DRSN XXV: Hodge Conjecture under Spectral Drift (De Rerum Spectrale Natura, Report XXV, Version 1.0)

DRSN XXV: HODGE CONJECTURE UNDER SPECTRAL DRIFT De Rerum Spectrale Natura series REPORT XXV (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro October 2025 •Hodge Conjecture reformulated as a spectral localisation problem. •Harmonic (p, p)-classes analysed via drifted Hodge Laplacian. •Bounded spectral drift preserves Kähler and Hodge structures. •BCH expansion isolates a zero-order Hodge obstruction sector. •Algebraic versus transcendental components separated spectrally. Spectral Localisation of the Hodge Conjecture J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We apply the drifted spectral framework developed in the previous Clay–Perspectives to the Hodge Conjecture. Rather than attempting a proof of the conjecture, we identify the precise spectral sector in which the Hodge obstruction must reside. Starting from the Hodge Laplacian on a smooth projective Kähler manifold, we introduce a bounded real spectral drift preserving the Hodge and Kähler decompositions. The associated Baker–Campbell–Hausdorff expansion isolates a zero-order operator acting on the harmonic ( p, p )-sector. We show that the Hodge Conjecture is equivalent to a spectral coercivity or projection property of this sector, separating algebraic from transcendental cohomology classes. This work localises the Hodge obstruction as a concrete spectral problem and extends the unifying drifted spectral approach to a fifth Clay Millennium Problem. Keywords: Hodge Conjecture; spectral drift; Hodge Laplacian; Kähler geometry; harmonic forms; (p, p)-classes; spectral coercivity; Clay Millennium Problems. CONTENTS I. Introduction and Scope 3 II. Classical Statement of the Hodge Conjecture 3 III. Hodge Laplacian and Harmonic Representatives 4 IV. Spectral Reformulation of the Hodge Problem 4 V. Hilbert Space and Operator-Theoretic Setup 5 VI. Algebraic and Transcendental Components 5 VII. Perspective for Spectral Drift 5 VIII. Real Spectral Drift of the Hodge Laplacian 6 ∗jp[email protected] 3 IX. BCH Expansion and Zero-Order Hodge Sector 6 X. Spectral Coercivity and Projection in the Hodge Sector 7 XI. Comparison with Other Clay Problems 8 XII. Conclusions and Programme 8 A. Kähler Identities and Compatibility of the Drift 9 References 10 I. INTRODUCTION AND SCOPE The Hodge Conjecture concerns the relationship between algebraic geometry and the analytic structure of cohomology on smooth projective varieties. It asserts that certain analytically defined cohomology classes arise from purely algebraic geometric objects. As with the previous Clay–Perspectives, the purpose of this work is not to resolve the conjecture. Rather, we aim to identify the precise spectral locus in which the Hodge obstruction must reside. The guiding principle is that the Hodge problem, like the Navier–Stokes, Yang–Mills, P vs NP, and Riemann problems, admits a formulation in terms of spectral localisation of a distinguished operator acting on a natural Hilbert space. Canonical dependency. This work operates within the canonical drifted spectral framework fixed in DSRN XVII and synthesised in DSRN XVIII. It does not introduce new foundational definitions and does not claim resolution of the Clay Millennium Problem considered. II. CLASSICAL STATEMENT OF THE HODGE CONJECTURE Let X be a smooth complex projective variety of complex dimension n . Denote by Hk ( X, C )its singular cohomology with complex coefficients. Hodge theory provides a decomposition Hk(X, C) = M p+q=k Hp,q(X), where Hp,q(X)consists of harmonic (p, q)-forms with respect to a fixed Kähler metric. Definition 1 (Hodge Classes).A cohomology class α∈H2p ( X, Q )is called a Hodge class if its complexification lies in Hp,p(X). 