Year: 2025
Type: article
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.
Source: Advances in Pure Mathematics
Author: Yong Seung Cho
Institution: Ewha Womans University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4413227988