Triple

T17330211
Position Surface form Disambiguated ID Type / Status
Subject Lefschetz hyperplane theorem E420792 entity
Predicate relatedTo P37 FINISHED
Object Hard Lefschetz theorem E551969 NE FINISHED

How this triple was built (2 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: Hard Lefschetz theorem | Statement: [Lefschetz hyperplane theorem, relatedTo, Hard Lefschetz theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Hard Lefschetz theorem
Context triple: [Lefschetz hyperplane theorem, relatedTo, Hard Lefschetz theorem]
  • A. Hard Lefschetz theorem chosen
    The Hard Lefschetz theorem is a fundamental result in algebraic geometry and Hodge theory that relates the cohomology groups of a compact Kähler manifold via repeated cup product with the Kähler class, yielding powerful symmetry and duality properties.
  • B. Lefschetz hyperplane theorem
    The Lefschetz hyperplane theorem is a fundamental result in algebraic geometry and topology that relates the topology (especially homology and homotopy groups) of a smooth projective variety to that of its hyperplane sections.
  • C. Hodge–Riemann bilinear relations
    The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
  • D. Lazarsfeld’s Positivity in Algebraic Geometry
    Lazarsfeld’s *Positivity in Algebraic Geometry* is a two-volume monograph that serves as a standard modern reference on the theory of positivity for line bundles and divisors in algebraic geometry, integrating techniques from cohomology, vanishing theorems, and birational geometry.
  • E. 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.
  • F. None of above.
  • G. Unsure - the case is ambiguous/there is not enough information to decide.

Provenance (3 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_69d889d3adc881909319f1edb8d2a956 completed April 10, 2026, 5:25 a.m.
NER Named-entity recognition batch_69e439d5c788819092bdc4d3de0ec958 completed April 19, 2026, 2:11 a.m.
NED1 Entity disambiguation (via context triple) batch_6a018c5025d08190ab2581a3b04ae661 completed May 11, 2026, 7:59 a.m.
Created at: April 10, 2026, 5:43 a.m.