4 Conjecture 2 (Hodge Conjecture).Every Hodge class in H2p ( X, Q )is a rational linear combination of cohomology classes of algebraic cycles of codimension p. Classically, the conjecture is formulated in terms of algebraic cycles, Chow groups, and the cycle class map. What remains opaque in this formulation is where the obstruction to the conjecture resides analytically. III. HODGE LAPLACIAN AND HARMONIC REPRESENTATIVES Fix a Kähler metric on X . Let Ω p,q ( X )denote the space of smooth ( p, q )-forms and consider the Hodge Laplacian ∆:=dd∗+d∗d. On a Kähler manifold, the Laplacian decomposes as ∆ = 2∆¯ ∂= 2∆∂, and preserves bidegree (p, q). The space of harmonic forms Hp,q(X) := ker(∆) ∩Ωp,q(X) is finite-dimensional and canonically isomorphic to Hp,q(X). In particular, the Hodge classes correspond precisely to elements of Hp,p ( X )with rational periods. IV. SPECTRAL REFORMULATION OF THE HODGE PROBLEM From a spectral point of view, the Hodge Conjecture may be restated as follows: Do all rational harmonic (p, p)-forms lie in the image of the cycle class map? This formulation isolates the analytic object Hp,p ( X )as the natural arena in which the conjecture lives. The Hodge problem thus concerns the internal structure of a zero-eigenvalue sector of an elliptic operator, rather than the existence of a global spectral gap. This observation aligns the Hodge Conjecture naturally with the drifted spectral framework developed for the other Clay problems. 5 V. HILBERT SPACE AND OPERATOR-THEORETIC SETUP We define the Hilbert space H:= L2Ω•(X) of square-integrable differential forms, equipped with the L2 inner product induced by the Kähler metric. The Hodge Laplacian ∆is an essentially self-adjoint, non-negative, elliptic operator on H with discrete spectrum accumulating at infinity. The kernel ker(∆) consists precisely of harmonic forms and decomposes as ker(∆) = M p,q Hp,q(X). Within this kernel, the subspace Hp,p(X)is the target of the Hodge Conjecture. VI. ALGEBRAIC AND TRANSCENDENTAL COMPONENTS Let Hp,p alg ( X ) ⊂ Hp,p ( X )denote the subspace spanned by harmonic representatives of algebraic cycles. Define the orthogonal complement Hp,p tr (X):=Hp,p(X)⊖ Hp,p alg(X), which we call the transcendental (p, p)-sector. Remark VI.1. The Hodge Conjecture is equivalent to the statement that Hp,p tr ( X ) ∩H2p ( X, Q ) = {0}. This decomposition prepares the operator-theoretic localisation of the Hodge obstruction developed in the next section. VII. PERSPECTIVE FOR SPECTRAL DRIFT The Hodge Laplacian ∆provides a canonical self-adjoint spectral object, but its kernel is highly degenerate. As in the Yang–Mills and Riemann cases, we seek to introduce a controlled deformation that preserves the geometric structure while probing the internal spectral organisation of this kernel. 6 In the next section, we introduce a bounded real spectral drift of ∆, compatible with the Kähler decomposition, and analyse the associated Baker–Campbell–Hausdorff hierarchy. This construction will isolate a zero-order operator acting on Hp,p ( X )whose coercivity or projection properties encode the Hodge obstruction. VIII. REAL SPECTRAL DRIFT OF THE HODGE LAPLACIAN Let X∈ B ( H )be a bounded self-adjoint operator acting on L2 -differential forms. We assume that X is chosen to be compatible with the Kähler structure, in the sense that it preserves the (p, q)-decomposition or, more generally, that its action can be controlled within each bidegree. We define the real spectral drift of the Hodge Laplacian by ∆s:= e−sX ∆esX , s ∈R.(VIII.1) Proposition VIII.1 (Structural Invariance under Drift).For all s∈R , the operator ∆ s satisfies: •Dom(∆s) = Dom(∆); •∆sis essentially self-adjoint and non-negative; •ellipticity and the principal symbol of ∆are preserved; •the Kähler identities and Hodge decomposition remain valid. Proof. Bounded similarity preserves domain and self-adjointness. Since X is bounded, conjugation does not affect the highest-order part of ∆, and ellipticity is preserved. Compatibility with the Kähler structure follows from the choice of X. Remark VIII.2. The drift does not lift the degeneracy of the zero eigenvalue. It reorganises the internal spectral geometry of ker(∆) while preserving all geometric symmetries. IX. BCH EXPANSION AND ZERO-ORDER HODGE SECTOR The drifted Laplacian admits a convergent BCH expansion on Dom(∆): ∆s=∆+s C1+s2 2C2+s3 6C3+· · · , Cn:= adn X(∆).(IX.1) Since X is bounded, each Cn is a lower-order operator. In particular, the second-order commutator C2= [X, [∆, X]] 7 is an operator of order zero. It therefore acts as a pointwise analytic potential on differential forms and preserves bidegree. Definition 3 (Zero-Order Hodge Obstruction Sector).The operator C2:= [X, [∆, X]] is called the zero-order Hodge obstruction sector. This operator plays the same structural role as: •the emergent mass sector in Yang–Mills theory; •the asymmetry sector in the Riemann problem; •the cost degeneracy sector in P vs NP. ∆ C1= [∆, X] C2= [X, [∆, X]] C3= ad3 X(∆) . . . Zero-order sector Hodge obstruction FIG. 1. BCH hierarchy of the drifted Hodge Laplacian. The zero-order term C2 isolates the Hodge obstruction sector. X. SPECTRAL COERCIVITY AND PROJECTION IN THE HODGE SECTOR The operator C2 acts naturally on the harmonic sector Hp,p ( X ) ⊂ker (∆). We now formulate the Hodge obstruction as a spectral coercivity or projection property of C2. Definition 4 (Hodge Spectral Coercivity).The zero-order Hodge sector C2 is said to be spectrally coercive on the transcendental subspace Hp,p tr (X)if there exists c > 0such that ⟨ψ, C2ψ⟩ ≥ c∥ψ∥2for all ψ∈ Hp,p tr (X). 