Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing.
I am a physics undergrad, trying to carry out some research work on topological solitons. I have been trying to read a paper that uses Kähler Manifolds. My guide just expects me to learn the mathematical definitions, without understanding it(or he expects me to study complex manifolds by scratch by myself), within 3-4 days, with exams going on, but I find this highly discomforting. So, it would be great if someone could tell me what is a Kähler manifold, highlighting the essential features of the definition and what they do, and intuitive explanations behind each. Why are they mathematically important? Also, some reasons as to why they are used in Physics?
The definition on Wikipedia is very obscure, linking you to 7-8 pages, and you forget what you are actually looking for. I have read the following definition from Nakahara:
A Kähler Manifold is an hermitian manifold, whose Kälher form is closed i.e. $d\Omega=0$.
After searching the internet, I know the following:
A Hermitian manifold is a complex manifold equipped with a metric $g$, such that $g_p(X,Y)=g_p(J_pX,J_p Y)$, where $p \in M$ and $X,Y \in T_pM$
Again the web tells me that, $J$ is a linear map between the tangent spaces at a point such that $J^2=-1$. Lastly, the Kähler form $\Omega$ is a tensor field whose action is given by $\Omega_p(X,Y)=(J_pX,Y)$.
Exact questions: This is what I would really really want to understand. What is the meaning and motivation for $J^2=-1$? What is the intuitive meaning and motivation for the definition of the Hermitian manifold, and the Kähler form? Most importantly, what does the Kähler form is closed really mean?
I am sorry for the long question, and would be delighted, even if I got a partial answer. Looking forward for the replies.
I am not looking for exact arguments, but an intuitive overall picture.
Background: I understand definitions of real manifolds, tangent spaces, and a differential forms. I have no intuition about exterior derivatives. I have a fair understanding of what is a complex manifold, and a few examples of Riemann surfaces.