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Some consequences of perversity of vanishing cycles. (English) Zbl 1070.14011

Let \(f\) be a nonconstant holomorphic function on a complex analytic space \(X\). For each \(x\in Y:=f^{-1}(0)\) denote by \(F_x\) the typical fibre of the Milnor fibration around \(x\), and by \(\widetilde{H}\) the reduced cohomology. The vanishing cohomology \(\widetilde{H}^j(F_x,{\mathbb Q})\) forms a constructible sheaf (the vanishing cohomology sheaf). The authors use perversity of the vanishing cycle complex to prove that the vanishing cohomology of lower degree at a point is determined by that of points near it. In the case when the hypersurface has simple normal crossings outside the point they calculate the order of vanishing explicitly. Applications are given to the size of the Jordan blocks of the monodromy operator.

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14B05 Singularities in algebraic geometry
32S20 Global theory of complex singularities; cohomological properties
32S25 Complex surface and hypersurface singularities
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32S55 Milnor fibration; relations with knot theory
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References:

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