 Sheaf of meromorphic functions on a complex manifold

The sheaf ${\mathcal{K}}_X$ assigns to an open subset of a complex manifold its ring of meromorphic functions, with ordinary restriction maps. Its multiplicative subsheaf ${\mathcal{K}}_X^{\ast }$ consists of the units of that ring: functions that are not identically zero on any connected component. Their reciprocals are again meromorphic functions.

This condition permits zeros and poles. For example $z$ is a unit in the meromorphic function ring of $\mathbb{C}$, with inverse $1/z$, although it is not a unit in the holomorphic function ring there. The subsheaf ${\mathcal{O}}_X^{\ast }$ consists instead of nowhere-zero holomorphic functions.^[ref]

On a disconnected open set, merely requiring a section to be nonzero is insufficient: a function equal to one on one component and zero on another restricts to the zero section on the latter. Requiring invertibility on every component makes the multiplicative collection a sheaf and is what the divisor sheaf on a complex manifold uses.