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I am wondering if anything is known about this. I couldn't find anything in the literature.

In '74 C. Fefferman published a solution to the following problem. Let $\sigma:D\rightarrow D$ be an automorphism of a strictly pseudoconvex domanain $D\subset\mathbb{C}^n$. Then $\sigma$ extends to a smooth map $\sigma:\overline{D}\rightarrow\overline{D}$.

My question: Is anything known about this problem for a strictly pseudoconvex domain $D\subset M$ in a complex manifold $M$?

I have an idea of how to prove it for $K_M >0$, i.e. positive canonical bundle. Fefferman's approach used the Bergman metric. The Bergman metric is nondegenerate for more general domains only if $K_M$ is very ample, which is too strong an assumption to be very interesting.

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