The standard way to define integration on a smooth manifold is to use partitions of unity, to extend to the case where the form you're integrating isn't supported on just one coordinate patch. Of course, in the analytic/holomorphic case, we don't have partitions of unity. So how do we do integration?

Furthermore, how does this affect the space $\mathcal{T}(M)$ of analytic vector fields (analytic global sections of the tangent bundle)? The usual extension lemma for smooth sections of a vector bundle depends on partitions of unity, so there doesn't seem to be any reason you should always be able to find an nonzero analytic section. Does it ever happen that $\mathcal{T}(M) = 0$?

Also in that vein - I remember needing to use partitions of unity to prove in an exercise that the space of 1-forms $\mathcal{T}^*(M)$ is actually the dual $C^\infty(M)$ module $\, \text{Hom}(\mathcal{T}(M),C^\infty(M))$ - because given a map $\mathcal{T}(M) \to C^\infty(M)$, it seems like you need some kind of extension lemma to construct a 1-form that induces it. Does this fail in the analytic case?

I imagine the answer to the previous two questions will involve sheaves, but I don't quite know enough about sheaves to frame them in the appropriate sheaf-theoretic way. Maybe per this question, I should ask if this means the sheaves are not soft and the sheaf cohomology doesn't vanish?

And in general, any good references (or just explanations you care to give) on the major differences between the smooth and analytic cases? It seems like most differential geometry books pay almost no attention to the analytic case.