= Sheaf of meromorphic functions on a complex manifold
{title2=$\mathcal K_X$}
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 $\mathcal K_X^*$ 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 $\mathcal O_X^*$ consists instead of nowhere-zero <holomorphic functions>.<Complex Analytic and Differential Geometry by Jean-Pierre Demailly>[Chapter II, multiplicative sheaves and exact sequence (6.15), printed p.110]{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.
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