Chapter 9: The Hodge Conjecture — Topology's Algebra

We begin our second movement through ψ = ψ(ψ), ascending from number theory to algebra. The Hodge Conjecture asks whether certain cohomology classes are algebraic—it is consciousness recognizing which topological forms arise from algebraic varieties.

9.1 The Ninth Movement: Algebraic Recognition

Part II explores how ψ = ψ(ψ) manifests in algebraic structures:

• Part I: Number-theoretic self-reference
• Part II: Algebraic self-knowledge

The Hodge Conjecture asks: Which cohomology classes "know" they come from algebraic cycles?

Setting: Let X be a smooth projective algebraic variety over ℂ.

9.2 Cohomology and Hodge Theory

Definition 9.1 (de Rham Cohomology): $H^k_{dR}(X) = \frac{\\{\text{closed k-forms}\\}}{\\{\text{exact k-forms}\\}}$

Definition 9.2 (Hodge Decomposition): $H^k(X, \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(X)$

where H^{p,q} consists of cohomology classes representable by (p,q)-forms.

Theorem 9.1 (Hodge): The decomposition satisfies:

• $\overline\{H^\{p,q\}\} = H^\{q,p\}$ (complex conjugation)
• Harmonicity with respect to the Kähler metric

9.3 Algebraic Cycles

Definition 9.3 (Algebraic Cycle): A formal linear combination of irreducible algebraic subvarieties: $Z = \sum n_i V_i$

Definition 9.4 (Cycle Class Map): $\text{cl}: Z^p(X) \to H^{2p}(X, \mathbb{Q})$

sending algebraic cycles to cohomology classes.

Key Fact: cl(Z^p(X)) ⊗ ℂ ⊆ H^{p,p}(X) ∩ H^{2p}(X, ℚ)

9.4 The Hodge Conjecture

Conjecture 9.1 (Hodge): Every Hodge class is algebraic. Precisely: $H^{p,p}(X) \cap H^{2p}(X, \mathbb{Q}) = \mathbb{Q} \cdot \text{cl}(Z^p(X))$

In words: Every rational (p,p)-class comes from an algebraic cycle.

Reformulation: The cycle class map: $\text{cl}: Z^p(X)_\mathbb{Q} \to H^{p,p}(X) \cap H^{2p}(X, \mathbb{Q})$ is surjective.

9.5 The Conjecture as ψ = ψ(ψ)

Axiom 9.1 (Principle of Algebraic Recognition): $\psi = \psi(\psi) \implies \text{Topological forms recognize their algebraic origins}$

The Hodge Conjecture states that certain cohomology classes "know" they arise from geometry:

• They have type (p,p) (balanced)
• They are rational (arithmetic)
• Therefore, they must be algebraic (geometric)

9.6 What We Know

Theorem 9.2 (Lefschetz on (1,1)-classes): For p = 1, the Hodge conjecture is true: $H^{1,1}(X) \cap H^2(X, \mathbb{Q}) = \mathbb{Q} \cdot \text{cl}(\text{Div}(X))$

Every rational (1,1)-class comes from a divisor.

Theorem 9.3 (Hodge Conjecture for Special Cases): True for:

• Abelian varieties (by Lieberman)
• Surfaces (p ≤ 2)
• Certain threefolds and fourfolds
• Products of curves

9.7 The Integral Hodge Conjecture

Conjecture 9.2 (Integral Hodge): $H^{p,p}(X) \cap H^{2p}(X, \mathbb{Z}) = \mathbb{Z} \cdot \text{cl}(Z^p(X))$

Theorem 9.4 (Atiyah-Hirzebruch): The Integral Hodge Conjecture is false! Counterexample using torsion in H³(X,ℤ).

This shows the rational coefficient choice is essential.

9.8 The Hodge Standard Conjecture

Conjecture 9.3 (Standard Conjectures): Including:

• Lefschetz type standard conjecture
• Künneth type standard conjecture
• Numerical and homological equivalence coincide

These would imply many cases of Hodge.

