Complex geometry

An almost complex manifold is a smooth manifold $X$ with a bundle endomorphism $J:TX\to TX$ satisfying $J^2=-I$.

The Nijenhuis tensor of an almost complex manifold isIt measures the failure of the $(0,1)$ tangent distribution to be closed under the Lie bracket.

An almost complex structure is integrable when it comes from holomorphic coordinate charts. Equivalently, its Nijenhuis tensor vanishes, or the $(0,1)$ vector fields are closed under Lie bracket.

The Newlander-Nirenberg theorem says that a smooth almost complex structure is integrable if and only if its Nijenhuis tensor vanishes.

A complex $n$-manifold has charts to ${\mathbb{C}}^n$ with holomorphic transition maps. It is a real $2n$-manifold with an integrable almost complex structure.

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

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

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

The complexified cotangent bundle decomposes into $(1,0)$ and $(0,1)$ parts. The Dolbeault operator is the type-$(0,1)$ component of the exterior derivative, $\bar{\partial }:{\Omega }^{p,q}\to {\Omega }^{p,q+1}$, and integrability gives ${\bar{\partial }}^2=0$.

Dolbeault cohomology is

The Hodge number of a compact complex manifold is $h^{p,q}={\dim }_{\mathbb{C}}{H}_{\bar{\partial }}^{p,q}(X)$.

A holomorphic vector bundle is a complex vector bundle whose local trivializations have holomorphic transition functions.

A holomorphic line bundle is a complex line bundle with holomorphic local trivializations and holomorphic nonvanishing transition functions.

A Dolbeault partial connection obeys ${\bar{\partial }}_E(fs)=\bar{\partial }f\otimes s+f{\bar{\partial }}_{E}s$. Its square is a tensorial $(0,2)$-form with values in $\mathrm{End}E$; it vanishes for the canonical partial connection of a holomorphic bundle.

A Hermitian metric on a holomorphic vector bundle $E\to X$ is a smoothly varying positive-definite Hermitian form $h_x$ on every complex fiber $E_x$.

A complex torus is a quotient ${\mathbb{C}}^n/\Lambda$ by a lattice $\Lambda \cong {\mathbb{Z}}^{2n}$. Translation-invariant complex forms descend to the quotient.

A Kähler manifold is a complex manifold with a positive real closed $(1,1)$-form $\omega$. The associated Riemannian metric is $g(u,v)=\omega (u,Jv)$.

The Lefschetz operator of a Kähler manifold is exterior multiplication by its Kähler form,Its formal adjoint is denoted by $\Lambda =L^{\ast }$ and lowers the bidegree of a differential form by $(1,1)$.

For the Lefschetz operator of a Kähler manifold and its adjoint, the Kähler identities include

The Dolbeault Laplacian is ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$. On a Kähler manifold, the Kähler identities imply

The Fubini-Study form is the standard Kähler form on complex projective space. With the integral normalization, $[{\omega }_{FS}/(2\pi )]=c_1(\mathcal{O}(1))$.

For a compact Kähler manifold,and every class has a unique harmonic representative of each type.

The canonical bundle of a complex $n$-manifold is the holomorphic line bundle $K_X={\bigwedge }^n(T^{1,0}X{)}^{\ast }$.

For a smooth hypersurface $X$ in a complex manifold $Y$,

The first Chern class classifies complex line bundles topologically. For a Hermitian holomorphic line bundle with Chern curvature $F_{\nabla }$, Chern-Weil theory gives $c_1(E)=[i{F}_{\nabla }/(2\pi )]$.

The Chern connection is the unique connection on a Hermitian holomorphic bundle compatible with both its metric and holomorphic structure. Its curvature has type $(1,1)$.

The Ricci form is the real curvature form $i{F}_{\nabla }$ of the Chern connection on $T^{1,0}X$. In complex dimension one, $\rho =K\omega =\frac{1}{2}\mathrm{Scal}\omega$.

The blowup replaces a point of a complex $n$-manifold by the projective space of complex tangent directions ${\mathbb{CP}}^{n-1}$. A biholomorphism carrying one center to another lifts to a biholomorphism of their blowups.

The exceptional divisor of the blowup of a complex manifold at a point $p$ is the fiber over $p$. In complex dimension $n$ it is naturally the projective space $\mathbb{P}(T_{p}X)\cong {\mathbb{CP}}^{n-1}$ of tangent directions at the center.

The strict transform of a subvariety $Y$ under a blowup $\sigma :\tilde{X}\to X$ is the closure of ${\sigma }^{-1}(Y\setminus Z)$, where $Z$ is the center. Its total transform also contains the exceptional divisor with the multiplicity with which $Y$ passes through $Z$.