A general calculation of the number of vanishing cycles
Let \(X\) be a reduced complex analytic space containing an open neighborhood \(U\) of the origin in \(\mathbb{C}^ n\) and let \(f : (U,0) \to (\mathbb{C})\) be an analytic function. Suppose that \(\mathbb{F}^ \bullet\) is a bounded complex of sheaves on \(X\) which is constructible with respect to a Whitney stratification \(\{S_ \alpha\}\) of \(X\). Let \(H\) be a hypersurface in \(U\) which transversely intersects the strata of \(\{S_ \alpha\}\) except, perhaps, at the origin. The author obtains an algebraic formula for calculating the difference between the Euler characteristics of the stalks cohomology at the origin of the sheaves of vanishing cycles along \(f\) and along the restriction \(f \mid_ H\). Thus, he generalizes a result of \textit{Le Dung Trang} [Ann. Inst. Fourier 23(1973), No. 4, 261- 270 (1974; Zbl 0293.32013)]. In the case where \(\mathbb{F}^ \bullet\) is a perverse sheaf on a complete intersection the formula (5.11a) of \textit{E. J. N. Looijenga} [`Isolated singular points on complete intersections', Lond. Math. Soc. Lectures Note Series 77, Cambridge (1984; Zbl 0552.14002)] is also generalized. As an application it is proved that the complex link of any isolated complete intersection singularity is never contractible.
scientific article; zbMATH DE number 735668
scientific article