Divisor on a complex manifold

A divisor on a complex manifold is a locally finite formal sumwhere the $Y$ are irreducible complex analytic hypersurfaces and $n_Y\in \mathbb{Z}$. Products of powers of local defining functions give local meromorphic equations $f_{\alpha }$ for $D$ whose ratios $f_{\alpha }/f_{\beta }$ are nowhere-vanishing holomorphic functions.

At a generic smooth point of an irreducible complex analytic hypersurface $Y$, a nonzero meromorphic function has the form $f=t^{m}u$, where $t$ is a reduced transverse local equation and $u$ is a holomorphic unit. The integer $m$ is its order along $Y$: positive for a zero and negative for a pole. The unique factorization domain property of the local ring of a complex manifold makes this independent of the equation and propagates it along $Y$. These orders are the coefficients of the principal divisor on a complex manifold of $f$.

The divisor on a complex manifold of a nonzero global meromorphic function records its zero and pole orders along irreducible analytic hypersurfaces. It is called principal. Its holomorphic line bundle associated to a divisor is trivial: choosing the same global meromorphic function as the local equation on every open set gives identity transition functions. In particular, two hyperplanes in Complex projective space have isomorphic associated bundles, since the quotient of their homogeneous linear equations is a global meromorphic function with their difference as divisor.

If $f_{\alpha }$ are local meromorphic equations for a divisor on a complex manifold $D$, glue holomorphic frames $e_{\alpha }$ byThe resulting holomorphic line bundle is denoted $[D]$. The expressions $f_{\alpha }{e}_{\alpha }$ glue to its canonical meromorphic section, whose divisor is $D$.

If $f_i$ and $g_i$ are local meromorphic equations for two divisors on a complex manifold, then $f_i{g}_i$ defines their sum. With local frames satisfying $e_j=(f_i/f_j)e_i$, the transition functions of a vector bundle for the tensor product are the products of the two ratios, exactly those for $f_i{g}_i$. Multiplication of local frames therefore gives the displayed holomorphic line bundle isomorphism.

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.