Relative Du Bois Complex in Singular Algebraic Geometry

• The relative Du Bois complex is a derived refinement of de Rham and Hodge structures, providing techniques for analyzing singularities in algebraic varieties.
• It is defined in both pair and family contexts, bridging classical de Rham theory with modern criteria for Du Bois singularities through cone constructions and log resolutions.
• The relative version for families over curves extends the classical de Rham complex, enabling base change compatibility and offering insights into local cohomology and duality.

Searching arXiv for papers on the relative Du Bois complex and closely related Du Bois/pair constructions. The relative Du Bois complex is a derived-category refinement of de Rham-theoretic and Hodge-theoretic structures for singular algebraic geometry. In current usage, the term appears in two closely related settings. For a closed immersion $Z\subseteq X$, one studies the pair complex

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$

whose degree-zero piece controls Du Bois pairs and related extension problems (Park, 2023). For a flat morphism $f:X\to C$ to a smooth complex curve, one studies a filtered complex $\underline{\Omega}_{X/C}^\bullet$ with graded pieces $\underline{\Omega}_{X/C}^p$, intended as a singular analogue of the relative de Rham complex (2307.07192). Both constructions extend the absolute Deligne–Du Bois complex $\underline{\Omega}_X^\bullet$, the standard replacement for $\Omega_X^\bullet$ on singular varieties (Kovács, 2011).

1. Absolute Du Bois theory as the foundation

The absolute Deligne–Du Bois complex is a filtered object $\underline{\Omega}_X^\bullet$ in a filtered derived category, with graded pieces

$$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$

If $\varepsilon_\bullet:X_\bullet\to X$ is a hyperresolution, then

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$0

so the complex is defined by descending smooth de Rham data from a hyperresolution (Kovács, 2011). It resolves the constant sheaf, restricts to opens, is functorial for proper morphisms, and for proper $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$1 its spectral sequence

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$2

degenerates at $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$3, recovering Deligne’s Hodge filtration (Kovács, 2011).

The basic singularity-theoretic criterion is the natural map

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$4

A variety has Du Bois singularities precisely when this map is a quasi-isomorphism (Kovács, 2011). On smooth varieties one has

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$5

so the absolute Du Bois complex is a genuine extension of the ordinary de Rham complex (Kovács, 2011).

A categorical reformulation replaces hyperresolutions by smooth poset schemes. If $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$6 is a smooth projective poset scheme such that

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$7

then

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$8

This gives an alternative way to compute the Du Bois complex and ties $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$9 to categorical resolutions (Lunts, 2010).

2. Pair-theoretic relative complexes

For a reduced pair $f:X\to C$0, the relative or pair Du Bois complex is defined by the distinguished triangle

$f:X\to C$1

equivalently by the cone formula above (Park, 2023). The degree-zero piece receives a natural morphism from the ideal sheaf,

$f:X\to C$2

and $f:X\to C$3 is a Du Bois pair exactly when this map is a quasi-isomorphism (Fujino et al., 2018). This package supplies the pair-theoretic analogue of the absolute criterion $f:X\to C$4.

When $f:X\to C$5 is smooth and $f:X\to C$6 is reduced, the pair complex admits a log-resolution description. If $f:X\to C$7 is a log resolution of $f:X\to C$8 with $f:X\to C$9 a simple normal crossings divisor, then Steenbrink’s realization gives

$\underline{\Omega}_{X/C}^\bullet$0

and the pair triangle becomes the bridge between ambient smooth geometry, the intrinsic Du Bois complex of $\underline{\Omega}_{X/C}^\bullet$1, and the logarithmic geometry of $\underline{\Omega}_{X/C}^\bullet$2 (Mustata et al., 2021). In the setting where $\underline{\Omega}_{X/C}^\bullet$3 has rational singularities, one also has a resolution-theoretic identification

$\underline{\Omega}_{X/C}^\bullet$4

for a log resolution $\underline{\Omega}_{X/C}^\bullet$5, together with a mixed-Hodge-module description through $\underline{\Omega}_{X/C}^\bullet$6 (Park, 2023).

These pair complexes are not formal accessories. They are used to prove descent theorems and birational criteria for Du Bois singularities. In particular, the functorial map

$\underline{\Omega}_{X/C}^\bullet$7

for a morphism of pairs $\underline{\Omega}_{X/C}^\bullet$8 underlies splitting criteria for Du Bois pairs and the descent of Du Bois singularities under proper morphisms and under cyclically pure maps (Fujino et al., 2018, Godfrey et al., 2022).

