Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations

TL;DR

This paper proves the nonexistence of certain holomorphic submersions between complex unit ball quotients and extends these results to more general settings, including finite volume quotients and equivariant maps, revealing rigidity phenomena.

Contribution

It establishes new nonexistence results for holomorphic submersions between complex ball quotients and generalizes these to equivariant and finite volume cases, highlighting rigidity properties.

Findings

Holomorphic submersions between compact ball quotients are only covering maps.

Nonexistence of holomorphic submersions extends to finite volume ball quotients for dimensions m>=2.

Any equivariant holomorphic submersion from a bounded symmetric domain factors through a canonical projection.

Abstract

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.