Intersection Du Bois Complex

• Intersection Du Bois Complex is a Hodge-module-theoretic refinement of ordinary Du Bois complexes, replacing constant sheaves with the intersection complex to capture intricate stratifications.
• It bridges ordinary cohomology and intersection cohomology by formulating an IC-type analogue that supports injectivity, vanishing, and duality theorems.
• The construction unifies diverse geometric intersections, including hyperresolutions, smooth poset schemes, and special fiber restrictions, to enhance our understanding of complex singularities.

Searching arXiv for recent and foundational papers on Du Bois complexes, intersection-complex analogues, and related base-change/hypersurface results. {"query":"all: \"Du Bois complex\" intersection complex Hodge modules Du Bois", "max_results": 10, "sort_by": "relevance"} The literature indicates that the expression intersection Du Bois complex has a restricted and nonuniform status. In the most precise current usage represented here, it denotes the Hodge-module-theoretic complexes

$$I\underline{\Omega}_X^p:=\operatorname{gr}_{-p}^F DR(IC_X\mathbf Q^H)[p-n],$$

obtained by replacing the constant Hodge module in the ordinary Du Bois construction with the intersection complex Hodge module. At the same time, several papers explicitly state that they do not define any object called an “intersection Du Bois complex,” and instead distinguish the ordinary Du Bois complex from the intersection complex $\IC_X$ and from intersection cohomology $\IH^\bullet(X)$ (Popa et al., 2024, Kim, 10 Jul 2025).

1. Terminological status and conceptual scope

The phrase is not standard across the subject. Recent work on finite morphisms and higher extension properties explicitly says that no object called an “intersection Du Bois complex” is being defined there, even though those papers work intensively with the ordinary Du Bois complex and with Hodge-module or perverse-sheaf constructions adjacent to intersection complexes (Kim, 10 Jul 2025, Tighe, 2023). A similar caution appears in work on pure subrings and Du Bois pairs, where the operative objects are $\underline{\Omega}_X^0$ and $\underline{\Omega}_{X,E}^0$, not an IC-type refinement (Godfrey et al., 2022).

This terminological instability reflects two different uses of the word “intersection.” In one use, “intersection” refers to the intersection complex $IC_X\mathbf Q^H$ of mixed Hodge module theory. In another, it refers to geometric intersections—hyperplane sections, complete intersections, fibers, or SNC intersection patterns—through which ordinary Du Bois complexes are analyzed. The Hodge-module-theoretic object $I\underline{\Omega}_X^p$ belongs to the first use; pair Du Bois complexes, hypersurface models, and base-change formulas belong to the second.

2. Ordinary Du Bois complexes as the precursor

For a complex variety $X$, the ordinary Du Bois package is the filtered complex $(\underline{\Omega}_X^\bullet,F)$, with graded pieces

$$\underline{\Omega}_X^p:=\operatorname{gr}_F^p\underline{\Omega}_X^\bullet[p].$$

If $\IC_X$0 is a hyperresolution, then

$\IC_X$1

For proper $\IC_X$2, the Hodge-to-de Rham spectral sequence

$\IC_X$3

degenerates at $\IC_X$4, and $\IC_X$5 is Du Bois precisely when the natural morphism

$\IC_X$6

is a quasi-isomorphism (Kovács, 2011).

A second construction replaces hyperresolutions by smooth poset schemes. If $\IC_X$7 is a smooth projective poset scheme satisfying

$\IC_X$8

then

$\IC_X$9

In particular, when $\IH^\bullet(X)$0 is built from smooth intersections of components, the ordinary Du Bois complex is already realized by a diagram encoding intersection strata (Lunts, 2010).

3. Definition of the intersection Du Bois complexes

The explicit IC-type analogue is formulated in mixed Hodge modules. In the derived category of mixed Hodge modules on an $\IH^\bullet(X)$1-dimensional complex variety $\IH^\bullet(X)$2, one has the composition

$\IH^\bullet(X)$3

Ordinary Du Bois complexes are recovered as

$\IH^\bullet(X)$4

The intersection analogue is then defined by

$\IH^\bullet(X)$5

These objects are called intersection Du Bois complexes in the 2024 injectivity-and-vanishing paper, which states explicitly that, from the point of view of Hodge modules and constructible sheaves, the passage from $\IH^\bullet(X)$6 to $\IH^\bullet(X)$7 is obtained by replacing the constant sheaf with the intersection complex (Popa et al., 2024).

This places the new complexes exactly at the interface between ordinary cohomology and intersection cohomology. The ordinary Du Bois complexes are the coherent-sheaf avatars of the Hodge filtration on the de Rham realization of $\IH^\bullet(X)$8; the intersection Du Bois complexes are the corresponding avatars for $\IH^\bullet(X)$9.

4. Comparison with ordinary Du Bois complexes

The mixed-Hodge-module construction comes equipped with natural comparison morphisms

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

where

$\underline{\Omega}_X^0$1

These maps place the intersection Du Bois complexes between the ordinary Du Bois complexes and their Grothendieck duals. The comparison becomes especially tight under higher-rationality hypotheses: if $\underline{\Omega}_X^0$2 is a normal variety with pre-$\underline{\Omega}_X^0$3-rational singularities, then

$\underline{\Omega}_X^0$4

Thus, in low degrees, pre-$\underline{\Omega}_X^0$5-rationality forces the IC-type and constant-sheaf-type constructions to coincide (Popa et al., 2024).