3. Relative Du Bois complexes for families over a smooth curve

The first explicit family-theoretic construction appears for flat morphisms

$\underline{\Omega}_{X/C}^\bullet$9

to a smooth complex curve. The motivating question, attributed to Steven Zucker, asks for a singular analogue of the relative de Rham complex whose Hodge-theoretic behavior would parallel the smooth case (2307.07192). Kovács–Taji answer this in dimension-one base by constructing a filtered object

$\underline{\Omega}_{X/C}^p$0

in the filtered derived category, starting from a filtered representative $\underline{\Omega}_{X/C}^p$1 of the absolute Du Bois complex and the differential $\underline{\Omega}_{X/C}^p$2. The construction is recursive: one defines maps

$\underline{\Omega}_{X/C}^p$3

and then cone complexes $\underline{\Omega}_{X/C}^p$4, with $\underline{\Omega}_{X/C}^p$5 representing $\underline{\Omega}_{X/C}^p$6 (2307.07192).

The resulting filtered object has graded pieces

$\underline{\Omega}_{X/C}^p$7

recovering the previously known relative differential objects $\underline{\Omega}_{X/C}^p$8 (2307.07192). It fits into distinguished triangles that mirror the absolute-to-relative de Rham sequence: $\underline{\Omega}_{X/C}^p$9 and, on graded pieces,

$\underline{\Omega}_X^\bullet$0

(2307.07192).

This relative complex has the expected formal properties. It restricts to opens on the total space, is compatible with restriction to an open subset of the base, and is functorial for morphisms of flat families over the same curve (2307.07192). Most importantly, if $\underline{\Omega}_X^\bullet$1 is smooth, then

$\underline{\Omega}_X^\bullet$2

so the construction genuinely extends the classical relative de Rham complex rather than replacing it by a different object (2307.07192).

4. Fiberwise base change and its limitations

Once $\underline{\Omega}_X^\bullet$3 is constructed, the central problem becomes fiberwise restriction. For a closed point $\underline{\Omega}_X^\bullet$4, with fiber $\underline{\Omega}_X^\bullet$5 and immersion $\underline{\Omega}_X^\bullet$6, the question is whether

$\underline{\Omega}_X^\bullet$7

or, more strongly,

$\underline{\Omega}_X^\bullet$8

This is Problem 5.2 in the original construction paper (2307.07192).

A partial answer is given by the generic base-change theorem. The key notion is a simultaneous relative hyperresolution: for $\underline{\Omega}_X^\bullet$9, this is a hyperresolution $\Omega_X^\bullet$0 such that every composite $\Omega_X^\bullet$1 is smooth (Ji et al., 4 Aug 2025). Under this hypothesis,

$\Omega_X^\bullet$2

because on each smooth $\Omega_X^\bullet$3 the relative Du Bois complex coincides with the ordinary sheaf of relative differentials (Ji et al., 4 Aug 2025). The paper then proves that if $\Omega_X^\bullet$4 admits such a simultaneous relative hyperresolution, then for every closed point $\Omega_X^\bullet$5,

$\Omega_X^\bullet$6

The proof uses base change on the hyperresolution, a Tor-independence lemma for effective Cartier divisors, and the standard derived base-change theorem (Ji et al., 4 Aug 2025).

For an arbitrary morphism $\Omega_X^\bullet$7, the same paper proves a generic statement: there exists a nonempty open subset $\Omega_X^\bullet$8 such that for every $\Omega_X^\bullet$9,

$\underline{\Omega}_X^\bullet$0

and likewise for the full filtered complex $\underline{\Omega}_X^\bullet$1 (Ji et al., 4 Aug 2025). Thus formation of the relative Du Bois complex commutes with base change to a general point of the curve.

The failure at special points is equally important. In a one-parameter degeneration

$\underline{\Omega}_X^\bullet$2

with $\underline{\Omega}_X^\bullet$3 smooth, $\underline{\Omega}_X^\bullet$4 smooth over $\underline{\Omega}_X^\bullet$5, and special fiber $\underline{\Omega}_X^\bullet$6 a simple normal crossings divisor, the paper shows

$\underline{\Omega}_X^\bullet$7

in top degree (Ji et al., 4 Aug 2025). The obstruction comes from the non-normal structure of the special fiber: for an SNC divisor,

$\underline{\Omega}_X^\bullet$8

where $\underline{\Omega}_X^\bullet$9 is the normalization, while $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$0 is a line bundle on $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$1 and therefore locally free (Ji et al., 4 Aug 2025). A common misconception is that failure of base change should signal singularities in the total space; this example shows the opposite. Here $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$2 is smooth, and the obstruction is entirely fiber-theoretic.

