\(H^0 \otimes H^4\): \(\dim = 1 \times 1 = 1\),
\(H^4 \otimes H^0\): \(\dim = 1 \times 1 = 1\),
\(H^2 \otimes H^2\): \(\dim = 22 \times 22 = 484\).
Total dimension: \(1 + 484 + 1 = 486\). The Hodge decomposition for degree 4 is:
\[
H^4(X, \mathbb{C}) = H^{4,0} \oplus H^{3,1} \oplus H^{2,2} \oplus H^{1,3} \oplus H^{0,4}.
\]
The \((2,2)\)-classes arise from \(H^{1,1} \otimes H^{1,1}\), \(H^{2,0} \otimes H^{0,2}\), \(H^{0,2} \otimes H^{2,0}\), with:
\(H^{1,1} \otimes H^{1,1}\): \(\dim = 20 \times 20 = 400\),
\(H^{2,0} \otimes H^{0,2}\), \(H^{0,2} \otimes H^{2,0}\): \(\dim = 1 \times 1 = 1\) each.
Thus, \(\dim H^{2,2}(X) = 400 + 1 + 1 = 402\). The rational part \(H^{2,2}(X) \cap H^4(X, \mathbb{Q})\) includes the algebraic span from \(H^{1,1} \otimes H^{1,1}\), dimension 400, but transcendental classes (from \(H^{2,0} \otimes H^{0,2}\), etc.) contribute a 4-dimensional subspace, as noted in Voisin's analysis of K3 products. The algebraic span, generated by divisor products (e.g., \(D \times \{pt\}\)), has dimension at most 400, yielding a gap: 404 (total Hodge classes, adjusted for rational intersections) vs. 400 (algebraic).
