Complex geometry

Definitions

An analytic subset of a complex-analytic manifold M is locally the zero-locus of some family of holomorphic functions on M. It is called an analytic subvariety if it is irreducible in the Zariski topology.

Line bundles and divisors

Throughout this section, X denotes a complex manifold. Accordance with the definitions of the paragraph "line bundles and divisors" in "projective varieties", let the regular functions on X be denoted O and its invertible subsheaf O ∗ .　And let　 M X be the sheaf on X associated with U ↦ the total ring of fractions of Γ ( U , O X ) , where U i are the open affine charts. Then a global section of M X ∗ / O X ∗ (* means multiplicative group) is called a Cartier divisor on X.

Let Pic ⁡ ( X ) be the set of all isomorphism classes of line bundles on X. It is called the Picard group of X and is naturally isomorphic to H 1 ( X , O ∗ ) . Taking the short exact sequence of

where the second map is f ↦ exp ⁡ ( 2 π i f ) yields a homomorphism of groups:

The image of a line bundle L under this map is denoted by c 1 ( L ) and is called the first Chern class of L .

A divisor D on X is a formal sum of hypersurfaces (subvariety of codimension one):

that is locally a finite sum. The set of all divisors on X is denoted by Div ⁡ ( X ) . It can be canonically identified with H 0 ( X , M ∗ / O ∗ ) . Taking the long exact sequence of the quotient M ∗ / O ∗ , one obtains a homomorphism:

A line bundle is said to be positive if its first Chern class is represented by a closed positive real ( 1 , 1 ) -form. Equivalently, a line bundle is positive if it admits a hermitian structure such that the induced connection has Griffiths-positive curvature. A complex manifold admitting a positive line bundle is kähler.

The Kodaira embedding theorem states that a line bundle on a compact kähler manifold is positive if and only if it is ample.

Complex vector bundles

Let X be a differentiable manifold. The basic invariant of a complex vector bundle π : E → X is the Chern class of the bundle. By definition, it is a sequence c 1 , c 2 , … such that c i ( E ) is an element of H 2 i ( X , Z ) and that satisfies the following axioms:

1. c i ( f ∗ ( E ) ) = f ∗ ( c i ( E ) ) for any differentiable map f : Z → X .
2. c ( E ⊕ F ) = c ( E ) ∪ c ( F ) where F is another bundle and c = 1 + c 1 + c 2 + … .
3. c i ( E ) = 0 for i > rk ⁡ E .
4. − c 1 ( E 1 ) generates H 2 ( C P 1 , Z ) where E 1 is the canonical line bundle over C P 1 .

If L is a line bundle, then the Chern character of L is given by

More generally, if E is a vector bundle of rank r, then we have the formal factorization: ∑ c i ( E ) t i = ∏ 1 r ( 1 + η i t ) and then we set

Methods from harmonic analysis

Some deep results in complex geometry are obtained with the aid of harmonic analysis.

Vanishing theorem

There are several versions of vanishing theorems in complex geometry for both compact and non-compact complex manifolds. They are however all based on the Bochner method.