Vanishing lattices and monodromy groups of isolated complete intersection singularities (Q1096055)

Let (X,0) be an isolated singularity of the complete intersection variety defined by \(X={\mathfrak f}^{-1}(0)\) where \({\mathfrak f}=(f_ 1,...,f_ k): ({\mathbb{C}}^{n+k},0)\to ({\mathbb{C}}^ k,0)\). For sufficiently small \(\epsilon\) and generic \({\mathfrak t}=(t_ 1,...,t_ k)\) \((0<t_ j\ll \epsilon)\), \(X_ t={\mathfrak f}^{-1}({\mathfrak t})\cap {\mathfrak B}_{\epsilon}\) is a smooth \((n-1)-\)connected n-dimensional complex manifold which is called the Milnor fiber of X. The homology group \(H=H_ n(X_ t,{\mathfrak Z})\) with the intersection pairing is called the Milnor lattice. Let \(\mu_+\) and \(\mu_-\) be the dimension of maximal positive, respectively negative, definite subspace of \(H_{{\mathfrak R}}=H\otimes {\mathfrak R}\) and let \(\mu_ 0\) be the rank of ker H. (X,0) is called to be parabolic if \(\mu_{-\epsilon}=0\) and \(\mu_ 0>0\) and hyperbolic if \(\mu_{-\epsilon}=1\) where \(\epsilon =(-1)^{n/2}\). The author proves that Theorem (3.1). Assume that (X,0) is not a hypersurface singularity. (i) Then (X,0) is parabolic if and only if (X,0) is of type \(\tilde D_{n+3}\) and (ii) The singularity (X,0) is hyperbolic if and only if (X,0) is of type \(T^ 2_{p,q,r,s}\) (2\(\leq p\leq r\), \(2\leq q\leq s\), \(3\leq s)\) or \(T^ n_{2,q,2,s}\) (2\(\leq q\leq s\), \(3\leq s).\) (iii) If (X,0) is neither parabolic nor hyperbolic, then \(\mu_{- \epsilon}\geq 2\).