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.
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 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 .
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 positive 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 Griffiths-positive curvature. A complex manifold admitting a positive line bundle is kähler.
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.
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.
- Bivector (complex)
- Deformation Theory#Deformations of complex manifolds
- Complex analytic space
- Several complex variables
- Complex projective space
- List of complex and algebraic surfaces
- Enriques–Kodaira classification
- Kähler manifold
- Stein manifold
- 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
- Holomorphic Higgs pairs
- This last condition is automatic for a noetherian scheme or a compact complex manifold.
- Kobayashi–Nomizu & 1996 Ch XII
- Huybrechts, Daniel (2005). Complex Geometry: An Introduction. Springer. ISBN 3-540-21290-6.
- Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523
- Hörmander, Lars (1990) , An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, ISBN 0-444-88446-7, MR 1045639, Zbl 0685.32001
- S. Kobayashi, K. Nomizu. Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2.