The degree-zero part is particularly concrete. For a resolution $\underline{\Omega}_X^0$6, one has

$\underline{\Omega}_X^0$7

This formula makes the intersection object more geometric than the ordinary $\underline{\Omega}_X^0$8, which can be subtler. In the local complete intersection case, the same paper combines this description with reflexivity statements to derive depth bounds and vanishing results for the IC-type complexes.

5. Injectivity and vanishing theory

The main 2024 results establish parallel injectivity and vanishing statements for ordinary and intersection Du Bois complexes. On the intersection side, the fundamental conjectural morphism is

$\underline{\Omega}_X^0$9

expected to be injective on cohomology for normal varieties with pre-$\underline{\Omega}_{X,E}^0$0-rational singularities. What is proved unconditionally is a comparison theorem: under normal pre-$\underline{\Omega}_{X,E}^0$1-rational singularities, dualizing $\underline{\Omega}_{X,E}^0$2 gives

$\underline{\Omega}_{X,E}^0$3

injective on cohomology. More sharply, this morphism is an isomorphism on $\underline{\Omega}_{X,E}^0$4-th cohomology for

$\underline{\Omega}_{X,E}^0$5

injective for

$\underline{\Omega}_{X,E}^0$6

and satisfies

$\underline{\Omega}_{X,E}^0$7

The full injectivity conjecture is proved when $\underline{\Omega}_{X,E}^0$8, when $\underline{\Omega}_{X,E}^0$9 has isolated singularities, and when $IC_X\mathbf Q^H$0 is a local complete intersection with $IC_X\mathbf Q^H$1-rational singularities (Popa et al., 2024).

The vanishing theory is parallel. In the same three settings, one obtains the expected depth-range vanishing for the cohomology sheaves of the intersection complexes. In particular, if $IC_X\mathbf Q^H$2 is a local complete intersection and

$IC_X\mathbf Q^H$3

then

$IC_X\mathbf Q^H$4

These results show that the IC-type theory is not merely formal: it supports its own injectivity conjectures, depth estimates, and vanishing theorems, although the general non-isolated, non-LCI case remains conjectural.

6. Adjacent “intersection” viewpoints in Du Bois theory

Even where no IC-type complex is named, several strands of the literature exhibit closely related intersection phenomena.

For pairs, if $IC_X\mathbf Q^H$5 is embeddable, $IC_X\mathbf Q^H$6 is closed, $IC_X\mathbf Q^H$7 has rational singularities, and

$IC_X\mathbf Q^H$8

is a log resolution with $IC_X\mathbf Q^H$9, then the pair Du Bois complex satisfies

$I\underline{\Omega}_X^p$0

where $I\underline{\Omega}_X^p$1 are the irreducible components of $I\underline{\Omega}_X^p$2. This identifies the pair Du Bois complex with the graded de Rham realization of Hodge-module intersection complexes on the rational locus, even though it is not itself called an “intersection Du Bois complex” (Park, 2023).

For hypersurfaces, the relation to intersection-complex-type objects appears through a canonical morphism of filtered right $I\underline{\Omega}_X^p$3-modules

$I\underline{\Omega}_X^p$4

underlying the composition

$I\underline{\Omega}_X^p$5

Theorem 3.1 in that paper states that $I\underline{\Omega}_X^p$6 is a $I\underline{\Omega}_X^p$7-homology manifold iff the unipotent monodromy part of the vanishing cycle complex vanishes; equivalently, the constant object and the intersection-complex-type object coincide. In the same hypersurface setting, the ordinary graded pieces $I\underline{\Omega}_X^p$8 are realized concretely by truncations and cones of the $I\underline{\Omega}_X^p$9-Koszul complex, showing how Du Bois theory meets IC-type Hodge-module structure without collapsing into it (Jung et al., 2021).

For restriction to fibers and Cartier intersections, the relative theory over a smooth complex curve gives a derived restriction formula

$X$0

for every $X$1 in a nonempty Zariski-open subset $X$2, and for every closed point when a simultaneous relative hyperresolution exists. The same paper proves that this can fail at special SNC fibers: if $X$3 is an SNC divisor in a smooth total space of relative dimension $X$4, then

$X$5

This suggests an intersection-fiber viewpoint in which the Du Bois complex of a special fiber remembers singular gluing data not captured by naive restriction of the relative complex (Ji et al., 4 Aug 2025).

Taken together, these constructions show that the subject has two stable cores. One is the precise IC-type object $X$6. The other is a broader constellation of pair, hypersurface, and restriction phenomena in which ordinary Du Bois complexes interact with intersection patterns, normalization, and Hodge-module realizations. The phrase intersection Du Bois complex is therefore most precise when reserved for the IC-Hodge-module construction, but it also points to a wider intersection-theoretic landscape inside Du Bois theory.