Triple

T19370556
Position Surface form Disambiguated ID Type / Status
Subject Cartan theorems A and B E484523 entity
Predicate influenced P9 FINISHED
Object Oka–Grauert principle
The Oka–Grauert principle is a fundamental result in complex geometry asserting that, for certain complex manifolds, topological and holomorphic classification problems coincide, allowing topological data to determine holomorphic structures.
E1372644 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: Oka–Grauert principle | Statement: [Cartan theorems A and B, influenced, Oka–Grauert principle]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Oka–Grauert principle
Context triple: [Cartan theorems A and B, influenced, Oka–Grauert principle]
  • A. Oka–Weil theorem
    The Oka–Weil theorem is a fundamental result in several complex variables that extends Runge’s approximation theorem by characterizing when holomorphic functions on certain compact sets in complex manifolds can be uniformly approximated by global holomorphic functions.
  • B. Kodaira vanishing theorem
    The Kodaira vanishing theorem is a fundamental result in algebraic geometry that gives conditions under which certain cohomology groups of ample line bundles on smooth projective varieties vanish, with deep implications for the classification of 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. 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.
  • E. 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.
  • 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: Oka–Grauert principle
Triple: [Cartan theorems A and B, influenced, Oka–Grauert principle]
Generated description
The Oka–Grauert principle is a fundamental result in complex geometry asserting that, for certain complex manifolds, topological and holomorphic classification problems coincide, allowing topological data to determine holomorphic structures.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Oka–Grauert principle
Target entity description: The Oka–Grauert principle is a fundamental result in complex geometry asserting that, for certain complex manifolds, topological and holomorphic classification problems coincide, allowing topological data to determine holomorphic structures.
  • A. Oka–Weil theorem
    The Oka–Weil theorem is a fundamental result in several complex variables that extends Runge’s approximation theorem by characterizing when holomorphic functions on certain compact sets in complex manifolds can be uniformly approximated by global holomorphic functions.
  • B. Kodaira vanishing theorem
    The Kodaira vanishing theorem is a fundamental result in algebraic geometry that gives conditions under which certain cohomology groups of ample line bundles on smooth projective varieties vanish, with deep implications for the classification of 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. 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.
  • E. 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.
  • 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_69d8e8d305088190ad13571532aa454c completed April 10, 2026, 12:10 p.m.
NER Named-entity recognition batch_69e619af33e481908643f8beb2f498dc completed April 20, 2026, 12:18 p.m.
NED1 Entity disambiguation (via context triple) batch_6a072419c4ec8190931c1918450f2a83 completed May 15, 2026, 1:48 p.m.
NEDg Description generation batch_6a0725c32c4c8190bb29a764695cd245 completed May 15, 2026, 1:55 p.m.
NED2 Entity disambiguation (via description) batch_6a0726ea260c8190b97e2b18fde36272 completed May 15, 2026, 2 p.m.
Created at: April 10, 2026, 1:35 p.m.