On the degree theory of holomorphic mappings (Q2744003)

In this note, some basic properties of the topological index of a (light) holomorphic map defined on a complex analytic space are proved. The standard homological or analytical approach of the topological degree theory does not seem to readily carry over to mappings defined on spaces with singularities. However, for (light) holomorphic mappings defined on a complex (analytic) space, a simple definition of the mapping index was given by W.~Stoll. It is shown that on complex manifolds this definition agrees with the notion of Brouwer degree, and some further properties (e.g., the product theorem) are given. It turns out that the Continuity Principle of the mapping index is essential to a general formulation of the Implicit Mapping Theorem which may be useful in situations involving branching phenomena. At least for normal complex spaces, the theorem is consistent with the classical Inverse and Implicit Function Theorems. In consequence of these results a Bezout type stability property of the intersection of parametrized projective algebraic hypersurfaces, and accordingly, the analytic accessibility of zeroes of a polynomial system are established. The latter gives a partial strengthening of a Li-Sauer-Yorke theorem. Other applications such as (i) a criterion for the openness, respectively, injectivity, of the limit map of a uniformly convergent sequence of holomorphic maps, and (ii) a generalized Poincaré-Bohl's theorem, are also given.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00052].