What exactly is a Kähler Manifold?

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.

Solutions Collecting From Web of "What exactly is a Kähler Manifold?"

First, as I think you know, Kähler manifolds are just special cases of Hermitian manifolds (complex manifolds with a Hermitian metric). There are a number of very different, but equivalent, ways to define Kähler manifolds. Here are a couple that might help:

  1. On a Riemannian manifold $X$, we can always choose ‘Riemann normal coordinates’ at any point $p \in X$. These are coordinates in which the metric takes its canonical form $g_{ab} = \delta_{ab}$ at $p$, and all its first derivatives vanish at $p$. On a general Hermitian manifold, it may not be possible to find holomorphic coordinates in which this is true.
    Kähler manifolds are exactly those manifolds on which we can always find a holomorphic change of coordinates which, at some given point, sets the metric to its canonical form, and its first derivatives to zero.

  2. Another characterisation of Kähler manifolds is as Hermitian manifolds for which the Christoffel symbols of the Levi-Civita connection are pure. In other words, $\Gamma^i_{jk}$ and $\Gamma^{\bar i}_{\bar j\bar k}$ may be non-zero, but all ‘mixed’ symbols like $\Gamma^{\bar i}_{jk}$ vanish. This means that (anti-)holomorphic vectors get parallel transported to (anti-)holomorphic vectors.

  3. Equivalent to the above is to say that $n$-dimensional Kähler manifolds are precisely $2n$-dimensional Riemannian manifolds with holonomy group contained in $U(n)$.

All of this can be found in Moroianu’s “Lectures on Kähler Geometry”, which is quite a nice concise book.

There is a lot more to be said about Kähler manifolds, but hopefully this gives you some intuition about them.

Suppose you have a real vector space (even dimensional, in this case the tangent space of the manifold), which you want to convert into a complex vector space. Then you want to say where the action of $i$ takes a vector $v$. In other words, you need a map $J$ such that $J^2 = -1$. If this J varies smoothly on the manifold, then you have an almost complex structure (note that pointwise you can always do this, but it may not fit together nicely globally).

The definition of the Hermitian structure incorporates the fact that the angle between $v$ and $w$ should be the same as the angle between $Jv$ and $Jw$, which you would want surely.

Saying that the Kahler form is closed actually converts it into a symplectic form which fits nicely with the Kahler structure, so all the tools of the vastly developed subject of symplectic geometry can be made to bear. You can look up what makes the symplectic form so important. One motivation can be found here (see Lawrence Crowell’s answer in particular).