Constructive Degenerations and the Algebraicity of Limiting Hodge

TL;DR

This paper introduces a constructive approach to the Hodge Conjecture by analyzing degenerations and limiting mixed Hodge structures, suggesting that algebraic classes can be obtained as limits of algebraic cycles through controlled degenerations.

Contribution

It develops a new criterion based on LMHS for when rational (p,p) classes become algebraic in degenerations, offering a potential pathway to prove the Hodge Conjecture.

Findings

Examples where vanishing cycles generate new algebraic classes

A proposed principle that all rational (p,p) classes can be limits of algebraic cycles

A framework connecting degenerations with algebraicity of Hodge classes

Abstract

We propose a novel constructive framework for approaching the Hodge Conjecture via explicit degenerations. Building on limiting mixed Hodge structures (LMHS), we formulate a criterion under which a rational class of type (p, p) on a smooth projective variety becomes algebraic in the limit of a semi-stable degeneration. We provide examples where vanishing cycles and monodromy explicitly generate new algebraic classes, and propose a general principle: every rational (p, p) class arises as the limit of algebraic cycles under controlled geometric degenerations. This viewpoint opens a new path toward an effective formulation of the Hodge conjecture.