Complex submanifold

A complex submanifold is locally the common zero set of holomorphic coordinates, equivalently a smooth submanifold whose tangent spaces are complex linear.

A closed analytic subset of a complex manifold is nonsingular when every point has a neighbourhood on which the subset is the zero set of a holomorphic map of full rank along that zero set. The holomorphic implicit function theorem then makes it a local coordinate subspace. This includes a closed complex submanifold, whereas an arbitrary open piece of a complex submanifold need not be globally closed in its ambient manifold.

Let $Y$ be a embedded submanifold of a complex manifold. If its tangent spaces are invariant under the almost complex structure induced by a complex atlas, choose ambient holomorphic coordinates making its tangent space at one point ${\mathbb{C}}^k\times 0$. The first $k$ coordinates restrict to a local real diffeomorphism by the inverse function theorem, so $Y$ is locally the graph of a smooth function $F$. Invariance of the tangent graph gives $dF(iv)=idF(v)$, the Cauchy-Riemann equations. Thus $F$ is holomorphic and these graph coordinates give the induced complex submanifold structure. This proves the result directly, without needing the general integrability theorem as an unproved step.

For a complex submanifold $Y\subseteq X$, the holomorphic normal bundle is the quotient holomorphic vector bundle

The derivative of a section defining a regular zero locus patches intrinsically along that locus because changes of trivialization introduce derivative terms multiplied by the vanishing section. Its tangent kernel and surjectivity give $N_V\cong E{|}_V$. Combining this with adjunction for a smooth submanifold gives $K_V=(K_M\otimes \det E){|}_V$.

For a smooth complex analytic hypersurface $Y\subset X$, its holomorphic normal bundle is naturally the restriction of its holomorphic line bundle associated to a divisor:If reduced local defining functions satisfy $f_i=g_{ij}{f}_j$, then $d{f}_i{|}_Y=g_{ij}d{f}_j{|}_Y$. Pairing a normal vector with $d{f}_i$ and using the reciprocal transition functions of $\mathcal{O}(Y)$ gives the isomorphism.

The dual of the tangent-normal sequence is the short exact sequence of holomorphic vector bundles

For a complex submanifold $Y\subset X$, the holomorphic conormal bundle is the annihilator of $T^{1,0}Y$ inside ${\Omega }_X^1{|}_Y$, or equivalently the dual of the holomorphic normal bundle. For a smooth analytic hypersurface with local equation $f$, it is generated by $df{|}_Y$. It is also the locally free analytic sheaf ${\mathcal{I}}_Y/{\mathcal{I}}_Y^2$.

For reduced local equations $f_j=u_{ji}{f}_i$ with $u_{ji}$ a holomorphic unit, the holomorphic line bundle associated to a divisor $[-Y]$ has local generators $f_i$ and transition $u_{ji}$. Differentiation along $Y$ gives $d{f}_j{|}_Y=u_{ji}{|}_{Y}d{f}_i{|}_Y$. Sending the restricted local generator to $d{f}_i{|}_Y$ gives the conormal isomorphism. The term $f_{i}d{u}_{ji}$ disappears only after restricting to $Y$.

A complex analytic hypersurface in an $n$-dimensional complex manifold is a closed analytic subset of pure complex codimension one. It is irreducible when it is not the union of two proper closed analytic subsets.

A local defining function for a complex analytic hypersurface $Y$ near $x$ is a holomorphic function $f$ on a neighbourhood $U$ such that $Y\cap U=\{f=0\}$. It may be chosen reduced, with each local irreducible factor occurring once. The local ring ${\mathcal{O}}_{X,x}$ of a complex manifold is a regular local ring and hence a unique factorization domain; each height-one prime of the hypersurface germ is principal, and the product of generators for its finitely many local branches gives $f$.

The local ring of a complex manifold is the unique factorization domain of convergent power series in local coordinates. A reduced local defining function of a complex analytic hypersurface has each local irreducible factor once. The analytic zero-set ideal is the radical of its principal ideal, and reducedness makes that principal ideal radical. Thus two reduced equations for the same germ generate the same ideal and differ by a unit. A globally irreducible hypersurface may have several local branches; reducedness includes each once. Without reducedness, $f$ and $f^2$ have the same zero set and need not differ by a unit.

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 meromorphic functions. Its subsheaf ${\mathcal{K}}_X^{\ast }$ consists of the nonzero meromorphic functions, while ${\mathcal{O}}_X^{\ast }$ consists of the nowhere-zero holomorphic functions.

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.