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 (1)
| Label | Occurrences |
|---|---|
| Donaldson–Uhlenbeck–Yau theorem canonical | 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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
Mumford–Takemoto stability ⓘ compact Kähler manifolds ⓘ holomorphic vector bundles ⓘ polystable vector bundles ⓘ slope stability ⓘ stability of vector bundles ⓘ |
| alsoKnownAs | Donaldson–Uhlenbeck–Yau correspondence NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Shing-Tung Yau NERFINISHED ⓘ Simon Donaldson NERFINISHED ⓘ |
| partOf | Kobayashi–Hitchin correspondence NERFINISHED ⓘ |
| provedBy |
Karen Uhlenbeck
NERFINISHED
ⓘ
Shing-Tung Yau NERFINISHED ⓘ Simon Donaldson NERFINISHED ⓘ |
| relatedTo |
Hermitian Yang–Mills connections
NERFINISHED
ⓘ
Kobayashi–Hitchin correspondence NERFINISHED ⓘ Narasimhan–Seshadri theorem NERFINISHED ⓘ Yang–Mills equations NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Kähler form ⓘ curvature of a connection ⓘ degree of a vector bundle ⓘ slope of a vector bundle ⓘ |
How these facts were elicited
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.