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On a smooth $2d$-dimensional real manifold $M$ with $\mathcal{C}^{\infty}$ boundary $\partial M$, the most common model for the boundary are via boundary charts $U$ which are homeomorphic to the upper half plane $\mathbb{R}^{2d}_{+}$ and which are compatible with the smooth structure.

If one wishes to define, in addition, a complex structure on $M$, then the only model of the boundary that I can find are via charts $V$ which are homeomorphic to open neighbourhood near the origin in \begin{equation} \{ z \in \mathbb{C}^{d} \ / \ \rho(z) \geq 0 \} \end{equation} where $\rho : \mathbb{C}^{d} \rightarrow \mathbb{R}$ is a $\mathcal{C}^{\infty}$ function which is also a locally defining fnction for $ \partial M \cap V $, i.e. $\partial M \cap V$ is the regular zero level set $\rho^{-1}(0)$. It was pointed out explicitly in What is the definition of a complex manifold with boundary? that one should not restrict to charts which are homeomorphic to $$\{ z \in \mathbb{C}^{d} \ / \ \text{Im}(z^{d}) \geq 0 \}$$

although I think this would somehow be a more natural (and useful) definition. A reason was briefly mentioned in the linked question, but I have not really been able to understand it. As there are very limited literature on such matter for beginner, I have the following questions:

  • Why is the second definition in the above ruled out? As mentioned above this seems to be a more natural chart to look at, moreover, one could always consider the underlying $\mathcal{C}^{\infty}$ structure and get a boundary chart from there, but just look at it as complex coordinates, the only problem is compatibility with the global holomorphic structure. In general, is it impossible for one to get a boundary coordinate like this?

  • If I have a metric $g$ which is say for instance Kahler, then I also need to consider the boundary behaviour of $g$ and the corresponding Kahler identities. There seems to be very limited reference on this subject matter. Is there a good reference that I might be able to make use of?

This seems like a long question, so many thanks in advanced for the helps!

Meagain
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1 Answers1

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To address your first question: there is a deep difference between the cases of one complex dimension and higher dimensions.

In one complex dimension, there is a version of the Riemann mapping theorem that says that any two simply connected planar domains are biholomorphic, and if the boundaries are piecewise smooth, then the biholomorphism extends smoothly to the smooth portions of the boundary. Thus if you're given a holomorphic chart of the first type, you can compose it with a biholomorphic mapping from (a suitable subset of) its image to a half-disk, which takes the boundary to the set $\{z: \operatorname{Im} z= 0\}$, and thus get a holomorphic chart of the second type. So in that case the two definitions are equivalent.

In higher dimensions, things are radically different. For $d\ge 2$, the boundary of a smoothly bounded domain in $\mathbb C^d$ inherits a CR structure, which is preserved by all biholomorphisms that are smooth up to the boundary. Thus, for example, at boundary points of a non-spherical ellipsoid, there will not be any charts of the second type.

I don't have my books handy, but you can probably find some details in the early chapters of the following books:

As to your second question, offhand I don't know any references for Kähler manifolds with boundary. But if the Kähler metric is smooth up to the boundary, then the Kähler identities will automatically be valid up to the boundary by continuity. There's probably a lot that can be said about the relationships among the induced boundary metric, the second fundamental form, and the CR structure on the boundary, but I've never looked into that.

Jack Lee
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