What can a complex manifold do for me that real manifolds can't.

[Spinnor]

What can a complex manifold of dimension N do for me that real manifolds of dimension 2N can't.

Edit, I guess the list might be long but consider only the main features.

Thanks for any help or pointers!

 

[Ben Niehoff]

Lots of things. For one, if your complex manifold is Kahler, then there is a huge shortcut for computing the Ricci tensor.

 

[Spinnor]

Similar question asked here,

http://math.stackexchange.com/quest...difference-between-real-and-complex-manifolds

 

[Spinnor]

From " Complex manifolds and mathematical physics"

http://www.researchgate.net/profile/Raymond_Wells/publication/38390006_Complex_manifolds_and_mathematical_physics/links/0c960528f84c35a7c9000000

"1. Introduction. In the past several years there have been some remarkable links forged between two rather distinct areas of research, namely complex manifold theory on the one hand, and mathematical physics on the other. Complex manifold theory has its roots in the theory of Riemann surfaces and in algebraic geometry, and has seen significant progress in this century based on the introduction of ideas from algebraic topology, differential geometry, partial differential equations, etc. Mathematical physics has been involved in this century in the developments of relativity theory, quantum mechanics, quantum electrodynamics, and quantum field theory, to mention some major developments. Most of these disciplines are formulated in forms of field equations, i.e. partial differential equations whose solutions (under some boundary conditions) represent physical or measurable quantities. The link mentioned above between complex manifold theory and mathematical physics is that in many cases, the solutions of a given field equation can be represented entirely in terms of complex manifolds, holomorphic vector bundles, or cohomology classes on open complex manifolds with coefficients in certain holomorphic vector bundles. In simplistic terms the field equations can be reduced to the Cauchy-Riemann equations by making suitable changes in the geometric background space. "

 

[blue_raver22]

I can understand the curiosity about the capabilities of complex manifolds compared to real manifolds. Complex manifolds are mathematical objects that extend the concept of real manifolds by introducing complex numbers. This added complexity allows for a wider range of mathematical operations and transformations that can be performed on the manifold.

One of the main features of complex manifolds is the ability to define holomorphic functions, which are complex-valued functions that are smooth and have a well-defined derivative at every point on the manifold. This property is not possible on real manifolds, as they only allow for real-valued functions. Holomorphic functions have many important applications in physics, engineering, and other fields, making complex manifolds a powerful tool for studying and solving complex problems.

Another key difference between complex and real manifolds is the ability to define complex structures. A complex structure is a set of rules that allows us to perform complex operations on a manifold, such as multiplication, division, and differentiation. This is not possible on real manifolds, as they only allow for real-valued operations. Complex structures play a crucial role in many areas of mathematics, including algebraic geometry and differential geometry, and allow for the study of complex phenomena that cannot be captured by real manifolds.

Furthermore, complex manifolds have a more intricate topology compared to real manifolds. This means that they have a more complex structure and can have features such as holes, handles, and non-orientability. This added complexity allows for a richer set of mathematical objects and transformations that can be studied on the manifold.

In terms of dimension, complex manifolds of dimension N have more degrees of freedom compared to real manifolds of dimension 2N. This means that they can capture more complex and higher-dimensional phenomena, making them useful in fields such as string theory, where higher-dimensional spaces are often studied.

In summary, complex manifolds offer a wide range of mathematical tools and techniques that are not possible on real manifolds. They allow for the study of complex phenomena and have applications in various fields of science and mathematics. In terms of dimension, complex manifolds of dimension N have more capabilities compared to real manifolds of dimension 2N, making them a valuable tool for understanding complex systems.