[For the entire collection see Zbl 0509.00008.]
Let \(f:\quad X^ n\to {\mathbb{P}}^ p\) be a finite morphism into a projective space with \(n\leq p\). The main purpose is to investigate relations between the fundamental group of X and the set of points of some fixed multiplicity. Using a new notion of multiplicity called stable multiplicity, \(m_ S(f)(x)\), the following theorem is proved. Theorem: Let \(f:\quad X^ n\to {\mathbb{P}}^ p,\) \(n\leq p\), be a finite morphism of a projective variety analytically irreducible at every point into a projective space. Suppose that there exists ad such that \(d(p-n+1)\leq n\) and every component of the set \(T_ S^{d+1}(f)=\{x\in X:m_ S(f)(x)\geq d+1\}\) has codimension \(>d(p-n+1)\) or is empty. Then X is simply connected.
The notion of stable multiplicity is given as follows. Let \(f:\quad (X^ n,x)\to ({\mathbb{C}}^ p,0)\) be a proper finite-to-one complex analytic map- germ. The topological multiplicity \(m_ T(f)(x)\) of f at x is defined to be \(m_ T(f)(x)=\lim_{\epsilon \to 0}\max \{\#(f^{-1}(y)\cap B_{\epsilon}(x)):y\in B_{e}(f(x))\}.\) Let \(F:\quad (X\times {\mathbb{C}}^ r,(x,0))\to ({\mathbb{C}}^ p\times {\mathbb{C}}^ r,0\times 0)\) be a stable unfolding of f in J. Mather’s sense, which always exists for a finite type map-germ. Then the stable multiplicity \(m_ S(f)(x)\) of f at x is defined to be \(m_ T(F)(x,0)\). Although when \(n=p\) the three definitions of multiplicities \(m_ T(f)(x)\), \(m_ S(f)(x)\) and dim Q(f) coincide, they differ when \(n<p\). It is pointed out that stable multiplicity has better properties than the other two in the following sense: Stable multiplicity has the additive property, although topological multiplicity does not: When \(n<p\), dim Q(f)(x) seems to count x too many times: Also \(m_ S(f)(x)\) seems to agree with the algebraic geometer’s intuition as to what the multiplicity should be. - After a theorem about the codimension of \(T_ S^{d+1}(f)\) and an existence theorem for \(T_ S^{d+1}(f)\) are proved, the main theorem is proved. For related results, see papers by J. P. Hansen [”A connectedness theorem for flagmanifolds and Grassmannians and singularities of morphisms to \({\mathbb{P}}^ m\)”, Ph. D. Thesis, Brown Univ. (Providence, R. I. 1980)], R. Lazarsfeld [”Branched coverings of projective space”, Ph. D. Thesis, Brown Univ., [Providence, R. I. 1980)] and the author and R. Lazarsfeld [Invent. Math. 59, 53-58 (1980; Zbl 0422.14010)].