TITLE:
Hodge Classes in Hodge Manifolds
AUTHORS:
Yong Seung Cho
KEYWORDS:
Hodge Manifold, Hodge Decomposition. Lefschetz Decomposition, Hodge Class, Algebraic Cycle Class, Polarization, Hodge Conjecture, Lefschetz Standard Conjecture
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.15 No.8,
August
11,
2025
ABSTRACT: The Hodge conjecture states that the Hodge group of Hodge classes is equal to the algebraic group generated by algebraic subvarieties on a Hodge manifold. To prove the conjecture, we introduce the Hodge structure, the Lefschetz decomposition, and polarization on the cohomology groups of the manifold, and we use mathematical induction on the degrees of the primitive cohomologies in the Lefschetz decomposition. We show that every Hodge class on a Hodge manifold is a rational linear combination of the cohomology classes of algebraic subvarieties of the manifold. The Lefschetz isomorphisms on the cohomology groups of a Hodge manifold are algebraic in the product space. As a consequence, we show that the inverses of the Lefschetz isomorphisms are also algebraic in the product space.