Proving Kähler, finding the Kähler form

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In summary, the conversation discusses the complex structure and Kahler form in relation to Kähler manifolds. It is mentioned that the Kahler form can be found by using the Riemannian metric and almost complex structure, and it will be closed if the complex structure is covariantly constant. The example given involves a metric that is already in a form that makes it easy to determine the complex structure, and the Kahler form can be found by grouping terms with the same signature.
  • #1
Mak182
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Hi!

I have a question about Kähler manifolds. Of course there are many books (I prefer Nakahara) and lecture notes on this topic, but as a physicist I need a very hands-on way of dealing with metrics, etc.

Given a metric, what is the simplest way to find the Kähler form and to prove the Kähler property? I mean, e.g. Nakahara describes Kähler, but is there something simpler one can do (perhaps just in some cases)?

An example I have encountered is
\begin{equation}
ds^2 = (r-u) \left(\frac{\text d r^2}{F(r)} -\frac{\text d u^2}{G(u)} \right) + \frac{1}{r-u} \left(F(r) (\text d t +u \text d z)^2 - G(u) (\text d t+ r \text d z)^2 \right) ~,
\end{equation}
where $F$ and $G$ may be any functions of one variable. This metric is claimed to be Kähler with Kähler form
\begin{equation}
\Omega = \text d (r+ u) \wedge \text d t + \text d (r u) \wedge \text d z ~,
\end{equation}
which is obviously closed.

Following Nakahara, one needs to change coordinates to make the metric Hermitian, but can the Kähler form be read off more directly?

Cheers!
 
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  • #2
Given a Riemannian metric ##g(X,Y)## and an almost complex structure ##J##, the Kahler form is defined as ##\Omega(X,Y) = g(JX,Y)## (eq (8.55) in Nakahara). The Kahler form will be closed if the almost complex structure is covariantly constant in the Riemannian metric. I'm not sure what complex structure is being used in the example you gave, but it's probably straightforward, if tedious, to work it out.
 
  • #3
Adding to what fzero said: You first need to find the complex structure and show that it is integrable. For a generic metric, this might be rather non-obvious. But in most actual applications, the metric is written in such a way that the complex structure is easy to see.

What one does, generally, is to write the metric in terms of orthonormal frames. Then you know that the almost-complex structure must simply rotate these frames. In your example, the metric is already a sum of four squares, so the hard work is already done.

Looking at your metric, it can have signature (++++), (++--), or (----), depending on the signs of F and G. In order to apply complex geometry, you must group things in pairs that have the same signature. So the two terms with ##F(r)## in front go together, and the two terms with ##G(u)## go together. From here it should be easy to postulate an almost-complex structure J. Then you apply standard techniques to show that J is integrable.

Finally, given J, the Kahler form is well-defined.
 

1. How do you prove that a manifold is Kähler?

To prove that a manifold is Kähler, you need to show that it is both symplectic and complex. This means that it must have a symplectic form, which is a non-degenerate 2-form that is closed and satisfies the compatibility condition with the complex structure, and a complex structure, which is a smooth endomorphism of the tangent bundle that squares to -1. Additionally, you must show that the symplectic form and complex structure are compatible, meaning that they commute with each other. If a manifold satisfies all of these conditions, it is considered Kähler.

2. What is the Kähler form?

The Kähler form is a closed, non-degenerate 2-form on a Kähler manifold that satisfies the compatibility condition with the complex structure. It is a fundamental object in Kähler geometry and is used to define the Kähler metric, which in turn is used to measure distances and angles on the manifold. The Kähler form plays a crucial role in many mathematical and physical theories, including complex geometry, algebraic topology, and string theory.

3. Can you explain the significance of proving Kähler?

Proving that a manifold is Kähler has several important implications. First, it allows us to use techniques and results from both symplectic geometry and complex geometry, making it a powerful tool for studying the geometry of a manifold. Additionally, Kähler manifolds have many interesting geometric properties, such as the existence of holomorphic vector fields and the fact that they are Ricci-flat. These properties have important applications in physics, such as in the study of Calabi-Yau manifolds in string theory.

4. What are some common techniques used to prove Kähler?

There are various techniques used to prove that a manifold is Kähler, depending on the specific context and problem at hand. Some common techniques include showing that the manifold is symplectomorphic to a known Kähler manifold, using symplectic reduction or deformation techniques, or constructing a Kähler potential explicitly. In general, proving Kähler requires a deep understanding of both symplectic and complex geometry, as well as advanced mathematical techniques.

5. Are there any open problems related to proving Kähler?

Yes, there are several open problems related to proving Kähler. One of the most famous is the existence of a Kähler metric with constant scalar curvature on a given Kähler manifold. This is known as the Calabi conjecture and is still unsolved in general. Other open problems include finding necessary and sufficient conditions for a manifold to be Kähler, and understanding the relation between Kähler geometry and other areas of mathematics, such as algebraic geometry and mathematical physics.

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