Triple

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).

Phrase: hard Lefschetz theorem | Statement: [Weil cohomology, hasAxiom, hard Lefschetz theorem]

Target entity: hard Lefschetz theorem
Context triple: [Weil cohomology, hasAxiom, 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. Lefschetz
  Lefschetz is a surname most notably associated with Solomon Lefschetz, a pioneering mathematician in algebraic topology and geometry.
• E. Hodge theory
  Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.