Lectures on the automorphism groups of Kobayashi-hyperbolic manifolds
Let \(M\) be a hyperbolic complex manifold of dimension \(n\geq 2\). The group \(\Aut(M)\) of holomorphic automorphisms of \(M\) is a real Lie group acting properly on \(M\). In this situation, in fact, \(\Aut(M)\) acts on \(M\) as a group of isometries with respect to the Kobayashi distance. Denote by \(d(M)\) the real dimension of \(\Aut(M)\). The goal of this book is to present a self-contained exposition of the classification of hyperbolic manifolds with ``large'' automorphism group obtained by the author and collaborators in several papers [see for example, \textit{A. V. Isaev} and \textit{S. G. Krantz}, J. Reine Angew. Math. 534, 187--194 (2001; Zbl 0974.32018); \textit{A. V. Isaev}, J. Geom. Anal. 15, No.~2, 239--259 (2005; Zbl 1095.32007); Geom. Funct. Anal. 17, No.~1, 192--219 (2007; Zbl 1120.32014); Geom. Topol. 12, No.~2, 643--711 (2008; Zbl 1143.32016)]. The manifolds under consideration are precisely the \(n\)-dimensional hyperbolic manifolds \(M\) satisfying the condition \[ n^2-1\leq d(M)\leq n^2+2n. \] The upper bound arises from a result of \textit{W. Kaup} [Invent. Math 3, 43--70 (1967; Zbl 0157.13401)] stating that, if \(d(M)>n^2\), then \(M\) is necessarily homogeneous and \(d(M)\leq n^2+2n\). Moreover, \(d(M)=n^2+2n\) if and only if \(M\) is biholomorphic to \(B^n\), the unit ball in \({\mathbb C}^n\). The lower bound \(n^2-1\) is suggested by the existence in \({\mathbb C}^2\) of an infinite family of pairwise non-equivalent Reinhardt domains, all with automorphism group of dimension \(2\). Such a family cannot be described by any formula, thus excluding the possibility of classifying 2-dimensional manifolds \(M\) with \(d(M)=2\) and in general \(n\)-dimensional manifolds \(M\) with \(d(M)=n^2-2\), for all values of \(n\). One should also observe that, for \(M\) not the unit ball, actually \(d(M)\leq n^2+2\) holds. By the properness of the action, the isotropy subgroup of an arbitrary point in \(M\) can in fact be identified with a closed subgroup of the unitary group \(U(n)\). Then, the jump in the dimensions of the closed subgroups of \(U(n)\) produces a jump in the values of \(d(M)\). In this way, the original classification problem reduces to considering \(n\)-dimensional hyperbolic manifolds \(M\) with \(d(M)\) equal to \[ n^2-1,\quad n^2 ,\quad n^2+1, \quad n^2+2. \] If \(M\) is homogeneous, then it is biholomorphic to a bounded homogeneous domain. By exploiting the Siegel realization of bounded homogeneous domains, it is shown that \(M\) falls into a short list: it is biholomorphic to \(B^{n-1}\times \Delta\) if \(d(M)=n^2+2\), to the 3-dimensional bounded symmetric domain of type IV if \(d(M)=n^2+1\), either to \(\Delta^3\) or to \(B^2\times B^2\) if \(d(M)=n^2\), and to the 4-dimensional bounded symmetric domain of type IV if \(d(M)=n^2-1\). (Here, \(\Delta\) denotes the unit disk in \({\mathbb C}\)). The classification of the non-homogeneous hyperbolic manifolds \(M\) takes most of the effort. The cases \(d(M)=n^2\), \(d(M)=n^2 -1\), with \(n\geq 3\), and \(d(M)=n^2-1=3\), with \(n=2\), are considered separately. The first step consists of determining the \(\Aut(M)\)-orbits which may occur. Then one needs to join them together to form hyperbolic manifolds. By examining the isotropy representation at an arbitrary point under the conditions \(d(M)=n^2\) or \(d(M)=n^2-1\), it is shown that the \(\Aut(M)\)-orbit though a point \(p\in M\) is either a complex or a real hypersurface. Only for \(n=2\), totally real orbits occur. If \(d(M)=n^2\), then \(M\) necessarily contains a real hypersurface orbit and at most two complex hypersurface orbits. The former are either spherical or Levi-flat, the latter are biholomorphic to \(B^{n-1}\). The next fundamental ingredient is a fine classification of the real hypersurface orbits as CR-manifolds and of their CR-automorphisms. The outcome of the classification of \(M\) is an explicit list consisting of seven cases. If \(d(M)=n^2-1\) and \(n\geq 3\), then \(M\) contains no real hypersurface orbit and it is biholomorphic to the product \(B^{n-1}\times S\), where \(S\) is a Riemann surface with \(d(S)=0\). The case \(n=2\) and \(d(M)=3\) is the most complicated. As we mentioned, the codimension 2 orbits which may appear (at most 2 of them) are either complex or totally real. Also the hypersurface orbits are either strongly pseudoconvex (not just spherical) or Levi-flat. Even in this case, a fundamental ingredient is the CR-classification of the real hypersurface orbits [see \textit{E. Cartan}, Annali di Mat. (4) 11, 17--90 (1933; JFM 58.1256.03) and \textit{A. Isaev}, J. Lie Theory 16, No.~3, 407--426 (2006; Zbl 1127.53043)]. The outcome of the classification of \(M\) in this case is a rather long explicit list. The final part of the book contains some considerations on possible extension of the above results to the case of proper actions on manifolds \(M\) which are not hyperbolic under the same conditions on \(d(M)\) [see \textit{A. Isaev} and \textit{N. G. Kruzhilin}, Isr. J. Math. 172, 193--252 (2009; Zbl 1175.32009)].
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