Several complex variables and analytic spaces Compact complex manifolds Concept
Complex manifold
Summary
A complex manifold is a space that looks locally like an open subset of , glued together by holomorphic maps. It is the setting for complex analysis beyond domains in , just as a smooth manifold is the setting for calculus. Riemann surfaces are the complex manifolds of dimension ; complex projective space and complex tori are basic compact examples. Compact complex manifolds behave very differently from smooth ones: for instance, on a compact connected complex manifold the only global holomorphic functions are the constants.
Where this leads
62 pages build on this one.
- Holomorphic functions on compact complex manifolds are constant
- Remmert–Stein theorem
- Deformation of complex structure
- q-convex manifold
- First Cousin problem
- Cartan's coherence theorem for ideal sheaves
- Kobayashi metric
- Andreotti–Grauert theorem
- Chow's theorem
- Dolbeault cohomology
- Kobayashi hyperbolic manifold
- Kodaira embedding theorem
- Class VII surface
- Bishop's theorem on analytic sets
- Positive line bundle
- Germ of an analytic set
- Kuranishi family
- Poincaré–Lelong formula
- Chern connection
- Dimension of an analytic set
- Kodaira–Nakano vanishing theorem
- Second Cousin problem
- Complexification of a real analytic manifold
- Stein manifold
- Siegel's theorem on meromorphic function fields
- Irreducible analytic germ
- Nakano positivity
- Holomorphic line bundle
- Holomorphic vector bundle
- Dolbeault isomorphism
- Inoue surface
- Algebraic dimension
- Hironaka's resolution for complex spaces
- Docquier–Grauert theorem
- Griffiths positivity
- Regular and singular points of an analytic set
- Complex geodesic
- Local parametrization theorem
- Riemann–Hilbert correspondence
- Holomorphic dynamics in several variables
- Oka manifold
- Plurisubharmonic exhaustion function
- Carathéodory metric
- Brody hyperbolicity
- Dolbeault operator
- Grauert's theorem on real analytic manifolds
- Hopf surface
- Grauert's solution of the Levi problem for manifolds
- Holomorphic fiber bundle
- Frankel conjecture
- Poincaré problem
- Positive current
- Levi-flat hypersurface
- Siu's semicontinuity theorem
- Complex structure on the six-sphere
- Brody's lemma
- Complex analytic variety
- Kodaira–Nirenberg–Spencer existence theorem
- Fujiki class C
- Multiplier ideal
- Kähler manifold
- Moishezon manifold
Further reading
About this page
| Tag | JSGWAN2RJV |
|---|---|
| URI | m2://JSGWAN2RJV |
| Kind | concept |
| Writing | W0 stub |
| MSC | 32Q99, 53C15 |
| Specializes | Smooth manifold |
| Same as |
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Results about this
- Holomorphic functions on compact complex manifolds are constant
- Kodaira embedding theorem
- Complex structure on the six-sphere
Used by
- Holomorphic functions on compact complex manifolds are constant
- Remmert–Stein theorem
- Deformation of complex structure
- q-convex manifold
- First Cousin problem
- Kobayashi metric
- Andreotti–Grauert theorem
- Dolbeault cohomology
- Kobayashi hyperbolic manifold
- Kodaira embedding theorem
- Class VII surface
- Bishop's theorem on analytic sets
- Positive line bundle
- Kuranishi family
- Poincaré–Lelong formula
- Chern connection
- Kodaira–Nakano vanishing theorem
- Second Cousin problem
- Complexification of a real analytic manifold
- Stein manifold
- Siegel's theorem on meromorphic function fields
- Nakano positivity
- Holomorphic line bundle
- Holomorphic vector bundle
- Dolbeault isomorphism
- Inoue surface
- Algebraic dimension
- Hironaka's resolution for complex spaces
- Docquier–Grauert theorem
- Griffiths positivity
- Regular and singular points of an analytic set
- Complex geodesic
- Riemann–Hilbert correspondence
- Holomorphic dynamics in several variables
- Oka manifold
- Plurisubharmonic exhaustion function
- Carathéodory metric
- Brody hyperbolicity
- Dolbeault operator
- Grauert's theorem on real analytic manifolds
- Hopf surface
- Grauert's solution of the Levi problem for manifolds
- Holomorphic fiber bundle
- Frankel conjecture
- Poincaré problem
- Positive current
- Levi-flat hypersurface
- Siu's semicontinuity theorem
- Complex structure on the six-sphere
- Brody's lemma
- Complex analytic variety
- Kodaira–Nirenberg–Spencer existence theorem
- Fujiki class C
- Multiplier ideal
- Kähler manifold
- Moishezon manifold