Where can I find a rigorous definition of the integration of a complex differential form over a complex mannifold

Question

I'm searching for a book that defines the integral of a complex differential form over a complex manifold.

I found it for the real case, but that's not want I need. In all the books I read so far they only used the integration of forms without defining it. I also read this post but all the recommended books gave only the real case or didn't define it exact.

Thanks for your help

I have not personally read it, but Lee has a book on complex manifolds. I'd look there. — Adil Raza, May 26 '25 at 17:25
The definition is on page 103 of my complex manifolds book. There's nothing fancy about it -- just integrate the real and imaginary parts of the form separately, and add them with a factor of $i$ in the appropriate place. — Jack Lee, May 26 '25 at 20:18

score 4 · Accepted Answer · answered May 26 '25 at 20:20

In a local chart you can integrate $(n,n)$ forms, i.e. forms of the form $$\sum_k \phi_k dz^k\wedge d\bar{z}^k $$ by noting the formula $dz^k\wedge d\bar{z}^k = -2i dx^k\wedge dy^k$ and by then using the definition of the integral for real differential forms. The general case is then handled by patching it together with a partition of unity, subordinate to some suitably chosen atlas. Note that complex manifolds are automatically orientable.

See, e.g. chapter 8 of https://www.math.stonybrook.edu/~cschnell/pdf/notes/complex-manifolds.pdf if you want to see the reason for this and a more detailed exposition.

Where can I find a rigorous definition of the integration of a complex differential form over a complex mannifold

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