Complex projective plane (${\mathbb{CP}}^2$)

The complex projective plane is the two-dimensional Complex projective space. Its second homology is generated by the class $H$ of a projective line, with $H^2=1$.

The fundamental class of a complex projective line generates $H_2({\mathbb{CP}}^2;\mathbb{Z})$. Two distinct projective lines are homologous and meet at one transverse point. In complex affine coordinates their two complex tangent lines concatenate to the complex orientation of the plane, so the signed intersection is $+1$. The integral intersection form therefore has matrix $(1)$, giving a unimodular intersection pairing. This computes a self-intersection through distinct representatives of the same class.