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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.

A Riemannian metric $g$ on a complex manifold is a Kähler metric when its complex-linear extension is a Hermitian form on each tangent space and its fundamental two-formis closed. Equivalently, $\omega$ is a positive real closed $(1,1)$-form, making $X$ a Kähler manifold.

The Lefschetz operator of a Kähler manifold and its adjoint areBecause $d\omega =0$ and $\omega$ has type $(1,1)$, both $\partial \omega$ and $\bar{\partial }\omega$ vanish. The graded Leibniz rule therefore gives $[L,\partial ]=[L,\bar{\partial }]=0$.

Writing formal adjoints with stars, define the three Laplacians byThe supplied Kähler identities identity $[\Lambda ,\partial ]=i{\bar{\partial }}^{\ast }$ gives, by complex conjugation and taking adjoints,Expanding these commutators and using $\partial \bar{\partial }+\bar{\partial }\partial =0$ shows that the mixed terms in $\Delta$ vanish and that ${\Delta }_{\partial }={\Delta }_{\bar{\partial }}$. Since $d=\partial +\bar{\partial }$, it follows that

The adjoint of $[L,\partial ]=0$ gives $[\Lambda ,{\partial }^{\ast }]=0$. Hencewhere the last equality is the adjoint of $\partial \bar{\partial }+\bar{\partial }\partial =0$. Thus $\Lambda$, and therefore every power ${\Lambda }^k$, commutes with ${\Delta }_{\partial }$. It follows that ${\Lambda }^k$ sends every $\partial$-harmonic $(p,q)$-form to a $\partial$-harmonic $(p-k,q-k)$-form whenever $0\le k\le \min (p,q)$.

For a rank-two holomorphic vector bundle on a nonempty complex manifold, this map from ${\mathrm{Pic}}_{\mathrm{hol}}(X)\times \mathbb{Z}$ to ${\mathrm{Pic}}_{\mathrm{hol}}(\mathbb{P}(E))$ is injective. Fibre degree detects $n$. If $p^{\ast }M$ is trivial, a nonvanishing holomorphic section is constant along every compact projective fibre in a local bundle chart, so it descends to a nonvanishing section of $M$. The proof requires no global section of $p$.

A smooth real function on a complex manifold is pluriharmonic if $\partial \bar{\partial }f=0$. A complex-valued function satisfies the same equation exactly when both real and imaginary parts are pluriharmonic. Restrictions to complex coordinate lines are harmonic functions. On a compact connected complex manifold, choose a maximum of $|f|$ and apply the harmonic maximum principle successively along coordinate lines; the function is constant on a polydisc, then on the manifold by open-and-closed propagation. Without connectedness, constancy is componentwise.

A polydisc is a product of open discs in $\mathbb{C}$. It is a contractible complex manifold, with vanishing positive-degree de Rham cohomology and vanishing Dolbeault cohomology in positive antiholomorphic degree.

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.

On a complex manifold, this sheaf consists of smooth differential forms of type (p, q). The type decomposition of the complexified tangent bundle gives ${\mathcal{A}}^k={⨁}_{p+q=k}{\mathcal{A}}^{p,q}$.

On a complex manifold, ${\Omega }_X^p$ is the sheaf of holomorphic degree-$p$ forms. It is the kernel of the Dolbeault operator on ${\mathcal{A}}^{p,0}$. The Dolbeault theorem computes its sheaf cohomology using smooth forms.

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.

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.

A complex manifold is Stein if its global holomorphic functions separate points, give local coordinates, and make it holomorphically convex: the holomorphic hull of every compact set is compact. Closed complex submanifolds of affine complex space are examples, including products of copies of $\mathbb{C}$ and ${\mathbb{C}}^{\ast }$. Cartan theorem B makes these spaces useful for acyclic covers in sheaf cohomology.

For a Hermitian metric on a complex manifold, the Chern connection on its holomorphic tangent bundle corresponds to a real metric-compatible connection preserving the almost complex structure. Its mixed torsion vanishes, and in holomorphic coordinates its remaining coefficients are ${\Gamma }_{ij}^k=h^{k\bar{l}}{\partial }_i{h}_{j\bar{l}}$. Symmetry in $i,j$ is equivalent to $\partial \omega =0$; its conjugate gives $d\omega =0$. Thus zero torsion is exactly the Kähler metric condition, and then uniqueness identifies this connection with the Levi-Civita connection.

On a compact connected complex manifold, pair nonzero holomorphic sections of a holomorphic line bundle and its holomorphic dual vector bundle. Their product is a global holomorphic function, hence constant by the maximum modulus principle. The identity theorem makes their nonvanishing loci dense, so this constant is nonzero. Both sections are consequently nowhere zero and trivialize their bundles. Conversely, the constant section of a trivial bundle and its dual provide the two sections.

A unitary representation of the fundamental group of a connected complex manifold gives a bundle $(\tilde{X}\times V)/{\pi }_{1}X$ with locally constant unitary transition matrices. Its induced flat unitary connection has $(0,1)$ part equal to its Dolbeault operator, so it is the Chern connection for the induced Hermitian metric. On a compact Riemann surface, zero curvature gives degree zero; the subbundle curvature inequality makes the bundle semistable.