On the Lie group structure of automorphism groups (Q2012345)

The author gives a sufficient condition for complex manifolds such that the automorphism groups become Lie groups. A classical result of H. Cartan says that the automorphism groups of bounded domains in \(\mathbb C^n\) are Lie groups. A key fact for its proof is that the isometry group of any Hermitian metric is a Lie group with respect to the compact-open topology. Using a similar approach, the author shows that for a Stein manifold \(M\), if \(M\neq Z\cup D\), then \(\text{Aut}(M)\) has a Lie group structure. Here, \(Z\) denotes the zero locus of the Bergman kernel and \(D\) denotes the degenerating locus of the Bergman pseudometric \(ds^2_{M\backslash Z}\). The author also gives a sufficient condition for \(M\neq Z\cup D\). In particular, the author discusses its connection with the notion of the ``core'' of a Stein domain.