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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)].

57R19 Algebraic topology on manifolds and differential topology
14J70 Hypersurfaces and algebraic geometry
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