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The cellular homology of complex projective space gives one integral generator $[{\mathbb{CP}}^j]$ in each degree $2j$, for $0\le j\le 2$, and zero in odd degrees. Since these groups are free, the Künneth theorem gives the homology cross product basisThus the ranks in degrees $0,2,4,6,8$ are $1,2,3,2,1$, respectively, and every other integral homology group vanishes.

Let $h$ be the normalized hyperplane class, with $⟨h^2,[{\mathbb{CP}}^2]⟩=1$, and write $a=p_1^{\ast }h$, $b=p_2^{\ast }h$. The cohomology ring of complex projective space and the Künneth theorem give $H^{\ast }({\mathbb{CP}}^2\times {\mathbb{CP}}^2;\mathbb{Z})=\mathbb{Z}[a,b]/(a^3,b^3)$. If $d$ is the diagonal embedding, then $d^{\ast }a=d^{\ast }b=h$. Hence each of $a^2,ab,b^2$ evaluates as one on the diagonal's fundamental class. These three classes are dual to the three degree-four product generators, so the diagonal class of complex projective space givesEquivalently its Poincare dual is $a^2+ab+b^2$. The signs are positive with the complex and product orientations.