Image structure for holomorphic map germs with higher-dimensional targets

Determine the structure of the image F(A_{X,p}^*) for non-constant holomorphic map germs F: (X,p) → (Y,y) when the target germ (Y,y) has complex dimension greater than two and the usual image is not well-defined as a set germ; in particular, ascertain whether an analogue of the blossom tree description developed for surface targets (dim(Y)=2) exists for dim(Y) > 2 and characterizes F(A_{X,p}^*).

Background

The paper addresses the difficulty that the image of an analytic map germ may fail to be locally open or even well-defined as a set germ. For target surface germs (dim(Y)=2), the authors propose defining the image of a map germ F as the subset of curve germs F_(A_{X,p}^) and give an algorithmic description based on successive blow-ups. This process yields a decorated finite tree—the blossom tree—that encodes the image’s structure.

They prove the finiteness of the algorithm, show uniqueness of the blossom tree, and define a complexity degree κ(F) reflecting the maximal length of chains of branches in the tree. The construction is extended to singular surface targets, producing a “forest” over the exceptional set of a desingularization. The authors explicitly state that extending this characterization beyond surfaces remains unresolved, identifying the higher-dimensional case (dim(Y) > 2) as an open problem.