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The codimension is the difference between the dimension of a manifold and that of its submanifold. For complex submanifolds, complex codimension one means real codimension two.

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 ${\mathcal{I}}_Y/{\mathcal{I}}_Y^2$.

A holomorphic embedding is a holomorphic map that is also a smooth embedding. Its image is a complex submanifold and the map is a biholomorphism onto that image.

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

Let $X$ be a nonempty positive-dimensional compact complex submanifold of Complex projective space. On a positive-dimensional connected component choose a hyperplane meeting, but not containing, that component. Its linear equation gives a holomorphic section of ${\mathcal{O}}_X(1)$ with a zero and with a nonzero value. For $m>0$, its $m$th power cannot be a section of a trivial holomorphic line bundle, since such sections are constant on the compact connected component. Negative powers are also nontrivial by duality. Every positive-dimensional compact projective component meets every hyperplane: otherwise its affine coordinate functions would all be constant.

The divisor determines the bundle through its local equations. Choose an open cover on which $V$ has a reduced holomorphic defining equation $f_i$, using $f_i=1$ on sets disjoint from $V$. Because $V$ is a complex submanifold of codimension of a submanifold one, its ideal is locally generated by a coordinate; consequently $f_i/f_j$ is a nowhere-zero holomorphic function on each overlap. Inside the sheaf of meromorphic functions on a complex manifold, consider the locally free sheaf of rank oneThese local descriptions agree on overlaps. With $e_i=1/f_i$, its transition functions of a vector bundle are specified byThe ratios satisfy the Čech cocycle condition, and the associated holomorphic line bundle associated to a divisor is $[V]$. The section $1=f_i{e}_i$ is holomorphic and vanishes to order one along $V$. Replacing $f_i$ by $u_i{f}_i$, with $u_i$ a holomorphic unit, replaces the frame by $u_i^{-1}{e}_i$ without changing the subsheaf of meromorphic functions. Refining the cover changes no local sections either. This proves choice independence up to isomorphism rather than merely producing a bundle for one cover.

For a complex manifold of dimension $n$, the canonical bundle isIt is the holomorphic line bundle of holomorphic top-degree differential forms. To compute it on Complex projective space, let $S=\mathcal{O}(-1)$ be the complex tautological line bundle, so that $S^{\ast }=[H]=\mathcal{O}(1)$ is the hyperplane line bundle. At a line $\ell \subset {\mathbb{C}}^{n+1}$, the tangent space is $\mathrm{Hom}(\ell ,{\mathbb{C}}^{n+1}/\ell )$. Tensoring the tautological quotient sequence by $S^{\ast }$ gives the Euler sequence on complex projective spaceTaking the top exterior power gives $\det T^{1,0}{\mathbb{P}}^n=\mathcal{O}(n+1)$; taking its dual bundle yieldsHere a negative tensor power means the corresponding positive tensor power of the dual bundle.

The proposed sections over $U$ form a sheaf: compatible maps into $L$ glue uniquely, and their projections remain $\varphi$. Addition and multiplication by holomorphic functions are defined in the fibres of $L$, making this a sheaf of ${\mathcal{O}}_X$-modules. If $e_i$ is a holomorphic local frame of $L$ on $W_i\subset Y$, every section over $U\subset {\varphi }^{-1}(W_i)$ has the unique expressionThus the sheaf is a locally free sheaf of rank one. Its transition functions of a vector bundle are those of $L$ composed with $\varphi$, so the corresponding pullback vector bundle is precisely ${\varphi }^{\ast }L$. This proves invertibility locally and compatibility globally.

For the local blowup of a complex manifold at a point, work on the projective chart $Y_i\ne 0$ and writeAll the incidence equations reduce to $x_j=u{t}_j$. Therefore $(u,(t_j{)}_{j\ne i})$ gives a holomorphic coordinate chart isomorphic to ${\mathbb{C}}^n$. On an overlap with $Y_{\ell }\ne 0$,which is a biholomorphism where $t_{\ell }\ne 0$. The incidence subset is closed in the product of two Hausdorff spaces and is covered by these charts; hence it is a complex manifold of dimension $n$. In each chart the exceptional fibre is $u=0$, so it is a smooth complex submanifold, with its induced projective coordinates givingAway from the origin the inverse of the projection is $x\mapsto (x,[x])$, which is holomorphic; the projection is therefore a biholomorphism outside $E$.

To find the canonical bundle, pull back the nowhere-zero top form on ${\mathbb{C}}^n$. In the chart above, its Jacobian determinant givesThe sign depends only on the ordering of the coordinates. This is a section of $K_X$ with divisor on a complex manifold exactly $(n-1)E$: it has that vanishing order in every exceptional chart and no zeros elsewhere. A nonzero meromorphic section of a holomorphic line bundle with divisor $D$ identifies its holomorphic line bundle with $[D]$, by sending the local generator $1/f_i$ of $\mathcal{O}(D)$ to the nowhere-zero frame obtained by dividing the section by $f_i$. ConsequentlyThis also includes $n=1$, when the local map is an isomorphism and the exponent is zero.

