Complex geometry

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In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis.

Throughout this article, "analytic" is often dropped for simplicity; for instance, subvarieties or hypersurfaces refer to analytic ones. Following the convention in Wikipedia, varieties are assumed to be irreducible.

1 Definitions
2 Line bundles and divisors
3 Complex vector bundles
4 Methods from harmonic analysis
5 Vanishing theorem
6 See also
7 Notes
8 See also
9 References

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.

Let $\operatorname{Pic}(X)$ 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 $H^1(X, \mathcal{O}^*)$ . Taking the short exact sequence of

$0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^* \to 0$

where the second map is $f \mapsto \exp (2\pi i f)$ yields a homomorphism of groups:

$\operatorname{Pic}(X) \to H^2(X, \mathbb{Z}).$

The image of a line bundle $\mathcal{L}$ under this map is denoted by $c_1(\mathcal{L})$ and is called the first Chern class of $\mathcal{L}$ .

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

$D = \sum a_i V_i, \quad a_i \in \mathbb{Z}$

that is locally a finite sum.^[1] The set of all divisors on X is denoted by $\operatorname{Div}(X)$ . It can be canonically identified with $H^0(X, \mathcal{M}^*/\mathcal{O}^*)$ . Taking the long exact sequence of the quotient $\mathcal{M}^*/\mathcal{O}^*$ , one obtains a homomorphism:

$\operatorname{Div}(X) \to \operatorname{Pic}(X).$

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 $\pi: E \to X$ is the Chern class of the bundle. By definition, it is a sequence $c_1, c_2, \dots$ such that $c_i(E)$ is an element of $H^{2i}(X, \mathbb{Z})$ and that satisfies the following axioms:^[2]

$c_i(f^*(E)) = f^*(c_i(E))$ for any differentiable map $f: Z \to X$ .
$c(E \oplus F) = c(E) \cup c(F)$ where F is another bundle and $c = 1 + c_1 + c_2 + \dots.$
$c_i(E) = 0$ for $i > \operatorname{rk}E$ .
$-c_1(E_1)$ generates $H^2(\mathbb{C}\mathbf{P}^1, \mathbb{Z})$ where $E_1$ is the canonical line bundle over $\mathbb{C}\mathbf{P}^1$ .