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Can anybody help me to be clear about this definition. I know the definition of a real manifold with boundary (as in Lee's book) and the definition of a complex manifold (locally diffeomophic to an open set in $\mathbb{C}^{n}$ and transition maps are holomorphic).

What is the definition of a complex manifold with boundary? I see it many times while reading about the complex-Monge Ampere equations on Kahler manifolds.

Binjiu
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    The boundary has the usual definition as a $(2n-1)$-dimensional manifold, coming from thinking of the original complex manifold as just a smooth manifold. It does, however, have extra structure. It is a CR manifold. – Ted Shifrin Oct 07 '16 at 18:56

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The usual definition is that $M$ is a complex manifold with boundary if and only if it has an atlas of biholomorphically compatible charts, each of which has as its image either an open subset of $\mathbb C^n$ or a set of the form $\{z\in U :f(z)\le 0\}$, where $U\subseteq\mathbb C^n$ is open and $f\colon U\to\mathbb R$ is a $C^\infty$ submersion.

It's important to allow such "curved" model boundaries instead of insisting that the image be a relatively open subset of a half-space, because most hypersurfaces in $\mathbb C^n$ are not biholomorphically equivalent to a plane such as $\{z: \operatorname{Im} z=0\}$.

Jack Lee
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  • As in your definition, a complex manifold with boundary is also a real manifold with boundary. However, I am confused that the transition maps may be biholomorphic between close sets in $\mathbb{C}^{n}$, it sounds weid to me. – Binjiu Oct 07 '16 at 22:01
  • A set of the form ${z\in U: f(z)\le 0}$ is generally not closed. However, it's typically not open either. What it means to say that a map defined on such a set is holomorphic is that, first of all, it is smooth (meaning it can be extended to a smooth map in a neighborhood of each point), and second, that it satisfies the Cauchy-Riemann equations wherever it's defined. – Jack Lee Oct 07 '16 at 22:03
  • Thanks a lot. I quite understand now. – Binjiu Oct 08 '16 at 09:08
  • But I am still getting confused about the Blocki's note of complex Monge_Ampere equations: http://link.springer.com/chapter/10.1007%2F978-3-642-36421-1_2 In the first place, it seems that he assumes $\partial M=\textrm{Ø}$, because he needs it in $dd^{c}$-lemma (page 105) and the equivalence between the Calabi conjecture and the Dirichlet problem for complex Monge-Ampere (page 106). Hence I guess that Yau solved the Calabi conjecture for a complex manifold "without boundary", and then others try to extend it to complex manifolds "with boundary". Is it right? – Binjiu Oct 08 '16 at 09:21
  • The Blocki article you referred to deals only with the case of empty boundary. I don't know why he uses the term "Dirichlet problem" -- the PDE problem he describes on page 106 is to find a solution to the complex Monge-Ampere equation on a compact manifold. There is no boundary in sight. – Jack Lee Oct 08 '16 at 21:21
  • Yes. That's what I mentioned. But he begins to deal with complex manifolds with boundary from the page 122. That's is, firstly, he gave a setting without boundary (the equivalence between the Calabi conjecture and the complex Monge-Ampere equation, page 106) then from section 7, we see problems working on manifolds with boundary. – Binjiu Oct 09 '16 at 00:49