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On the connectivity of complex affine hypersurfaces. (English) Zbl 0771.57010
Improving a result of M. Kato [Topology 16, 33-50 (1977; Zbl 0367.14009)], the author obtains lower bounds for the connectivity of a reduced algebraic hypersurface $$X\subseteq\mathbb{C}^ n$$. Let $$X$$ be defined by the equation $$f(x)=0$$, and let $$\overline X$$ be the closure in the projective space. Let $$\Sigma(f)$$ denote the intersection of the singular set of $$\overline X$$ with the hyperplane at infinity. Among the homogeneous components of $$f$$ that do not vanish identically, let $$f_ d$$ and $$f_ e$$ be those of the two highest degree $$e<d$$, and define the set $$S(f)\subseteq\Sigma(f)$$ by the equations $$f_ e(x)=0=\nabla f_ d(x)$$.
The author shows that $$X$$ is $$(n-2-\dim \Sigma(f))$$-connected. In fact, if $$e>0$$, then he shows that each fiber $$f^{-1}(c)$$ es even $$(n-2-\dim S(f))$$-connected. He obtains (improvements of) several known results as corollaries. For example, the fibers of $$f$$ are all connected if $$f_ d$$ is square free (first proved by G. Angermüller [Manuscr. Math. 54, 349-359 (1985; Zbl 0587.14001)]), and the generic fiber $$f^{-1}(c)$$ has the homotopy type of a bouquet of $$n$$-spheres if $$n\neq 2$$ and the set $$\Sigma(f)$$ is finite, and if moreover $$f$$ has only isolated singularities; the last assertion sharpens a result of S. A. Broughton [Proc. Symp. Pure Math. 40, Part I, 167-178 (1983; Zbl 0526.14010)].

##### MSC:
 57R19 Algebraic topology on manifolds and differential topology 14J70 Hypersurfaces and algebraic geometry
##### Keywords:
connectivity; algebraic hypersurface
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