Donaldson–Uhlenbeck–Yau theorem
E613807
The Donaldson–Uhlenbeck–Yau theorem is a fundamental result in differential and algebraic geometry that characterizes when a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian–Einstein metric, linking geometric stability with the existence of such metrics.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Donaldson–Uhlenbeck–Yau correspondence | 1 |
| Donaldson–Uhlenbeck–Yau theorem canonical | 1 |
| Hermitian Yang–Mills connections | 1 |
| Yau's theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6708999 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Donaldson–Uhlenbeck–Yau theorem Context triple: [Simon Donaldson, knownFor, Donaldson–Uhlenbeck–Yau theorem]
-
A.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
-
B.
Yamabe problem
The Yamabe problem is a fundamental question in differential geometry concerning whether every compact Riemannian manifold admits a metric of constant scalar curvature within a given conformal class.
-
C.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
-
D.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
E.
Kähler–Ricci flow
Kähler–Ricci flow is a geometric evolution equation that deforms Kähler metrics on complex manifolds according to their Ricci curvature, playing a central role in complex differential geometry and the study of canonical metrics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Donaldson–Uhlenbeck–Yau theorem Target entity description: The Donaldson–Uhlenbeck–Yau theorem is a fundamental result in differential and algebraic geometry that characterizes when a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian–Einstein metric, linking geometric stability with the existence of such metrics.
-
A.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
-
B.
Yamabe problem
The Yamabe problem is a fundamental question in differential geometry concerning whether every compact Riemannian manifold admits a metric of constant scalar curvature within a given conformal class.
-
C.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
-
D.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
E.
Kähler–Ricci flow
Kähler–Ricci flow is a geometric evolution equation that deforms Kähler metrics on complex manifolds according to their Ricci curvature, playing a central role in complex differential geometry and the study of canonical metrics.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| about |
Hermitian–Einstein metrics
ⓘ
Mumford–Takemoto stability ⓘ compact Kähler manifolds ⓘ holomorphic vector bundles ⓘ polystable vector bundles ⓘ slope stability ⓘ stability of vector bundles ⓘ |
| alsoKnownAs |
Donaldson–Uhlenbeck–Yau correspondence
ⓘ
linked to:
Donaldson–Uhlenbeck–Yau theorem
|
| appliesTo |
complex projective manifolds as special cases of compact Kähler manifolds
ⓘ
holomorphic principal bundles in certain generalizations ⓘ |
| characterizes |
existence of Hermitian–Einstein metrics
ⓘ
slope polystability of holomorphic vector bundles ⓘ |
| domain | holomorphic vector bundle over a compact Kähler manifold ⓘ |
| field |
algebraic geometry
ⓘ
complex geometry ⓘ differential geometry ⓘ gauge theory ⓘ |
| generalizationOf | Narasimhan–Seshadri theorem ⓘ |
| hasConsequence |
construction of moduli spaces of stable vector bundles
ⓘ
equivalence between moduli of stable bundles and moduli of Hermitian–Einstein connections ⓘ links algebraic stability with differential-geometric equations ⓘ |
| implies |
Hermitian–Einstein bundles are polystable
ⓘ
linked to:
Hermitian–Einstein metrics
stable bundles admit Hermitian–Einstein metrics ⓘ |
| influenced |
modern gauge-theoretic approaches in algebraic geometry
ⓘ
study of moduli of vector bundles on complex manifolds ⓘ |
| involves | nonlinear elliptic partial differential equations ⓘ |
| namedAfter |
Karen Uhlenbeck
ⓘ
Shing-Tung Yau ⓘ Simon Donaldson ⓘ |
| partOf | Kobayashi–Hitchin correspondence ⓘ |
| provedBy |
Karen Uhlenbeck
ⓘ
Shing-Tung Yau ⓘ Simon Donaldson ⓘ |
| relatedTo |
Hermitian Yang–Mills connections
ⓘ
linked to:
Donaldson–Uhlenbeck–Yau theorem
Kobayashi–Hitchin correspondence ⓘ Narasimhan–Seshadri theorem ⓘ Yang–Mills equations ⓘ
linked to:
Yang–Mills theory
|
| statesThat | a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian–Einstein metric if and only if it is polystable ⓘ |
| timePeriod | 1980s ⓘ |
| usesConcept |
Chern connection
ⓘ
Kähler form ⓘ curvature of a connection ⓘ degree of a vector bundle ⓘ slope of a vector bundle ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Donaldson–Uhlenbeck–Yau theorem Description of subject: The Donaldson–Uhlenbeck–Yau theorem is a fundamental result in differential and algebraic geometry that characterizes when a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian–Einstein metric, linking geometric stability with the existence of such metrics.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Donaldson–Uhlenbeck–Yau theorem
linked to: Donaldson–Uhlenbeck–Yau theorem
linked to: Donaldson–Uhlenbeck–Yau theorem