Sheaf of meromorphic functions on a complex manifold (${\mathcal{K}}_X$)

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 ${\mathcal{K}}_X^{\ast }$ 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.

The quotient sheaf ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$ records the zero and pole orders of local nonzero meromorphic functions. Each of its global sections is naturally a divisor on a complex manifold on $X$.

The connecting homomorphism ofsends a divisor on a complex manifold $D$ to its holomorphic line bundle associated to a divisor $\mathcal{O}(D)$ in the Picard group. Its kernel consists exactly of the principal divisors of global nonzero meromorphic functions.

At a point $x$ of a Riemann surface, the principal part of a meromorphic function is the negative-degree part of its Laurent series in a local coordinate. It is equivalently a germ in the quotient ${\mathcal{K}}_x/{\mathcal{O}}_x$.

The principal part at a pole is the finite sum of negative-degree terms of its Laurent series. Prescribing such parts is the problem solved by the Mittag-Leffler theorem.

On a Riemann surface, the quotient sheaf $\mathcal{P}={\mathcal{M}}_X/{\mathcal{O}}_X$ records the finite negative Laurent series tails of meromorphic germs. At $x$, a local coordinate $t$ identifies its stalk with ${⨁}_{m\ge 1}\mathbb{C}{t}^{-m}$. As a sheaf it is the direct sum of sheaves ${⨁}_x(i_x{)}_{\ast }{\mathcal{P}}_x$. Global sections prescribe locally finite families of principal parts, possibly infinite on a noncompact surface. The connecting homomorphism to $H^1(X,{\mathcal{O}}_X)$ is exactly the obstruction to a global meromorphic function with those parts and no extra poles.

The Mittag-Leffler problem asks for a global meromorphic function with prescribed principal parts at a discrete collection of points. For a finite collection, those principal parts define a Čech cocycle for the sheaf $\mathcal{O}$; vanishing of its class in ${\check{H}}^1(X,\mathcal{O})$ is exactly the condition that the problem has a solution.