In the case of a real smooth manifold the tangent space (at a point) as the set of functionals that take directional derivatives in the local coordinates is an invariant object, in the sense that it does not depend on the coordinates (although the correspondence between the direction as a vector in $\mathbb{R}^{n}$ and a tangent vector depends). However, the set of the functionals that take second derivatives $\sum a_{ij}\frac{d^2}{dx_idx_j}$ depends on the coordinates. In order to make it independent we have to consider all functionals of the form $\sum a_{ij}\frac{d^2}{dx_idx_j}+\sum a_{k}\frac{d}{dx_k}$.
It seems that for the complex case the set of "second derivatives" of the form $\sum a_{ij}\frac{d^2}{dz_id\overline{z_j}}$ is independent on the coordinates, and moreover looks like a tensor product of the holomorphic and antiholomorphic tangent spaces. I understand that this is somehow related to the fact that the exterior derivative decomposes into complex-linear and anilinear parts, but as you can see I don't really know what I am talking about.
Is the last statement correct? Any notation or a name for a map that takes an element in $T^{1,0}$ and an element in $T^{0,1}$ and produces the corresponding "second derivative"?
What is the big picture? Any intuition behind the difference between the real and the complex case?
Where can I read about... Wirtinger calculus on manifolds, I suppose?
Sorry for this dilettantism, I am frantically trying to fill a gap in my education.
Thank you