Geometric Interpretation of the Hodge Conjecture via Dual Complex Numbers

This paper presents a geometric reinterpretation of the Hodge conjecture using the dual complex number system introduced in Morimoto (2026). The dual complex framework separates the real and imaginary components of complex geometry into two explicit complex planes, allowing the $(p,q)$-type structure of Hodge theory to be expressed through angle pairs (θ₁,θ₂). Within this setting, the discrepancy between algebraic cycles and harmonic forms is represented by the angle difference \[Δθ = θ_1 - θ_2,\] and its magnitude, the torsion quantity \[τ = |Δθ|,\] provides a geometric measure of non-algebraicity. The multiplication structure of dual complex numbers aligns naturally with the wedge product in Hodge theory, enabling a consistent angle-based description of $(p,q)$-forms. Under the condition Δθ = 0, the angle structures coincide, indicating compatibility between algebraic and analytic components; when Δθ ≠ 0, the mismatch reflects the failure of a Hodge class to correspond to an algebraic cycle. This approach offers a new geometric perspective on the Hodge conjecture by evaluating algebraicity through angle structures in the dual complex plane. 本論文は,対複素数体系における二重複素平面構造および偏角・偏角差・捩れ量の概念を用いて,Hodge 予想の幾何的側面を再解釈するための枠組みを与えるものである。対複素数は,実部複素平面と虚部複素平面という二枚の複素平面に同時に現れる「二重点」として定義され,それぞれに独立な偏角構造と偏角差構造を持つ。この二重構造により,従来の複素多様体上の $(p,q)$-型分解に対応する「側性」と「捩れ」を,角度的・幾何的量として明示的に記述することが可能となる。 本研究では,まず対複素数の可換的基礎層における六構造(実部複素平面・虚部複素平面・実数偏角平面・虚数偏角平面・実数偏角差・虚数偏角差)