Definition of complex hyperbolic geometry

Question

I am trying to read about complex hyperbolic geometry.But I couldnot find a basic definition for it. Is it just the special case of hyperbolic geometry where we work with complex numbers in the model. Or is there something more to it? Can someone please give me a specific definition of complex hyperbolic geometry?

Answer 1

I think, you are asking about complex-hyperbolic space. See for instance "Notes on Complex Hyperbolic Geometry" by John Parker or "Complex Hyperbolic Geometry" by Bill Goldman for the detailed definitions. In short, yes, complex-hyperbolic space is a complex analogue of the real-hyperbolic space.