  In this article we prove first of all the nonexistence of holomorphic
submersions other than covering maps between compact quotients of complex unit
balls, with a proof that works equally well in a more general equivariant
setting. For a non-equidimensional surjective holomorphic map between compact
ball quotients, our method applies to show that the set of critical values must
be nonempty and of codimension 1. In the equivariant setting the line of
arguments extend to holomorphic mappings of maximal rank into the complex
projective space or the complex Euclidean space, yielding in the latter case a
lower estimate on the dimension of the singular locus of certain holomorphic
maps defined by integrating holomorphic 1-forms. In another direction, we
extend the nonexistence statement on holomorphic submersions to the case of
ball quotients of finite volume, provided that the target complex unit ball is
of dimension m>=2, giving in particular a new proof that a local biholomorphism
between noncompact m-ball quotients of finite volume must be a covering map
whenever m>=2. Finally, combining our results with Hermitian metric rigidity,
we show that any holomorphic submersion from a bounded symmetric domain into a
complex unit ball equivariant with respect to a lattice must factor through a
canonical projection to yield an automorphism of the complex unit ball,
provided that either the lattice is cocompact or the ball is of dimension at
least 2.
