H2(Y,Q)NS(Y)QT(Y),H^2(Y,\mathbb Q) \;\cong\; \operatorname{NS}(Y)\otimes\mathbb Q \;\oplus\; T(Y),H2(Y,Q)NS(Y)QT(Y),
where NS(Y)\operatorname{NS}(Y)NS(Y) denotes the Nron--Severi group (the algebraic classes of codimension 111) and T(Y)T(Y)T(Y) denotes the transcendental lattice, the orthogonal complement of NS(Y)\operatorname{NS}(Y)NS(Y) inside H2(Y,Q)H^2(Y,\mathbb Q)H2(Y,Q). Set
:=rankNS(Y)(120).\rho \;:=\; \operatorname{rank}\operatorname{NS}(Y) \qquad (1\le\rho\le 20).:=rankNS(Y)(120).
Then dimQNS(Y)Q=\dim_{\mathbb Q}\operatorname{NS}(Y)\otimes\mathbb Q = \rhodimQNS(Y)Q= and
dimQT(Y)=22.\dim_{\mathbb Q} T(Y) \;=\; 22-\rho.dimQT(Y)=22.
4. Contribution of algebraic products and the maximal-algebraic case
A natural algebraic subspace of H2,2(X)H^{2,2}(X)H2,2(X) is provided by the image of the external product of divisor classes:
Im(cl1(CH1(Y)Q)cl1(CH1(Y)Q))H2,2(X).\operatorname{Im}\big( \operatorname{cl}_1(\mathrm{CH}^1(Y)\otimes\mathbb Q)\otimes \operatorname{cl}_1(\mathrm{CH}^1(Y)\otimes\mathbb Q)\big) \;\subseteq\; H^{2,2}(X).Im(cl1(CH1(Y)Q)cl1(CH1(Y)Q))H2,2(X).
Under the decomposition 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), the subspace generated by products of algebraic divisor classes is naturally identified with
NS(Y)QNS(Y)Q,\operatorname{NS}(Y)\otimes\mathbb Q \;\otimes\; \operatorname{NS}(Y)\otimes\mathbb Q,NS(Y)QNS(Y)Q,
whose dimension is 2\rho^22 over Q\mathbb QQ.