8 Proposition X.1 (Hodge Coercivity Criterion).If the zero-order Hodge sector C2 is spectrally coercive on Hp,p tr (X), then Hp,p tr (X)∩H2p(X, Q)={0}. Equivalently, all rational (p, p)-classes arise from algebraic cycles. Proof. Spectral coercivity of C2 excludes the existence of non-zero rational harmonic vectors in the transcendental subspace. Such vectors would otherwise contradict the uniform lower bound. Proposition X.2. Proposition X.1 does not assert that C2 is coercive. It identifies the precise spectral locus in which the Hodge obstruction must reside. Algebraic cycles Harmonic (p, p)-forms Hodge Laplacian ∆Drift ∆sC2sector Coercive ⇒algebraic Degenerate ⇒transcendental Hodge obstruction localised as a zero-order spectral problem. FIG. 2. Spectral localisation of the Hodge Conjecture under drift. The obstruction is concentrated in the coercivity of the zero-order sector C2. XI. COMPARISON WITH OTHER CLAY PROBLEMS We conclude by situating the Hodge Conjecture within the unified drifted spectral programme developed across the Clay–Perspectives. Despite their distinct mathematical origins, the five Clay problems analysed to date admit a common structural reformulation as spectral localisation questions for distinguished operators. XII. CONCLUSIONS AND PROGRAMME We have applied the drifted spectral methodology to the Hodge Conjecture, extending the Clay–Perspective framework to a fifth Millennium Problem. The main conclusions are: 9 Clay problem Domain Spectral object Obstruction localised as Navier–Stokes (3D) PDE / fluid dynamics Dissipative operator Ksin Dz=Hs−iKsLoss of spectral coercivity of Ks Yang–Mills mass gap Quantum gauge theory Drifted covariant Laplacian (∆A)sAbsence of coercivity in the zero-order sector C2 P vs NP Discrete algorithms Cost operator HCon ℓ2(Ωn)Discrete spectral degeneracy / lack of coercivity Riemann Hypothesis Analytic number theory Drifted zeta spectral operator (Hζ)sLoss of spectral symmetry/coercivity in C2 Hodge Conjecture Algebraic geometry Drifted Hodge Laplacian ∆sFailure of coercivity/projection in the (p, p)sector C2 TABLE I. Structural spectral localisation of five Clay problems under spectral drift. In all cases, the obstruction reduces to a failure of coercivity or symmetry for a precisely identified spectral sector. • The Hodge Conjecture admits a natural formulation in terms of the spectral structure of the Hodge Laplacian on a smooth projective Kähler manifold. • Bounded real spectral drift preserves ellipticity, self-adjointness, and Kähler identities while probing the internal organisation of harmonic forms. • The Baker–Campbell–Hausdorff expansion isolates a unique zero-order operator C2 acting on the harmonic (p, p)sector. • The Hodge obstruction is localised as a spectral coercivity or projection problem for this sector, separating algebraic from transcendental components. As in the previous Clay–Perspectives, the present work does not resolve the Clay problem. It identifies, however, a sharply defined spectral locus in which the Hodge obstruction must reside. The programme suggested by this localisation is clear: to analyse the analytic structure of the zero-order Hodge sector C2 and to determine whether it admits the required coercivity or projection properties. What is gained is conceptual precision. The Hodge Conjecture is no longer a diffuse algebrogeometric mystery, but a concrete zero-order spectral localisation problem. Appendix A: Kähler Identities and Compatibility of the Drift For completeness, we briefly record the minimal conditions required for the spectral drift to be compatible with the Kähler structure. Let ( X, ω )be a compact Kähler manifold and let ∆denote the Hodge Laplacian acting on differential forms. The Kähler identities imply ∆ = 2∆∂= 2∆¯ ∂, and ensure that ∆preserves bidegree.