DOI: 10.6084/m9.figshare.30121507
title: Algebraic discriminator and TYZ theory

description: This paper presents a complete proof of the Hodge Conjecture, a central challenge in algebraic geometry. The proof revolves around a reductio ad absurdum, demonstrating a structural contradiction arising from the assumption that there exists a (p,p)-type Hodge class of rational numbers that cannot be expressed as an algebraic cycle. The existence of such a class implies the existence of a nonzero primitive transcendental component α_tr.
The core of the proof is to introduce a new analytical device for determining algebraic or transcendental properties of a given (p,p)-form α. We define a new Hermitian form ⟨·,·⟩_α using the form α on the vector space H^0(M, L^k) of global regular sections over powers L^k of a ample line bundle L.
* Algebraic discriminator
This Hermitian form, which we will call **'Algebraic Discriminator'** in this paper, has a stable positive-definite property when α is algebraic, but it has a structurally unstable indefinite property when it includes a non-zero transcendental component α_tr.
We prove that transcendental states cause indefiniteness by constructing a special regular section called a **'peak section'**. A peak section is a section that can concentrate its L², mass in an arbitrarily small specific region on the manifold, and its existence is strictly guaranteed by the asymptotic properties of the Szegő kernel, a result known as the Tian-Yau-Zelditch (TYZ) expansion [7, 8, 9]. The final contradiction is derived through a topological argument. The set of algebraic classes is dense in the Hodge class space. Furthermore, the set of all forms α such that the algebraic discriminant ⟨·,·⟩_α is **positive semi-definite** is topologically **closed**. Since all algebraic classes lie in this closed set, the harmonic representative of any Hodge class, which is a limit point of the convergent sequence of algebraic classes, must also lie in this closed set. However, if this representative has a nonzero transcendental component, the discriminant is indefinite, with negative eigenvalues, and thus can never be positive semi-definite. This contradiction implies that the initial assumption is false, and therefore leads to the conclusion that every rational Hodge class must be algebraic.
* Peak Section
 It is a key tool for proving the pathological properties of transcendental states. It is a new concept of regular sections whose existence is rigorously proven using the asymptotic properties of the Szegő Kernel (aka Tian-Yau-Zelditch expansion). The peak sections have the property that their L²-mass can be concentrated in an arbitrarily small specific region on the manifold in a desired proportion. Using these sections, we show that the inner product defined in the transcendental form $\alpha_{tr}$ is necessarily indefinite, having both positive and negative values.
* Degeneracy locus
 This is a topological concept introduced to resolve the circular logic of proofs. It is defined as the set of all states α for which the "algebraic discriminator" degenerates, i.e., its determinant det(G(α)) becomes 0.
 &gt; Degeneracy locus K_p = { α | det(G(α)) = 0 }
 &gt; This set has the crucial property that it is topologically a **closed set**. A "closed set" means that the limit points of a convergent sequence of points inside the set can never go outside the set.
 &gt;
The proof is completed by combining these three tools. First, using the "peak section," we show that the existence of a nonzero transcendental form negates the "algebraic discriminant," which in turn causes the determinant of the discriminant to be zero. This implies that all transcendental states lie within the "degeneracy locus."
Finally, all algebraic classes lie outside the "degeneracy locus" because the determinant of the discriminant is non-zero. Given the density of the algebraic classes, a convergent sequence can be constructed from them. However, since the "degeneracy locus" is a closed set, the limits of such a convergent sequence cannot fall within the locus. Therefore, the harmonic representative of any rational Hodge class must lie outside the bound, which means that the class cannot have a transcendental component. This proves the Hodge conjecture.