Let YYY be a complex K3 surface and X=YYX=Y\times YX=YY. In V.A we factored the rational second cohomology as
H2(Y,Q)=NS(Y)QT(Y),H^2(Y,\mathbb Q)\;=\;\operatorname{NS}(Y)\otimes\mathbb Q\;\oplus\;T(Y),H2(Y,Q)=NS(Y)QT(Y),
and observed that in the maximal-algebraic case =rankNS(Y)=20\rho=\operatorname{rank}\operatorname{NS}(Y)=20=rankNS(Y)=20 one has dimQT(Y)=2\dim_{\mathbb Q}T(Y)=2dimQT(Y)=2, so that
T(Y)QT(Y)T(Y)\otimes_{\mathbb Q} T(Y)T(Y)QT(Y)
contributes a 444-dimensional summand of H2,2(X)H^{2,2}(X)H2,2(X) not accounted for by NS(Y)NS(Y)\operatorname{NS}(Y)\otimes\operatorname{NS}(Y)NS(Y)NS(Y). Section V.B analyses the nature of these transcendental classes and formulates their interpretation and the constructive approaches one might pursue to realize them algebraically.
1. Nature of the transcendental summand
The summand T(Y)T(Y)T(Y) is by definition the orthogonal complement (for the intersection form) of the algebraic part NS(Y)\operatorname{NS}(Y)NS(Y) inside H2(Y,Q)H^2(Y,\mathbb Q)H2(Y,Q). Elements of T(Y)T(Y)T(Y) are cohomology classes that are not Poincar dual to algebraic divisors on YYY. They carry the genuinely transcendental Hodge information of the surface: Hodge classes or Hodge-theoretic phenomena that cannot be detected by the Nron--Severi lattice alone.
On the product X=YYX=Y\times YX=YY the tensor-square T(Y)T(Y)H2(Y)H2(Y)H4(X)T(Y)\otimes T(Y)\subset H^2(Y)\otimes H^2(Y)\cong H^4(X)T(Y)T(Y)H2(Y)H2(Y)H4(X) sits inside the (2,2)(2,2)(2,2)-part of the Hodge decomposition. Thus the four "missing" rational (2,2)(2,2)(2,2)-classes are intrinsically built from transcendental data on each copy of YYY. Concretely, they are not linear combinations of external products of divisor classes; they are classes whose geometric origin, if it exists, must come from correspondences or constructions that reflect the transcendental Hodge structure.
2. Why these classes are delicate for algebraicity
There are several mathematical reasons the T(Y)T(Y)T(Y)\otimes T(Y)T(Y)T(Y) piece is delicate:
Lack of obvious algebraic cycles producing those tensors. External products of divisors exhaust NSNS\operatorname{NS}\otimes\operatorname{NS}NSNS but do not produce mixed tensors lying purely in TTT\otimes TTT. Hence one must look for cycles whose cohomology classes have nontrivial projection onto the transcendental subspace.
Galois/Mumford--Tate rigidity. Transcendental classes are constrained by the Hodge structure and its symmetry group (Mumford--Tate group). Unless there is extra endomorphism structure (e.g. complex multiplication (CM) or unusually small Mumford--Tate group), there are severe symmetry obstructions to producing algebraic correspondences that land exactly on the transcendental tensors.
Interaction with variational Hodge theory. Transcendental classes often vary nontrivially in moduli (they generate nontrivial variation of Hodge structure). Algebraicity would require these classes to be fixed (or to vary inside an algebraic Hodge locus) in a way compatible with a family of algebraic cycles; this is a strong constraint.
These features explain why even a small-dimensional transcendental summand (here: dimension 444) can represent a substantial conceptual obstacle for the Hodge Conjecture.
