The monodromy groups of critical points of functions. II (Q791721)

[For part I see ibid. 67, 123-131 (1982; Zbl 0468.57024).] This article is devoted to a description of the monodromy group of a critical point of a holomorphic function in an even number of variables. The main result of the first part of this article (cited above) reduces the description of the monodromy group to the description of some finite group of automorphisms of some finite \({\mathbb{Z}}_{2\lambda}\)-module, where this module and the natural number \(\lambda\) are determined by the skew-symmetric form. This article is devoted to the description of this finite group. We shall prove that there are invariant subsets of this finite group in the \({\mathbb{Z}}_{2\lambda}\)-module, and any automorphism of the \({\mathbb{Z}}_{2\lambda}\)-module for which these sets are invariant belongs to this finite group (theorem 1). The definition of invariant sets see in section 5. For critical points in two variables: \(\lambda =1\); the \({\mathbb{Z}}_ 2\)-module is the homology group \(H_ 2\) of the nonsingular local level manifold with coefficients in \({\mathbb{Z}}_ 2\); and the invariant subsets are the following: (1) subset of all vanishing vectors; (2) subset of all nonvanishing vectors not belonging to the kernel of the skew-symmetric form and (3) each vector from the kernel of the skew-symmetric form (see theorem 3). - In the case of critical point of a function in an arbitrary even number of variables these invariant subsets are defined by the skew-symmetric form and the set of vanishing vectors in the homology group \(H_ 2\) with coefficients in \({\mathbb{Z}}_ 2.\) In spite of a uniform formulation of the main result of this article (theorem 1) for all critical points in an even number of variables the proof of this result consists of two, very nonsimilar cases. The first case consists of the critical points \(A_{\mu}\) and \(D_{\mu}\). For these critical points theorem 1 was proved by \textit{A. N. Varchenko} (1979, unpublished). This proof is given in an addition to this article. The second case is the main case. It contains all the other critical points. The critical points \(E_ 6\) plays a surprising role in the proof of theorem 1 in the main case. All critical points of the main case adjoin to \(E_ 6\). In the purely arithmetical proof the critical point \(E_ 6\) is used in the following way. Firstly, in the six-dimensional lattice corresponding to \(E_ 6\) there is a symplectic basis which consists of vanishing vectors (lemma 2). Therefore in the lattice corresponding to each critical point of the main case, there is a six- dimensional sublattice with this property. Secondly, the standard Dynkin diagram of \(E_ 6\) has three ''tails'', two of which are sufficiently long. This property is useful for the description of the vanishing vectors. This article contains also the description of vanishing vectors in \(H_ 2\). The vector \(x\in H_ 2\) is a vanishing vector iff this vector does not belong to Ker \(\phi {}_ 2\), and \(q(x)=1\), where \(\phi_ 2\) is a skew-symmetric form (intersection form) in \(H_ 2\) and q is a quadratic Wajnzyb function (see proposition 1). The function q is defined by the formulas: \((1)\quad q(e_ i)=1\) for some feebly distinguished basis \(e_ 1,...,e_{\mu}\) in \(H_ 2\); \((2)\quad q(x+y)=q(x)+q(y)+\phi_ 2(x,y)\) for all \(x,y\in H_ 2\). A similar statement was independently proved by \textit{W. A. M. Janssen} [''Skew- symmetric vanishing lattices and their monodromy groups'', Math. Ann. 266, 115-133 (1983)].