Hodge Cycle Explorer: Geometry & Algebraic Varieties Bridge

Hodge (p,q) Diamond Spectrum H²(X, ℚ)

Period Matrix & Integrals ∫_γ ω_i

Integration of harmonic (1,1)-forms over traced topological homology loops:

Cycle (γ)	Re(∫ ω)	Im(∫ ω)	Rational Ratio

Algebraic Subvariety Validation

Target cycle γ is generated by linear combinations of rational algebraic subvarieties: [C] = 1/2 [C₁] + 1/2 [C₂].

LaTeX Proof Representation

\begin{align*} \omega &\in H^{1,1}(X) \cap H^2(X, \mathbb{Q}) \\ \int_{\gamma_{A1}} \omega &= 1.4140 \quad (\approx \sqrt{2}) \\ \int_{\gamma_{B1}} \omega &= 0.7070 \quad (\approx 1/\sqrt{2}) \end{align*}

Proof Surface Identity: Hodge Cycle Explorer Completed State

Export Proof Sheet & Manifold State