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Deformations, local freeness, and base change for higher Du Bois singularities

This paper establishes that strict higher Du Bois singularities are invariant under small deformations and proves a corresponding base change theorem for the relative Du Bois complex, thereby generalizing local-freeness results and proving Hodge number constancy while demonstrating that the strictness condition is essential.

Original authors: Haoming Ning

Published 2026-08-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Haoming Ning

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to build a skyscraper, but instead of steel and concrete, you are building with the invisible, mathematical "shape" of space itself. In the world of algebraic geometry, mathematicians study these shapes, but they often encounter "cracks" or "kinks" where the surface isn't perfectly smooth. These are called singularities. For decades, mathematicians have had a special toolkit to measure the hidden "holes" and "loops" inside these shapes, known as Hodge theory. However, when a shape has a crack, the standard tools often break or give confusing answers. To fix this, a special set of rules called "Du Bois singularities" was invented. Think of these rules as a way to say, "Even though this part of the building is crumpled, we can still trust our measurements if we look at it the right way."

Now, imagine you aren't just building one skyscraper, but a whole neighborhood of them, where each building is a tiny variation of the last, like a family of shapes growing and changing. A major question for architects of these mathematical worlds is: If one building in the family has a "good" crumpled shape (a specific type of singularity), does that guarantee the whole neighborhood is safe? Or, if you change the blueprint slightly, does the "goodness" of the shape survive? This paper tackles exactly that. It asks whether these special, well-behaved shapes stay well-behaved when they are deformed (stretched or squished) or when we swap the ground they stand on (a process called base change). The answer turns out to be a delicate "yes, but only if we are very strict about the rules."

The author, Haoming Ning, proves that a specific, very strict version of these "good" shapes, called strict-m-Du Bois singularities, are incredibly stable. If you have a family of shapes where one member is strictly "good," then the whole family nearby is also "good," and the mathematical measurements (Hodge numbers) stay exactly the same no matter how you look at them. This is a big deal because it allows mathematicians to predict the behavior of complex shapes with confidence.

However, the paper also delivers a stern warning. The author shows that if you try to relax the rules just a little bit—moving from "strict" to a slightly looser definition called "m-Du Bois"—the whole system collapses. The paper constructs specific, counter-intuitive examples where a shape looks "good" in one spot, but the moment you try to deform it or change the background, the "goodness" vanishes, and the measurements go haywire. It's like finding a bridge that looks sturdy until you drive a single car across it, at which point it turns out to be made of paper.

The key innovation here is the invention of a new mathematical tool called the right relative Du Bois complex. Think of the old tools as a left-handed glove that fits perfectly for some hands but not others. The author designs a "right-handed" version that fits much better for these specific families of shapes. This new tool acts as a bridge, proving that for the strictest kind of shapes, the measurements of the whole family perfectly match the measurements of the individual pieces. But for anything less strict, the bridge doesn't hold, and the measurements fail to match up.

In short, this paper draws a very sharp line in the sand. It proves that for the most robust, "strict" version of these singularities, the universe of shapes is predictable and stable: deformations work, and measurements stay constant. But it also proves that this stability is a fragile gift; if you try to cut corners and use a weaker definition, the predictability disappears entirely. The paper doesn't just say "it works"; it rigorously demonstrates why it works for the strict cases and exactly how it fails for the looser ones, ensuring that future mathematicians know exactly which rules to follow to keep their mathematical skyscrapers standing tall.

Technical Summary: Deformations, Local Freeness, and Base Change for Higher Du Bois Singularities

Problem Statement
The paper addresses three fundamental questions regarding the behavior of higher Du Bois singularities in families of complex varieties:

Base Change: For a dominant morphism $f: X \to B$ to a smooth curve, does the relative Du Bois complex $\Omega^p_{X/B}$ base change to the absolute Du Bois complex $\Omega^p_{X_b}$ of a fiber $X_b$ ? Specifically, does the natural map $\Omega^p_{X/B} \otimes^L_{\mathcal{O}_X} \mathcal{O}_{X_b} \xrightarrow{\sim} \Omega^p_{X_b}$ hold?
Deformation Invariance: If a Cartier divisor $Z \subseteq X$ satisfies a higher Du Bois condition, does the ambient space $X$ satisfy the corresponding condition in a neighborhood of $Z$ ?
Local Freeness: For a flat proper family, are the higher direct image sheaves $R^i f_* \Omega^p_{X/B}$ locally free and compatible with base change?

While these properties are well-understood for local complete intersection (lci) singularities, their validity in the general setting remains open. The paper investigates whether the various refinements of higher Du Bois conditions (pre-, weakly-, m-, and strict-m-Du Bois) satisfy these properties, with a specific focus on identifying the sharpest condition under which they hold.

