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Vanishing cohomology of singularities of mappings. (English) Zbl 0839.32017
The authors study stabilisations \(f\) of map germs \(f_0 : (\mathbb{C}^n, O) \to (\mathbb{C}^p, O)\), where \((n,p)\) are assumed to be in Mather’s range of nice dimensions and moreover \(n < p\). In this case the image \(Y\) of the map \(f\) plays the same rôle in the theory of singularities of mappings as the Milnor fibre in the theory of isolated complete intersection singularities.
Important topological and geometrical information about the image \(Y\) is contained in the multiple point spaces \(D^k (f)\) (in the source).
The authors construct an alternating semi-simplicial resolution of the constant sheaf \(Z\) over \(Y\), which relates the topology of \(Y\) to that of the multiple point spaces of \(f\). When the corank of \(f_0\) is equal to 1 and \(2 \leq k \leq p/(p - n)\), then each \(D^k (f_0)\) is an icis and therefore \(D^kf\) has the homotopy type of a wedge of spheres in the middle dimension. This has important consequences for the degeneration of the spectral sequence for hypercohomology of the corresponding rational complex. This makes explicit calculations easy.
The case \(p = n + 1\) has extra interest. The image is now a hypersurface and has itself the homotopy type of a bouquet of spheres. The image multiple point schemes have rational cohomology only in the middle dimension.
The last two sections of the paper concern the case that \(f_0\) is quasi-homogeneous. The authors define a mixed Hodge structure on \(Y\) and compute Betti numbers and Hodge numbers of \(Y\) in terms of the quasi-homogeneous type of \(f_0\).

MSC:
32S30 Deformations of complex singularities; vanishing cycles
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
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