How does one create a partition of unity for a complex manifold? As far as I am aware, partitions of unity for smooth manifolds require the use of smooth functions with compact support (e.g. bump functions). However, for a complex manifold, the transition maps#Transition_maps) have to be not only smooth, but holomorphic. And by the Identity Theorem any holomorphic function with compact support is identically zero. **Question:** How does one surmount this obstacle when working with complex manifolds? Naively it seems to me that because of this the only possible complex manifold to define would be the open unit disk in $\mathbb{C}^n$, i.e. a manifold with only one chart, since Wikipedia says that the atlas of a complex manifold consists of charts to the open unit disk in $\mathbb{C}^n$.

Note that complex manifolds are in particular smooth manifold (holomorphic $\Rightarrow$ smooth). Thus one can define partition or unity as in the smooth case. We cannot have something like "holomorphic" partition of unity, of course.