Du Bois singularity¶

Classify a reduced characteristic-zero scheme by requiring its structure sheaf to agree quasi-isomorphically with degree zero of the Du Bois complex, equivalently through a log-resolution criterion.

Version: v2 · 2026-09-19 · History
Domain-specific #: 1721
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Algebraic Geometry → Mathematics

Core Idea¶

A reduced scheme X has Du Bois singularities when the natural map from O_X to the zeroth graded Du Bois complex is a quasi-isomorphism; Schwede's criterion recognizes the property using a log resolution and reduced inverse image. A filtered derived de Rham object records cohomological behavior of a singular variety, while the log-resolution criterion compares O_X with Rπ_*O_E. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

No faithful explanation at this level. All three generators agree that any child-level picture reduces a Du Bois singularity to a spot that merely looks mildly pinched or pointy, whereas the class is defined by an exact derived-sheaf quasi-isomorphism condition that visual mildness does not track or distinguish from rational or log-canonical classes.

No faithful explanation at this level. Two of three generators judged that at age ten it still collapses into 'a mild kind of bumpy point', teaching visible gentleness instead of the precise cohomological comparison and blurring it with neighboring rational and log-canonical classes.

Cohomologically Mild Singularities

In algebraic geometry, shapes defined by polynomial equations can have singular points where they aren't smooth. Mathematicians classify how 'mild' a singularity is using tools from cohomology, which records global and local structure. Du Bois singularities are one such mild class. The definition uses the Du Bois complex, an object built to capture the Hodge-theoretic cohomology of singular shapes; a shape has Du Bois singularities when the simplest piece of that complex agrees, in a precise derived sense, with the shape's ordinary functions. Schwede gave a practical test: resolve the singularities, look at the reduced preimage E, and check whether the functions on the shape match the derived functions pushed down from E. It's a different condition from neighboring classes like rational or log-canonical singularities.

Du Bois singularities are a class of mild singularities of reduced schemes in characteristic zero, defined cohomologically. To a singular variety X one attaches the Du Bois complex, a filtered object that plays the role of the de Rham complex and records the cohomological (Hodge-theoretic) behavior of X despite its singularities. X has Du Bois singularities when the natural map from the structure sheaf O_X to the zeroth graded piece of that complex is a quasi-isomorphism, meaning an isomorphism in the derived category, not merely on sections. Schwede's criterion makes this testable with a log resolution: with π a suitable log resolution and E the reduced inverse image, X is Du Bois iff the natural map O_X → Rπ_*O_E is a quasi-isomorphism. The condition is independent of the chosen resolution. The hypotheses matter: X must be reduced and in the characteristic-zero setting, and checking a neighboring property such as rationality or log canonicity is not the same as checking this one. The class is used to place singularities within Hodge-theoretic and minimal-model hierarchies and, with further hypotheses, to derive cohomological and vanishing results.

Scope of Application¶

Du Bois singularity belongs to complex and characteristic-zero algebraic geometry and is useful where the analyst can specify a reduced separated scheme of finite type in characteristic zero, often embedded as a closed subscheme of a smooth ambient scheme, then evaluate the structure sheaf is recovered by the designated derived Du Bois or log-resolution object. The scope is broad within that domain but bounded by the need for the canonical map O_X→̲Ω_X^0 is a quasi-isomorphism, equivalently under Schwede's hypotheses O_X→Rπ_*O_E is. The standard theory in the cited sources assumes reduced finite-type schemes over a characteristic-zero field; extension claims require separate justification.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the structure sheaf is recovered by the designated derived Du Bois or log-resolution object the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because Du Bois is an eponym for a singularity class, not a numerical invariant.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: hyperresolutions, filtered derived categories, log resolutions, exceptional divisors, derived pushforwards, and interactions with other singularity classes. Du Bois singularity compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: a reduced separated scheme of finite type in characteristic zero, often embedded as a closed subscheme of a smooth ambient scheme. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the canonical map O_X→̲Ω_X^0 is a quasi-isomorphism, equivalently under Schwede's hypotheses O_X→Rπ_*O_E is independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of complex and characteristic-zero algebraic geometry because they reuse a reduced separated scheme of finite type in characteristic zero, often embedded as a closed subscheme of a smooth ambient scheme, A filtered derived de Rham object records cohomological behavior of a singular variety, while the log-resolution criterion compares O_X with Rπ_*O_E., and verify the scheme and resolution hypotheses, identify the reduced inverse image E, and establish the derived quasi-isomorphism. A theorem, diagnostic, or modeling warning can travel when those roles remain literal.

Relationships to Other Abstractions¶

Current abstraction Du Bois singularity Domain-specific

Parents (1) — outgoing typed relations

Du Bois singularity is a kind of Classification Prime

Du Bois singularity is literally a rule-defined category of schemes.

Hierarchy path (1) — routes to 1 parentless root

Du Bois singularity → is a kind of Classification

Neighborhood in Abstraction Space¶

Du Bois singularity sits in a crowded region of the domain-specific corpus (32^nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Sheaf Theory & Algebraic Geometry (12 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25