Complex geometry
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 aspects 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.
Definitions
An analytic subset of a complexanalytic manifold M is locally the zerolocus 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
This section may be confusing or unclear to readers. In particular, because of the use of symbols and without definitions. Is the subsheaf of nonvanishing functions of the sheaf of holomorphic functions on X?. (May 2014) 
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 and its invertible subsheaf . And let be the sheaf on X associated with the total ring of fractions of , where are the open affine charts. Then a global section of (* means multiplicative group) is called a Cartier divisor on X.
Let 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 . Taking the short exact sequence of
where the second map is yields a homomorphism of groups:
The image of a line bundle under this map is denoted by and is called the first Chern class of .
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 . It can be canonically identified with . Taking the long exact sequence of the quotient , one obtains a homomorphism:
A line bundle is said to be if its first Chern class is represented by a closed positive real form. Equivalently, a line bundle is positive if it admits a hermitian structure such that the induced connection has 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 is the Chern class of the bundle. By definition, it is a sequence such that is an element of and that satisfies the following axioms:^{}
 for any differentiable map .
 where F is another bundle and
 for .
 generates where is the canonical line bundle over .
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: 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 noncompact complex manifolds. They are however all based on the Bochner method.
See also
 Bivector (complex)
 Deformation Theory#Deformations of complex manifolds
 Complex analytic space
 GAGA
 Several complex variables
 Complex projective space
 List of complex and algebraic surfaces
 Enriques–Kodaira classification
 Kähler manifold
 Stein manifold
 Pseudoconvexity
 Kobayashi metric
 Projective variety
 Cousin problems
 Cartan's theorems A and B
 Hartogs' extension theorem
 Calabi–Yau manifold
 Mirror symmetry
 Hermitian symmetric space
 Complex Lie group
 Hopf manifold
 Hodge decomposition
 Kobayashi–Hitchin correspondence
 Lelong number
 Multiplier ideal
References
 Huybrechts, Daniel (2005). Complex Geometry: An Introduction. Springer. ISBN 3540212906.
 Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 9780471050599, MR 1288523
 Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: NorthHolland, ISBN 0444884467, MR 1045639, Zbl 0685.32001
 S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2.
 E. H. Neville (1922) Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions, Cambridge University Press.
