A central simplification for codimension 111 cycles is provided by the Lefschetz (1,1)(1,1)(1,1)-theorem, which states:
H1,1(X)H2(X,Z)=Pic(X),H^{1,1}(X) \cap H^2(X,\mathbb{Z}) \;=\; \operatorname{Pic}(X),H1,1(X)H2(X,Z)=Pic(X),
for any smooth projective variety XXX over C\mathbb{C}C. In other words, every integral cohomology class of type (1,1)(1,1)(1,1) is the class of an algebraic divisor. This result establishes the Hodge Conjecture for divisors and shows that, in codimension 111, the conjecture is not only true but structurally guaranteed by the geometry of projective varieties.
1. Application to EEE \times EEE
For X=EEX = E \times EX=EE, we computed in Section III.A that
h1,1(X)=4.h^{1,1}(X) = 4.h1,1(X)=4.
A basis for H1,1(X)H^{1,1}(X)H1,1(X) can be represented by the classes of the following divisors:
a. Horizontal divisor: Dh=E{p}D_h = E \times \{p\}Dh=E{p}.
b. Vertical divisor: Dv={p}ED_v = \{p\} \times EDv={p}E.
c. Diagonal divisor: ={(x,x)EE}\Delta = \{(x,x) \in E \times E\}={(x,x)EE}.
d. Anti-diagonal or correction class: DhDv\Delta - D_h - D_vDhDv, completing the basis.
By the Lefschetz (1,1)(1,1)(1,1)-theorem, these four classes span the entire group H1,1(X)H2(X,Z)H^{1,1}(X) \cap H^2(X,\mathbb{Z})H1,1(X)H2(X,Z). Thus:
NS(X)QH1,1(X)H2(X,Q),\operatorname{NS}(X) \otimes \mathbb{Q} \;\cong\; H^{1,1}(X)\cap H^2(X,\mathbb{Q}),NS(X)QH1,1(X)H2(X,Q),
where NS(X)\operatorname{NS}(X)NS(X) is the Nron--Severi group of XXX.
2. Closure in the CAS-6 framework
