I know the technical difference between an almost complex manifold and a complex manifold, namely in the former the almost complex structure $J$ may not be integrable while in the later it is. However, in an almost complex manifold one can do many things that can be done in a complex manifold: namely one can still make the Dolbeault decomposition of tensors, and also of the exterior derivative as well as define holomorphic functions. I wonder what are the key differences between the two scenarios, namely what things can be done in a complex manifold that cannot be done in an almost complex manifold because of $J$ being non-integrable.
By the way, could someone recommend me a good book about complex differential geometry?
Thanks.
Best Answer
Here are some things which make sense on complex manifolds, but not almost complex manifolds:
Complex coordinates $(z_1, \ldots, z_n)$. In particular, vectors like $\frac{\partial}{\partial z_k}$ only make sense on a complex manifold.
On complex manifolds, we have $\overline{\partial}^2 = 0$. This gives rise to the Dolbeault complex $$\Omega^{p,0}(M) \xrightarrow{\overline{\partial}} \Omega^{p,1}(M) \xrightarrow{\overline{\partial}} \cdots$$ and therefore to the notion of Dolbeault cohomology $H^{p,q}_{\overline{\partial}}(M)$. That is, Dolbeault cohomology does not make sense on an almost complex manifold. In particular, note that we need $\overline{\partial}^2 = 0$ to ensure that we actually have $$ \overline{\partial}\text{-exact} \implies \overline{\partial}\text{-closed.}$$ For this reason, it's not really worth having notions of "$\overline{\partial}$-exact" or "$\overline{\partial}$-closed" on a manifold which is merely almost complex.
For reasons as in the previous point, the $\overline{\partial}$-Poincaré Lemma and Local $i\partial \overline{\partial}$-Lemma only work on complex manifolds.
Holomorphic vector bundles require both the total space and the base to be complex manifolds. Note that we need holomorphic vector bundles to talk about the Chern connection.
There are probably one or two more points that I'm missing (e.g., regarding Hodge stars and Laplacians (when a Hermitian metric is present)), but I think the above should be the main differences.
As for textbook recommendations, I like these two:
Andrei Moroianu, "Lectures on Kahler Geometry."
Daniel Huybrechts, "Complex Geometry."