{X}$. Let $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}G(R_{F}𝒟_{X})$ be the derived category of (cohomologically bounded and coherent complexes of) graded $R_{F}𝒟_{X}$-modules. Then the Rees module construction gives an equivalence of categories

according to [Sai88, Prop. 2.1.16]. The cohomology modules of an object in $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}F(𝒟_{X})$ are therefore in general not filtered $𝒟_{X}$-modules, but graded $R_{F}𝒟_{X}$-modules.

A graded $R_{F}𝒟_{X}$-module is called strict if it is isomorphic to the Rees module of a coherent filtered $𝒟_{X}$-module. A complex $K∈\operatorname{D}_{\mathit{coh}}^{\mathit{b}}G(R_{F}𝒟_{X})$ is called strict if all of its cohomology modules $ℋ^{j}K$ are strict.

The functor that takes a coherent filtered $𝒟_{X}$-module $(\mathcal{M},F_{\textbullet}\mathcal{M})$ to the underlying $𝒟_{X}$-module $\mathcal{M}$ extends uniquely to an exact functor

Indeed, if we denote by $z∈R_{F}𝒟_{X}$ the degree-one element obtained from $1∈F_{1}𝒟_{X}$, then the functor is simply the derived tensor product with $R_{F}𝒟_{X}/(1-z)R_{F}𝒟_{X}$. Similarly, the functor that takes a coherent filtered $𝒟_{X}$-module $(\mathcal{M},F_{\textbullet}\mathcal{M})$ to the coherent graded $\operatorname{Sym}𝒯_{X}$-module $\operatorname{gr}_{\textbullet}^{F}\mathcal{M}$ extends uniquely to an exact functor

This time, the functor is given by the derived tensor product with $R_{F}𝒟_{X}/zR_{F}𝒟_{X}$. Lastly, for every $p∈ℤ$, the functor that takes a coherent filtered $𝒟_{X}$-module $(\mathcal{M},F_{\textbullet}\mathcal{M})$ to the complex of coherent $𝒪_{X}$-modules $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M})$ extends uniquely to an exact functor

Indeed, by [Sai88, Prop. 2.2.10], the de Rham functor (which Saito denotes by the symbol $\widetilde{DR}$) defines an equivalence of categories between $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}F(𝒟_{X})$ and the derived category of filtered differential complexes $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}F^{f}(𝒪_{X},\operatorname{Diff})$, and $\operatorname{gr}_{p}^{F}$ of a filtered differential complex is by construction a (cohomologically bounded and coherent) complex of $𝒪_{X}$-modules [Sai88, §2.2.4].

3.2. Direct image functor for filtered 𝒟𝒟𝒟-modules

Now suppose that $f\colon X\textrightarrow Y$ is a proper holomorphic mapping between complex manifolds. In this setting, one can construct a direct image functor

see [Sai88, §2.3.5] for the precise definition. This functor is compatible with the functor $\operatorname{gr}_{p}^{F}\operatorname{DR}$ in the following manner [Sai88, §2.3.7].

Let $f\colon X\textrightarrow Y$ be a proper holomorphic mapping between complex manifolds. For every $p∈ℤ$, one has a natural isomorphism of functors

as functors from $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}G(R_{F}𝒟_{X})$ to $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}(𝒪_{Y})$.

By [Sai88, Lem. 2.3.6], the de Rham functor exchanges the direct image functor $f_{+}\colon\operatorname{D}_{\mathit{coh}}^{\mathit{b}}G(R_{F}𝒟_{X})\textrightarrow\operatorname{D}_{\mathit{coh}}^{\mathit{b}}G(R_{F}𝒟_{Y})$ and the direct image functor

for filtered differential complexes. But the latter commutes with taking $\operatorname{gr}_{p}^{F}$, as is clear from the construction in [Sai88, §2.3.7]. ∎

In the case of a single coherent filtered $𝒟_{X}$-module, this says that

as objects of the derived category $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}(𝒪_{Y})$.

3.3. Direct image theorem, pure case

We now assume that the proper holomorphic mapping $f\colon X\textrightarrow Y$ is actually projective. Then we have the following important “direct image theorem” due to Saito [Sai88, §5.3].

Let $f\colon X\textrightarrow Y$ be a projective morphism between complex manifolds, and let $ℓ∈H²(X,ℤ(1))$ be the first Chern class of a relatively ample line bundle. If $M∈\operatorname{HM}(X,w)$ is a polarisable Hodge module $X$, then:

The complex $f_{+}(R_{F}\mathcal{M})$ is strict, and each $ℋⁱf_{+}(R_{F}\mathcal{M})$ is the filtered $𝒟_{Y}$-module underlying a polarisable Hodge module $Hⁱf_{*}M∈\operatorname{HM}(Y,w+i)$.

is an isomorphism between Hodge modules of weight $w-i$.

Any polarisation on $M$ induces a polarisation on $\bigoplus_{i}Hⁱf_{*}M$ in the Hodge-Lefschetz sense (= on primitive parts with respect to the action of $ℓ$). ∎

One consequence of Theorem 4.11 is a version of the Decomposition Theorem for those filtered $𝒟$-modules that underlie polarisable Hodge modules.

Let $f\colon X\textrightarrow Y$ be a projective morphism between complex manifolds. Let $M∈\operatorname{HM}(X,w)$ be a polarisable Hodge module on $X$, and let $M_{i}=Hⁱf_{*}M∈\operatorname{HM}(Y,w+i)$. Write $(\mathcal{M},F_{\textbullet}\mathcal{M})$ respectively $(\mathcal{M}_{i},F_{\textbullet}\mathcal{M}_{i})$ for the underlying filtered $𝒟$-modules. Then

in the derived category $\operatorname{D}_{\mathit{coh}}^{\mathit{b}}G(R_{F}𝒟_{Y})$.

