math6 said:Hi friends !
Let E be a holomorphic vector bundle over a complex manifold M . We identify M with the zero section of E .
i would like to know waht's mean "" We identify M with the zero section of E ".
thnx :)
A holomorphic vector bundle is a mathematical object that combines the concepts of a vector space and a continuous family of vector spaces over a topological space. It is a type of manifold that is locally modeled on a complex vector space.
Some of the main properties of a holomorphic vector bundle include holomorphic transition functions, a holomorphic structure sheaf, and a holomorphic tangent bundle. It also has a well-defined notion of holomorphic sections and holomorphic maps between bundles.
The main difference between a holomorphic vector bundle and a smooth vector bundle is that holomorphic vector bundles are defined over complex manifolds and have holomorphic transition functions, while smooth vector bundles are defined over smooth manifolds and have smooth transition functions.
Holomorphic vector bundles have many applications in mathematics, including in algebraic geometry, complex analysis, and differential geometry. They are also used in physics, particularly in the study of quantum field theory and string theory.
The holomorphic structure of a vector bundle can be determined by studying its holomorphic transition functions and holomorphic sections. Additionally, there are various tools and techniques, such as sheaf cohomology, that can be used to study the holomorphic structure of a vector bundle.