Two examples concerning hyperbolic quotients (Q1111738)

For M a compact complex variety and \(k_ M\) its Kobayashi pseudodistance, let R be the equivalence relation \(k_ M^{-1}(0)\) and p: \(M\to M/R\). Every holomorphic map \(M\to N\) factors through p, and every holomorphic \(M\to N\) with hyperbolic N factors through q: \(M\to M_ n\) (the hyperbolic quotient of M). Kobayashi asked: \(p=q?\) A first example shows that M/R in general does not admit a complex structure such that p becomes holomorphic, and a second example provides a holomorphic p with M/R not hyperbolic. So the answer is no. The author also establishes that for a proper holomorphic map \(\phi\) : \(M\to N\) between complex manifolds, if N and all the fibers of \(\phi\) are hyperbolic, then so is M.