Editing Complex geometry (section)

== Techniques in complex geometry ==

Due to the rigidity of holomorphic functions and complex manifolds, the techniques typically used to study complex manifolds and complex varieties differ from those used in regular differential geometry, and are closer to techniques used in algebraic geometry. For example, in differential geometry, many problems are approached by taking local constructions and patching them together globally using partitions of unity. Partitions of unity do not exist in complex geometry, and so the problem of when local data may be glued into global data is more subtle. Precisely when local data may be patched together is measured by [[sheaf cohomology]], and [[Sheaf_(mathematics)|sheaves]] and their [[cohomology groups]] are major tools.

For example, famous problems in the analysis of several complex variables preceding the introduction of modern definitions are the [[Cousin problems]], asking precisely when local meromorphic data may be glued to obtain a global meromorphic function. These old problems can be simply solved after the introduction of sheaves and cohomology groups.

Special examples of sheaves used in complex geometry include holomorphic [[line bundle]]s (and the [[Divisor (algebraic geometry)|divisor]]s associated to them), [[holomorphic vector bundle]]s, and [[coherent sheaves]]. Since sheaf cohomology measures obstructions in complex geometry, one technique that is used is to prove vanishing theorems. Examples of vanishing theorems in complex geometry include the [[Kodaira vanishing theorem]] for the cohomology of line bundles on compact Kähler manifolds, and [[Cartan's theorems A and B]] for the cohomology of coherent sheaves on affine complex varieties. 

Complex geometry also makes use of techniques arising out of differential geometry and analysis. For example, the [[Hirzebruch-Riemann-Roch theorem]], a special case of the [[Atiyah-Singer index theorem]], computes the [[holomorphic Euler characteristic]] of a holomorphic vector bundle in terms of characteristic classes of the underlying smooth complex vector bundle.