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On the geometry of cusps for \(SU(n,1)\). (English) Zbl 0855.32022

de Bartolomeis, Paolo (ed.) et al., Manifolds and geometry. Proceedings of a conference, held in Pisa, Italy, September 1993. Cambridge: Cambridge University Press. Symp. Math. 36, 112-131 (1996).
Let \(U_n : = \{(z,w) \in \mathbb{C}^{n - 1} \times \mathbb{C} \mid \text{Im} (w) > |z |^2\}\). A standard cup for \(SU (n,1)\) is the germ at infinity of the compactification of the quotient of \(U_n\) by a freely acting discrete subgroup of \(\operatorname{Aut} (U_n)\) which leaves all horospheres invariant, and contains a finite index subgroup, which is a cocompact lattice in the Heisenberg group of translations of \(U_n\).
It is proved that a germ \((X,0)\) of a normal analytic space with an isolated singularity at 0 and \(X \backslash \{0\}\) carrying a complex-hyperbolic metric, which is complete and 0, is biholomorphically isometric to a standard cusp for \(SU(n,1)\).
On the other hand, if \((X,0)\) is biholomorphic to a standard cusp for \(SU(n,1)\) and \(ds^2\) is a complex-hyperbolic metric on \(X \backslash \{0\}\), then \(ds^2\) is complete at 0 and \((X \backslash \{0\},\;ds^2)\) is biholomorphically isometric to a standard cusp for \(SU(n,1)\).
For the entire collection see [Zbl 0840.00037].

MSC:

32S70 Other operations on complex singularities
32C20 Normal analytic spaces
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
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