Principal part of a meromorphic function

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.