Complex manifold

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

Holomorphic normal bundle ( $N_{Y / X}$ )

For a complex submanifold

Y \subseteq X

, the holomorphic normal bundle is the quotient holomorphic vector bundle

N_{Y / X} = TX ∣_{Y} / T Y .

Holomorphic conormal sequence

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

0 ⟶ N_{Y / X}^{*} ⟶ Ω_{X}^{1} ∣_{Y} ⟶ Ω_{Y}^{1} ⟶ 0.

Dolbeault operator ( $\overset{ˉ}{\partial}$ )

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,

\overset{ˉ}{\partial} : Ω^{p, q} \to Ω^{p, q + 1}

, and integrability gives

\overset{ˉ}{\partial}^{2} = 0

Dolbeault cohomology ( $H_{\overset{ˉ}{\partial}}^{p, q} (X)$ )

Dolbeault cohomology is

H_{\overset{ˉ}{\partial}}^{p, q} (X) = ker (\overset{ˉ}{\partial} : Ω^{p, q} \to Ω^{p, q + 1}) / im (\overset{ˉ}{\partial} : Ω^{p, q - 1} \to Ω^{p, q}) .

Hodge number ( $h^{p, q}$ )

The Hodge number of a compact complex manifold is

h^{p, q} = dim_{C} H_{\overset{ˉ}{\partial}}^{p, q} (X)

Holomorphic vector bundle

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

Holomorphic line bundle

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

Dolbeault partial connection ( $\overset{ˉ}{\partial}_{E}$ )

A Dolbeault partial connection obeys

\overset{ˉ}{\partial}_{E} (f s) = \overset{ˉ}{\partial} f \otimes s + f \overset{ˉ}{\partial}_{E} s

. Its square is a tensorial

(0, 2)

-form with values in

End E

; it vanishes for the canonical partial connection of a holomorphic bundle.

Hermitian metric on a holomorphic vector bundle ( $h$ )

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}

Complex torus

A complex torus is a quotient

C^{n} /Λ

by a lattice

Λ ≅ Z^{2 n}

. Translation-invariant complex forms descend to the quotient.

Kähler manifold

A Kähler manifold is a complex manifold with a positive real closed

(1, 1)

-form

ω

. The associated Riemannian metric is

g (u, v) = ω (u, J v)

Lefschetz operator of a Kähler manifold ( $L$ )

The Lefschetz operator of a Kähler manifold is exterior multiplication by its Kähler form,

Lα = ω \land α .

Its formal adjoint is denoted by

Λ = L^{*}

and lowers the bidegree of a differential form by

(1, 1)

Kähler identities

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

[Λ, \partial] = i \overset{ˉ}{\partial}^{*}, [Λ, \overset{ˉ}{\partial}] = - i \partial^{*}, [L, \partial^{*}] = i \overset{ˉ}{\partial}, [L, \overset{ˉ}{\partial}^{*}] = - i \partial .

Dolbeault Laplacian ( $Δ_{\overset{ˉ}{\partial}}$ )

The Dolbeault Laplacian is

Δ_{\overset{ˉ}{\partial}} = \overset{ˉ}{\partial} \overset{ˉ}{\partial}^{*} + \overset{ˉ}{\partial}^{*} \overset{ˉ}{\partial}

. On a Kähler manifold, the Kähler identities imply

Δ_{d} = 2 Δ_{\partial} = 2 Δ_{\overset{ˉ}{\partial}} .

Fubini-Study form ( $ω_{FS}$ )

The Fubini-Study form is the standard Kähler form on complex projective space. With the integral normalization,

[ω_{FS} / (2 π)] = c_{1} (O (1))

Hodge decomposition theorem for compact Kähler manifolds

For a compact Kähler manifold,

H_{d R}^{k} (X; C) = ⨁_{p + q = k} H_{\overset{ˉ}{\partial}}^{p, q} (X),

and every class has a unique harmonic representative of each type.

Canonical bundle ( $K_{X}$ )

The canonical bundle of a complex

n

-manifold is the holomorphic line bundle

K_{X} = ⋀^{n} (T^{1, 0} X)^{*}

Adjunction formula

For a smooth hypersurface

X

in a complex manifold

Y

K_{X} ≅ (K_{Y} \otimes O_{Y} (X)) ∣_{X} .

First Chern class ( $c_{1} (E)$ )

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 π)]

Chern connection

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)

Ricci form of a Kähler manifold ( $ρ$ )

The Ricci form is the real curvature form

i F_{\nabla}

of the Chern connection on

T^{1, 0} X

. In complex dimension one,

ρ = K ω = \frac{1}{2} Scal ω

Blowup of a complex manifold at a point

The blowup replaces a point of a complex

n

-manifold by the projective space of complex tangent directions

CP^{n - 1}

. A biholomorphism carrying one center to another lifts to a biholomorphism of their blowups.

Exceptional divisor ( $E$ )

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

P (T_{p} X) ≅ CP^{n - 1}

of tangent directions at the center.

Strict transform ( $Y$ )