5. Local cohomology, duality, and derived descriptions

The pair-theoretic relative complex admits a mixed-Hodge-theoretic interpretation in terms of local cohomology. If $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$3 is smooth of dimension $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$4, $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$5 is a closed subscheme, and $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$6 is the inclusion, then

$$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$7

If $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$8 is an lci of pure codimension $$\underline{\Omega}_X^p:=Gr_F^p\underline{\Omega}_X^\bullet[p].$$9, this simplifies to

$\varepsilon_\bullet:X_\bullet\to X$0

These formulas identify the Du Bois complex of $\varepsilon_\bullet:X_\bullet\to X$1 with Hodge-theoretic data carried by local cohomology along the embedding $\varepsilon_\bullet:X_\bullet\to X$2 (Mustata et al., 2021).

The same framework yields a criterion for local cohomological dimension. For every positive integer $\varepsilon_\bullet:X_\bullet\to X$3,

$\varepsilon_\bullet:X_\bullet\to X$4

so the graded pieces of the Du Bois complex control a purely local invariant of the pair $\varepsilon_\bullet:X_\bullet\to X$5 (Mustata et al., 2021). This is one of the clearest places where the relative Du Bois complex acts as an organizing object rather than as a formal replacement for differential forms.

Local algebraic consequences of the Du Bois package also pass through pair complexes. If $\varepsilon_\bullet:X_\bullet\to X$6 is a local ring essentially of finite type over $\varepsilon_\bullet:X_\bullet\to X$7 and $\varepsilon_\bullet:X_\bullet\to X$8 is Du Bois, then

$\varepsilon_\bullet:X_\bullet\to X$9

is surjective for every $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$00 (Ma et al., 2016). More generally, for a pair $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$01, if the reduced pair is Du Bois, then local cohomology of $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$02 surjects onto the local cohomology of its reduction (Ma et al., 2016). The mechanism is the pair triangle

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$03

together with duality for $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$04 (Ma et al., 2016).

On the dual side, if $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$05 is Du Bois then

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$06

and under finite-length hypotheses on local cohomology the low-degree truncation of the dualizing complex becomes as simple as possible. Concretely, if $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$07 is essentially of finite type over a field of characteristic zero, $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$08 is Du Bois, and $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$09 has finite length for $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$10, then

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$11

is quasi-isomorphic to a complex of $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$12-vector spaces (Bhatt et al., 2015). This identifies a precise way in which the Du Bois condition controls the dualizing complex through the derived dual of $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$13.

6. Higher variants, finite-morphism formalism, and open directions

The graded pieces $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$14 support higher versions of Du Bois singularities. In one direction, $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$15-Du Bois singularities are defined by the requirement that

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$16

or, in broader formulations, by vanishing conditions on the higher cohomology sheaves $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$17 together with reflexivity of $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$18 (Friedman et al., 2022, Shen et al., 2023). This higher structure has a genuinely relative consequence: if $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$19 is a flat proper family and one fiber has $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$20-Du Bois lci singularities, then near that point

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$21

is locally free and compatible with arbitrary base change for $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$22 and all $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$23 (Friedman et al., 2022). In this range, the ordinary relative Kähler differentials behave as the low Hodge pieces of a relative Du Bois package. The same paper derives constancy of low Hodge numbers in families and unobstructedness results for singular Calabi–Yau varieties (Friedman et al., 2022).

A different kind of relative formalism appears under finite morphisms. For a finite group quotient $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$24, one has

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$25

in the filtered derived category, and therefore

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$26

for every $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$27 (Kim, 10 Jul 2025). More generally, for a finite surjective morphism $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$28 between normal varieties, there exists a morphism

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$29

such that

$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$30

is an isomorphism (Kim, 10 Jul 2025). Thus each graded piece downstairs is a direct summand of the pushforward upstairs. This yields descent of pre-$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$31-Du Bois and pre-$$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$32-rational singularities, and inequalities for local cohomological defect under finite maps (Kim, 10 Jul 2025).

Several structural directions remain open. The curve case uses specific one-dimensional features: the available construction of $$\underline{\Omega}_{X,Z}^\bullet:=\operatorname{Cone}\bigl(\underline{\Omega}_X^\bullet\to \underline{\Omega}_Z^\bullet\bigr)[-1],$$33 itself is tailored to a smooth curve, and the generic base-change theorem uses the fact that closed fibers are effective Cartier divisors (2307.07192, Ji et al., 4 Aug 2025). The literature explicitly leaves open the extension of the family-theoretic relative Du Bois complex to arbitrary smooth bases, as well as necessary-and-sufficient criteria for base change at special points of a curve (2307.07192, Ji et al., 4 Aug 2025). The existing results therefore describe a theory that is already rich and technically effective, but still incomplete: pair-theoretic relative complexes are well integrated with log resolutions and mixed Hodge modules, while the full family-theoretic theory is presently best understood over one-dimensional smooth bases.