An elliptic curve \(E\) has dimension 1, with Hodge numbers \(h^{0,0} = h^{1,1} = h^{2,0} = 1\) and \(h^{1,0} = h^{0,1} = 1\). For \(X = E \times E\), a surface of dimension 2, we compute the cohomology using the Knneth decomposition. The total degree-2 cohomology is:
\[
H^2(X, \mathbb{Q}) = H^2(E \times E, \mathbb{Q}) \cong (H^1(E, \mathbb{Q}) \otimes H^1(E, \mathbb{Q})) \oplus (H^2(E, \mathbb{Q}) \otimes H^0(E, \mathbb{Q})) \oplus (H^0(E, \mathbb{Q}) \otimes H^2(E, \mathbb{Q})).
\]
Since \(\dim H^0(E, \mathbb{Q}) = \dim H^2(E, \mathbb{Q}) = 1\) and \(\dim H^1(E, \mathbb{Q}) = 2\), we compute:
\(H^1(E, \mathbb{Q}) \otimes H^1(E, \mathbb{Q})\): Dimension \(2 \times 2 = 4\),
\(H^2(E, \mathbb{Q}) \otimes H^0(E, \mathbb{Q})\): Dimension \(1 \times 1 = 1\),
\(H^0(E, \mathbb{Q}) \otimes H^2(E, \mathbb{Q})\): Dimension \(1 \times 1 = 1\).
Thus, \(\dim H^2(X, \mathbb{Q}) = 4 + 1 + 1 = 6\). The Hodge decomposition gives:
\[
H^2(X, \mathbb{C}) = H^{2,0}(X) \oplus H^{1,1}(X) \oplus H^{0,2}(X),