The first isomorphism is a formal consequence of (4.11.2). The second isomorphism follows because the complex $f_{+}(R_{F}\mathcal{M})$ is strict. ∎

3.4. Direct image theorem, mixed case

In the case of mixed Hodge modules, there are some additional results, having to do with the weight filtration. We summarise them in the following theorem [Sai90, Thm. 2.14 and Prop. 2.15].

Let $f\colon X\textrightarrow Y$ be a projective morphism between complex manifolds, and let $M∈\operatorname{MHM}(X)$ be a graded-polarisable mixed Hodge module on $X$.

The complex $f_{+}(R_{F}\mathcal{M})$ is strict, and each $ℋⁱf_{+}(R_{F}\mathcal{M})$ is the filtered $𝒟_{Y}$-module underlying a graded-polarisable mixed Hodge module $Hⁱf_{*}M∈\operatorname{MHM}(Y)$.

One has a convergent weight spectral sequence

and each differential $d_{1}\colon E_{1}^{p,q}\textrightarrow E_{1}^{p+1,q}$ is a morphism in $\operatorname{HM}(Y,q)$.

The weight spectral sequence degenerates at $E_{2}$, and one has

One can use this result to control the range in which the Hodge filtration on the direct image of a graded-polarisable mixed Hodge module is nontrivial.

Let $f\colon X\textrightarrow Y$ be a projective morphism between complex manifolds, and let $M∈\operatorname{MHM}(X)$ be a graded-polarisable mixed Hodge module on $X$. Suppose that the underlying filtered $𝒟_{X}$-module $(\mathcal{M},F_{\textbullet}\mathcal{M})$ satisfies $F_{m-1}\mathcal{M}=0$. Then one has

for every $i∈ℤ$, where $c=\dim Y-\dim X$.

One can deduce this from the construction of the direct image functor in [Sai88, §2.3]. Here we outline another proof based on Theorem 4.11 and Theorem 4.13.

We first deal with the case where $M∈\operatorname{HM}(X,W)$ is a polarisable pure Hodge module. By Proposition 4.10 and Corollary 4.12, we have for every $p∈ℤ$ an isomorphism

where $(\mathcal{M}_{i},F_{\textbullet}\mathcal{M}_{i})$ is the filtered $𝒟_{Y}$-module underlying $Hⁱf_{*}M∈\operatorname{HM}(Y,w+i)$. Since $F_{m-1}\mathcal{M}=0$, we get $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M})=0$ for all $p≤m-1-\dim X$, and $\operatorname{gr}_{p}^{F}\operatorname{DR}(\mathcal{M}_{i})$ is therefore acyclic as long as $p≤m-1-\dim X$. According to Lemma 4.4, this is enough to conclude that $F_{m+c-1}\mathcal{M}_{i}=F_{m-1-\dim X+\dim Y}\mathcal{M}_{i}=0$ for every $i∈ℤ$.

Now suppose that $M∈\operatorname{MHM}(X)$ is a graded-polarisable mixed Hodge module. The underlying $𝒟_{X}$-module of the Hodge module $\operatorname{gr}_{w}^{F}M∈\operatorname{HM}(X,w)$ is $\operatorname{gr}_{w}^{W}\mathcal{M}$, with the induced Hodge filtration; because $F_{m-1}\mathcal{M}=0$, we have $F_{m-1}\operatorname{gr}_{w}^{F}\mathcal{M}=0$. Since we already have the result in the pure case, the assertion now follows by looking at the spectral sequence in (4.13.2). ∎

4. Non-characteristic restriction to hypersurfaces

We briefly review the non-characteristic restriction of a mixed Hodge module to a hypersurface. For a more general discussion of non-characteristic restriction, see [Sai88, §3.5] or [Sch16, §8].

Let $X$ be a complex manifold, and let $D⊆X$ be a smooth hypersurface. The inclusion $i_{D}\colon D↪X$ gives rise to the following morphisms between cotangent bundles:

Given a regular holonomic left $𝒟_{X}$-module $\mathcal{M}$ on $X$, let $\operatorname{Ch}(\mathcal{M})⊆T^{\ast}X$ denote its characteristic variety. We say that $D⊆X$ is non-characteristic for $\mathcal{M}$ if $p_{1}^{-1}\operatorname{Ch}(\mathcal{M})$ is finite over its image in $T^{\ast}D$.

As explained for example in [Sch16, §8], $D⊆X$ is non-characteristic for $\mathcal{M}$ if and only if $D$ is transverse to every stratum in a Whitney stratification of $X$ that is adapted to the perverse sheaf $\operatorname{DR}(\mathcal{M})$. In particular, generic hyperplane sections (in $ℙ^{n}$ or $ℂ^{n}$) are always non-characteristic.

The following result of Saito [Sai90, Lem. 2.25] describes what happens to mixed Hodge modules under non-characteristic restriction to smooth hypersurfaces.

Let $X$ be a complex manifold, and let $M∈\operatorname{MHM}(X)$ be a graded-polarisable mixed Hodge module on $X$, with underlying filtered $𝒟_{X}$-module $(\mathcal{M},F_{\textbullet}\mathcal{M})$. Suppose that $i_{D}\colon D↪X$ is a smooth hypersurface that is non-characteristic for $\mathcal{M}$. Then there is a graded-polarisable mixed Hodge module $H^{-1}i_{D}^{*}M∈\operatorname{MHM}(D)$, whose underlying filtered $𝒟_{D}$-module is isomorphic to

and whose de Rham complex is quasi-isomorphic to

Moreover, if $M$ is pure of weight $w$, then $H^{-1}i_{D}^{*}M$ is again pure of weight $w-1$.