Triple
T6716267
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Salomon Bochner |
E153275
|
entity |
| Predicate | notableFor |
P22
|
FINISHED |
| Object |
Bochner–Kodaira–Nakano identity
The Bochner–Kodaira–Nakano identity is a fundamental formula in complex differential geometry relating the Laplacian on differential forms to curvature terms, with key applications to vanishing theorems and Hodge theory.
|
E613409
|
NE FINISHED |
How this triple was built (4 steps)
Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.
NER
Named-entity recognition
gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Bochner–Kodaira–Nakano identity | Statement: [Salomon Bochner, notableFor, Bochner–Kodaira–Nakano identity]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bochner–Kodaira–Nakano identity Context triple: [Salomon Bochner, notableFor, Bochner–Kodaira–Nakano identity]
-
A.
Kähler identities
Kähler identities are fundamental commutation relations in Kähler geometry that link the Lefschetz operator, its adjoint, and the Dolbeault operators, playing a key role in Hodge theory and complex differential geometry.
-
B.
Differential Analysis on Complex Manifolds
"Differential Analysis on Complex Manifolds" is a foundational mathematical monograph that systematically develops the theory of differential and complex geometry on complex manifolds.
-
C.
Cartan theorems A and B
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
-
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.
Dolbeault cohomology classes
Dolbeault cohomology classes are equivalence classes of differential forms on a complex manifold defined using the ∂̄-operator, encoding the manifold’s complex-analytic and geometric structure.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg
Description generation
gpt-5.1
Instruction
Generate a one-sentence description of the target entity. You are given a context triple in the form (subject, predicate, object), where the object is the target entity. # Instructions Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. Avoid repeating the information from the triple, unless really essential. # Response Format Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Bochner–Kodaira–Nakano identity Triple: [Salomon Bochner, notableFor, Bochner–Kodaira–Nakano identity]
Generated description
The Bochner–Kodaira–Nakano identity is a fundamental formula in complex differential geometry relating the Laplacian on differential forms to curvature terms, with key applications to vanishing theorems and Hodge theory.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Bochner–Kodaira–Nakano identity Target entity description: The Bochner–Kodaira–Nakano identity is a fundamental formula in complex differential geometry relating the Laplacian on differential forms to curvature terms, with key applications to vanishing theorems and Hodge theory.
-
A.
Kähler identities
Kähler identities are fundamental commutation relations in Kähler geometry that link the Lefschetz operator, its adjoint, and the Dolbeault operators, playing a key role in Hodge theory and complex differential geometry.
-
B.
Differential Analysis on Complex Manifolds
"Differential Analysis on Complex Manifolds" is a foundational mathematical monograph that systematically develops the theory of differential and complex geometry on complex manifolds.
-
C.
Cartan theorems A and B
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
-
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.
Dolbeault cohomology classes
Dolbeault cohomology classes are equivalence classes of differential forms on a complex manifold defined using the ∂̄-operator, encoding the manifold’s complex-analytic and geometric structure.
- F. None of above. chosen
Provenance (5 batches)
The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.
| Step | Stage | Batch ID | Status | When |
|---|---|---|---|---|
| creating | Elicitation | batch_69c68809b4608190a2509ddb5ab87f05 |
completed | March 27, 2026, 1:37 p.m. |
| NER | Named-entity recognition | batch_69c6d125db3c8190aad28919226a16da |
completed | March 27, 2026, 6:49 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69c700993128819081614ccfa68d7320 |
completed | March 27, 2026, 10:11 p.m. |
| NEDg | Description generation | batch_69c70262f3e48190b544be536ee0b674 |
completed | March 27, 2026, 10:19 p.m. |
| NED2 | Entity disambiguation (via description) | batch_69c70311418c8190a902cf21187fdc51 |
completed | March 27, 2026, 10:22 p.m. |
Created at: March 27, 2026, 2:07 p.m.