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Vanishing cycles of matrix singularities. (English) Zbl 1470.32086

Summary: We study local singularities of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices, that is, of mappings of smooth manifolds to the matrix spaces. Our main object is the vanishing topology of the pre-images of the hypersurface \(\Delta\) of all degenerate matrices in assumption that the dimension of the source is at least the codimension of the singular locus of \(\Delta\) in the ambient space.
We start with showing that the complex link of \(\Delta\) is homotopic to a sphere of the middle dimension and give a geometric interpretation of such spheres. This allows us to define vanishing cycles on the singular Milnor fibre of a matrix family, that is, on the local inverse image of \(\Delta\) under a generic perturbation of the family. According to Lê and Siersma, such a fibre is a wedge of middle-dimensional spheres. We prove that in some important cases, which include the Damon-Pike conjecture and all simple matrix singularities, the number \(\mu_{\Delta}\) of the spheres in the wedge is equal to the relevant Tjurina number \(\tau\) of the family.
We introduce two kinds of bifurcation diagrams for matrix families, and prove a Lyashko-Looijenga type theorem for the larger diagrams for all simple matrix families.
Making the first steps towards understanding the monodromy of matrix singularities, we define an intersection form on the singular Milnor fibres of corank 2 symmetric matrix families, which yields a complete description of the monodromy of such fibres. A modification of this approach reveals a quite unexpected relationship between certain Shephard-Todd groups, simple odd functions and simple corank 3 matrix singularities, those forming a sporadic part of the entire simple matrix classification obtained by Bruce and Bruce-Tari.
We conclude with a general \(\mu_{\Delta} = \tau\) conjecture for matrix singularities.

MSC:

32S05 Local complex singularities
32S30 Deformations of complex singularities; vanishing cycles
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
58K15 Topological properties of mappings on manifolds
32S55 Milnor fibration; relations with knot theory
58K05 Critical points of functions and mappings on manifolds
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