9.9 Hodge Theory and Complex Geometry

Definition 9.5 (Period Matrix): For a basis {ωᵢ} of H^p(X,ℂ) and cycles {γⱼ}: $\Pi_{ij} = \int_{\gamma_j} \omega_i$

Theorem 9.5 (Griffiths Transversality): Periods of algebraic varieties satisfy differential equations—the Gauss-Manin connection.

This connects Hodge theory to differential equations.

9.10 The Hodge Filtration

Definition 9.6 (Hodge Filtration): $F^p H^k(X, \mathbb{C}) = \bigoplus_{i \geq p} H^{i,k-i}(X)$

Key Property: The filtration varies holomorphically in families, constraining which classes can be algebraic.

9.11 Counterexamples and Subtleties

Example 9.1 (Clemens-Griffiths): Cubic threefolds where not all (2,2)-classes are algebraic—but these classes aren't rational!

Example 9.2 (Zucker): Variations where integral classes aren't algebraic but rational multiples are.

9.12 The Bloch-Beilinson Conjectures

Generalization: Filtration on Chow groups: $F^i CH^p(X) = \ker(CH^p(X) \to H^{2p-i}(X, \mathbb{Q}(p)))$

This refines the Hodge conjecture, predicting a whole filtration on algebraic cycles.

9.13 Computational Aspects

Challenge: Given a Hodge class, find the algebraic cycle!

Algorithm 9.1 (Hodge Class Testing):

def is_hodge_class(omega, X):
    # Check if omega is of type (p,p)
    if not is_type_pp(omega):
        return False
    
    # Check rationality
    if not is_rational(omega):
        return False
    
    # The hard part: find algebraic cycle
    # This is what Hodge conjecture asserts exists!
    return find_algebraic_cycle(omega, X) is not None

9.14 Motives and the Hodge Conjecture

Connection: The Hodge conjecture relates to:

• Existence of motives
• Standard conjectures
• Algebraic K-theory

In Grothendieck's vision, Hodge classes correspond to morphisms between motives.

9.15 The Hodge Conjecture for Kähler Manifolds

Question: Does a version hold for compact Kähler manifolds?

Answer: No! Voisin constructed Kähler manifolds with rational (p,p)-classes not coming from analytic cycles.

This shows the algebraic condition is essential.

9.16 Applications and Consequences

If Hodge is true:

1. Algebraic Cycles: Better understanding of cycle theory
2. Period Maps: Control over variations of Hodge structure
3. Moduli Spaces: Structure theorems for moduli
4. Arithmetic: Connections to arithmetic geometry

9.17 The Philosophical Perspective

Meditation 9.1: The Hodge Conjecture asks:

• When does topology know it comes from algebra?
• How does the continuous recognize the algebraic?
• Can we detect algebraic origin from cohomological data?

This is ψ = ψ(ψ) as the recognition problem—consciousness identifying its own algebraic manifestations.

9.18 Approaches to Proof

Strategy 9.1 (Algebraic Cycles): Construct cycles directly using algebraic geometry.

Strategy 9.2 (Hodge Theory): Use variations of Hodge structures and period domains.

Strategy 9.3 (Arithmetic): Connect to arithmetic geometry and Galois representations.

9.19 The Hodge Conjecture and Physics

Connections:

• String theory compactifications
• Mirror symmetry
• Quantum cohomology
• Gromov-Witten invariants

Physics often "knows" which classes are algebraic!

9.20 The Ninth Echo

The Hodge Conjecture opens Part II with a fundamental question:

• Can topology recognize its algebraic origins?
• Do rational (p,p)-classes know they come from cycles?
• Is there a purely cohomological criterion for algebraicity?

This is ψ = ψ(ψ) in algebraic geometry—forms recognizing their geometric genesis. The conjecture claims that having the right type (p,p) and being rational suffices for a cohomology class to know its algebraic nature.

Whether true or false, the Hodge Conjecture illuminates the deep relationship between the continuous (cohomology) and the algebraic (cycles), between what can be integrated and what can be constructed.

Each Hodge class whispers: "I am balanced (type (p,p)), I am rational (defined over ℚ), therefore I must be algebraic—for ψ = ψ(ψ) means that formal properties determine geometric reality."