# Complex integration over a surface

I am trying to define complex integration over a surface. I have found some notes and books here and there but nothing that defines it rigorously.

So suppose we have the complex integral of a function $$f(z)$$ over a surface $$A$$ of $$\mathbb{C}$$. My thought is that we can define:

$$\int \int_A f(z) dz \wedge d\bar{z}$$ as $$\int \int_B f(\gamma(t,s)) | J(\gamma) | dt \wedge ds$$, where $$\gamma: B \subset \mathbb{R}^2 \to A \subset \mathbb{R}^2$$ is a parametrization of the surface and $$J(\gamma)$$ is its Jacobian matrix.

Now I have also seen that $$dz \wedge d\bar{z} = -2i dx\wedge dy$$ where $$z=x+iy$$.

So, suppose we want to calculate the area of a given square $$A=[a,b]\times [ai,bi]$$ in $$\mathbb{C}$$. Then:

$$\int \int_A dz \wedge d\bar{z} = (-2i) \int_{ai}^{bi} \int_{a}^b dxdy = 2(b-a)^2,$$

and we get this annoying $$2$$ on the integral, where is my mistake ? (My point is to understand complex integration over surfaces not just this concrete example). Thanks!

You cannot define the integral of a function over a (Riemann) surface without having a measure, or, more conveniently, a $$2$$-form. If this is a complex $$1$$-manifold, i.e., a Riemann surface, you want to integrate a form of type $$(1,1)$$, as in your example.
The standard area $$2$$-form in the plane is $$\omega=\dfrac i2 dz\wedge d\bar z = dx\wedge dy$$, using coordinates $$z=x+iy$$. Your square should be $$[a,b]\times [a,b]$$ (no $$i$$, as $$y$$ is real here). So your integral is in fact $$-2i$$ times the area, and, as I said, you correct by multiplying by $$i/2$$.
A general Riemann surface will need a Kähler form $$\omega$$ analogous to the form $$\omega$$ we had on $$\Bbb C$$. This is equivalent to having a hermitian metric (just as we have the flat Euclidean metric on the plane). One integrates a complex $$2$$-form no differently from the integration of real $$2$$-forms on smooth $$2$$-manifolds.