For a general complex manifold, choose a coordinate neighbourhood $U$ of $P$, with $P$ sent to $0$, and replace $U$ by the inverse image of this coordinate neighbourhood in the local blowup of a complex manifold at a point. Glue this space to $M\setminus \{P\}$ along $U\setminus \{P\}$ using the preceding biholomorphism. The resulting charts give a complex manifold $\tilde{M}$ and a holomorphic map $\varphi$; the exceptional fibre is ${\mathbb{P}}^{n-1}$ and the map is a biholomorphism off that fibre. For completeness, the local projection is proper because its inverse image over a compact subset is closed in that subset times compact Complex projective space. This ensures the gluing is Hausdorff: separate points with distinct images downstairs, and points over $P$ inside the local blowup. The atlas is second countable as well.

The top exterior power of the differential defines a global holomorphic bundle map ${\varphi }^{\ast }{K}_M\to K_{\tilde{M}}$. Equivalently, it is a section of $K_{\tilde{M}}\otimes ({\varphi }^{\ast }{K}_M{)}^{-1}$. The same local Jacobian determinant has vanishing order $n-1$ on $E$, and the differential is invertible elsewhere. Applying the holomorphic line bundle associated to a divisor construction gives the canonical bundle formula for a point blowup

The two opens are $U_1=\mathbb{C}\times {\mathbb{C}}^{\ast }$ and $U_2={\mathbb{C}}^{\ast }\times \mathbb{C}$; their intersection is $({\mathbb{C}}^{\ast }{)}^2$. There is just one degree-one term in the Čech cochain complex, and no degree-two term. The Čech coboundary sends $(g_1,g_2)$ to $g_2-g_1$, soThe denominator denotes the sum of the restricted function spaces.

To make the quotient explicit, expand a holomorphic function on the intersection in a normally convergent two-variable Laurent series,The terms with $m\ge 0$ extend to $U_1$. Of the remaining terms, those with $n\ge 0$ extend to $U_2$. Both subseries converge normally on their stated domains, by the coefficient bounds from the Cauchy integral formula. The unique remaining representative is the doubly negative partNo nonzero series of this form lies in the denominator, since a function on $U_1$ has no negative $z_1$ exponents and a function on $U_2$ has no negative $z_2$ exponents.

Writing $w_i=z_i^{-1}$, these representatives are exactly $w_1{w}_{2}F(w_1,w_2)$ with $F$ an entire holomorphic function on ${\mathbb{C}}^2$. To see that $F$ is entire, integrate for the coefficients on arbitrarily small product circles: $|a_{-i,-j}|\le M(r_1,r_2)r_1^i{r}_2^j$. Choosing $r_i{R}_i<1$ gives absolute convergence for $|w_i|\le R_i$, for every finite pair $R_i$. Conversely every such entire $F$ supplies a normally convergent representative on the intersection. Therefore the quotient consists of convergent doubly negative Laurent series, not merely finite Laurent polynomials:This gives the holomorphic first cohomology of punctured complex two-space; for instance $1/(z_1{z}_2)$ represents a nonzero class.

For the comparison with sheaf cohomology, $U_1,U_2$ and their intersection are Stein manifolds. They can be realized as closed complex submanifolds of affine complex spaces by adding equations $z_i{w}_i=1$ for their nonzero coordinates. Cartan theorem B makes their higher cohomology with coefficients in the sheaf of holomorphic functions vanish. Thus the cover is acyclic for ${\mathcal{O}}_X$, and the acyclic cover theorem givesThe individual cover members are Stein; their union has the nonzero cohomology just computed.

The restriction of $J$ to the $J$-invariant bundle $TY$ is an almost complex structure. Its Nijenhuis tensor is the restriction of the ambient Nijenhuis tensor because vector fields tangent to an embedded submanifold have tangent Lie bracket. The ambient tensor vanishes since $X$ is a complex manifold, so the Newlander-Nirenberg theorem makes the induced structure on $Y$ integrable. The inclusion has complex-linear differential and is therefore holomorphic; hence $Y$ is a complex submanifold.

For a complex submanifold, the holomorphic normal bundle isIf $Y$ is a smooth hypersurface, taking top exterior powers in the holomorphic conormal sequencegivesThe normal bundle of a hypersurface is $N_{Y/X}\cong {\mathcal{O}}_X(Y){|}_Y$, so the adjunction formula is

On $X={\mathbb{CP}}^n\times {\mathbb{CP}}^m$, a bihomogeneous polynomial of bidegree $(d_1,d_2)$ is a section of the holomorphic line bundle $\mathcal{O}(d_1,d_2)$. Its zero locus is smooth precisely when the section is transverse to the zero section, equivalently when $F$ and all of its homogeneous first partial derivatives have no common projective zero. Sincethe adjunction formula yieldsThus a smooth $F$ with $d_1=n+1$ and $d_2=m+1$ defines a complex submanifold with trivial canonical bundle.

On the affine chart $Z_i\ne 0$, divide the homogeneous polynomial $F$ by $Z_i^d$ to obtain a holomorphic function $f_i$ of two affine coordinates. The hypothesis $\mathrm{grad}F(p)\ne 0$ says that at every zero at least one affine partial derivative of $f_i$ is nonzero; the radial derivative contributes nothing on $F=0$ by Euler's homogeneous identity. The holomorphic implicit-function theorem therefore makes $f_i^{-1}(0)$ a complex submanifold of complex codimension one. These local loci agree on chart overlaps, so $X$ is a complex one-dimensional submanifold of ${\mathbb{CP}}^2$.

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.