Methodology
The author introduces and utilizes an alternative construction of the relative Du Bois complex, termed the right relative Du Bois complex ( $\Omega^{p,+}_{X/B}$ ), contrasting it with the standard left relative Du Bois complex ( $\Omega^{p,-}_{X/B}$ ) defined in prior literature (e.g., Kovács, Kovács–Taji).

Construction: The right complex is built "upward" from $p=0$ via cone constructions involving the wedge map with the pullback of the canonical bundle of the base, whereas the left complex is constructed "downward" from the dimension.
Comparison: The paper establishes that the right complex admits a more direct comparison with the absolute Du Bois complex of a fiber. A key technical tool is a commutative diagram (Theorem 4.11) relating $\Omega^{p,+}_{X/B}$ , $\Omega^p_X$ , and $\Omega^p_{X_b}$ , which facilitates the analysis of base change.
Injectivity Theorems: To prove deformation invariance, the paper extends the Kovács–Schwede injectivity theorem for Kähler differentials. Specifically, it proves that for a variety with strict- $(m-1)$ -Du Bois singularities, the map $H^k(D_X(\Omega^p_X)) \to H^k(D_X(\Omega^p_X))$ is injective for $p \le m$ (Theorem 5.2). This relies on analyzing Kähler differentials under cyclic covers and utilizing Kato's logarithmic differentials ( $\Omega^p_X(\log^K H)$ ).
Deformation to Base Change: The paper demonstrates a bidirectional link: deformation invariance of strict-m-Du Bois singularities implies base change for the right complex, and conversely, assuming base change allows for the derivation of deformation results for weaker singularity classes.

Key Contributions and Results

Deformation Invariance of Strict-m-Du Bois Singularities:
The paper proves that if a Cartier divisor $Z \subseteq X$ has strict-m-Du Bois singularities, then $X$ has strict-m-Du Bois singularities in a neighborhood of $Z$ (Theorem 6.5). This is established via the extension of the injectivity theorem and Elkik-type arguments.
Base Change Theorem:
The main result answers a question of Kovács–Taji: If a fiber $X_b$ has strict-m-Du Bois singularities, it satisfies base change for the right relative Du Bois complex. Specifically, for all $p \le m$ :
$\Omega^{p,+}_{X/B} \otimes^L_{\mathcal{O}_X} \mathcal{O}_{X_b} \simeq_{qis} \Omega^p_{X_b}$
Furthermore, if $X_b$ has strict-m-Du Bois singularities for all $m \in \mathbb{N}$ , then $X_b$ satisfies base change for the left relative Du Bois complex for all $p$ (Theorem 7.3). For a fixed $m$ , the left complex does not necessarily satisfy base change even if the fiber is strict-m-Du Bois.
Sharpness and Failure of Weaker Conditions:
The paper exhibits that the "strict" condition is essentially sharp.
- Right Base Change: A fiber satisfies right-m-base change if and only if it is strict-m-Du Bois (Proposition 9.3).
- Left Base Change: Left-m-base change fails for fibers that are merely m-Du Bois or weakly-m-Du Bois. Explicit counterexamples (Example 9.5, Proposition 9.7) show that even for fibers that are Du Bois (0-Du Bois) or weakly-m-Du Bois, base change can fail in degree 0 or higher.
- Deformation Failure: The paper constructs examples (Proposition 9.14) where a Cartier divisor $Z$ is 1-Du Bois, but the total space $X$ is not even pre-1-Du Bois, demonstrating that deformation invariance fails for the m-Du Bois condition.
Local Freeness and Constancy of Hodge Numbers:
Using the base change results, the paper generalizes the local-freeness theorem of Friedman–Laza beyond the lci setting. For a flat proper family over a smooth curve, if the fiber $X_b$ has strict-m-Du Bois singularities, then $R^i f_* \Omega^p_{X/B}$ is locally free and compatible with base change for $0 \le p \le m$ (Theorem 8.7).
Additionally, via a curve-slicing argument, the paper proves the constancy of Hodge numbers $h^{p,q}$ for families over arbitrary bases, provided the fibers are strict-m-Du Bois (Theorem 8.8).

Significance
The paper establishes that strict-m-Du Bois singularities constitute the most suitable generalization of higher Du Bois conditions beyond the lci setting. While this condition is more restrictive than other existing notions (such as m-Du Bois or weakly-m-Du Bois), it is the only one among them that guarantees the three desirable properties of deformation invariance, base change, and local freeness of cohomology sheaves. The introduction of the right relative Du Bois complex provides a new technical framework that simplifies the comparison between relative and absolute complexes, resolving open questions regarding the stability of Hodge-theoretic invariants in families of singular varieties. The results clarify the limitations of weaker singularity definitions, showing that without the strictness condition (or the condition holding for all $m$ for the left complex), fundamental properties like base change and deformation invariance fail even for relatively mild singularities.

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