In complex geometry, we have $(p,q)$form $\in$ $\wedge^{p,q}T^*X$, I wonder how to define their integration on submanifold, or top-form on all manifold. For instance, in the Riemann surface book I read, the author let me calculate an integration:
Prove $(1,1)$-form $\omega=\frac{i}{2\pi}(1+|z|^2)^{-2}dz\wedge d\bar{z}$ could be extend as a smooth $(1,1)$-form on Riemann sphere $S$, and $\int_S\omega=1$.
In fact, I have no idea on all parts of this question, but the crucial problem is the author even do not wrote how to integral a complex differential form on a complex manifold. I wonder how to calculate it.