Goryunov 49 48 ane complex crystallographic reection

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Unformatted text preview: is always of finite index in O(M ) except in the case of hyperbolic unimodal singularities with (p, q, r ) = (2, 3, 7), (2, 4, 5), (3, 3, 4) (see Example 4.11). There is a generalization of Brieskorn’s theorem 8.9 to the case of hyperbolic singularities due to E. Looijenga [74], [75]. 9. Symmetries of singularities 9.1. Eigen-monodromy groups. Suppose we have a holomorphic map f : X → S as in section 8.1. Suppose also that a finite group G acts on all fibres of the map in a compatible way. This means that there is an action of G on X which leaves n fibres invariant. Then the cohomology groups Hc (Xs , C) become representation spaces for G, and we can decompose them into irreducible components n Hc ( X s , C ) = n Hc ( X s , C ) χ . χ∈Irr(G) One checks that the monodromy map decomposes too and defines the χ-monodromy map n ρχ0 : π1 (S ; s0 ) → GL(Hc (Xs , C)χ ). s Let E be a real vector space equipped with a bilinear form (v, w), symmetric or ¯ skew-symmetric. Let EC be its complexification with the conjugacy map v → v . We extend the bilinear form on E to EC by linearity. It is easy to see that it satisfies (¯, y ) = (x, y ). Next we equip EC with a hermitian form defined by x¯ x, y = (x, y ), ¯ i(x, y ) ¯ if (x, y ) is symmetric, otherwise. n We apply this to EC = Hc (Xs , C) with the bilinear map defined by the cupproduct. The χ-monodromy map leaves the corresponding hermitian form invariant and defines a homomorphism n ρs0 : π1 (S ; s0 ) → U(Hc (Xs , C)χ ). We are interested in examples when the image of this homomorphism is a complex reflection group. 9.2. Symmetries of singularities. Assume that the germ of an isolated hypersurface singularity (X, 0) can be represented by a polynomial f which is invariant with respect to some finite subgroup G of GL(n +1, C). One can define the notion of a G-equivariant deformation of (X, 0) and show that a semi-universal G-equivariant deformation of (X, 0) can be given by the germ of the map τ (9.1) ΦG : (Cn+τ , 0) → (Cτ , 0), (z, u) → (f (z ) + ui gi , u1 , . . . , uτ −1 ) , i=1 G,χ where (g1 , . . . , gτ ) is a basis of the subspace Jf =0 of relative invariants of the algebra (8.6). In the case when G is cyclic, we can choose gi ’s to be monomials. The equivariant discriminant ∆G is defined the same as in the case of the trivial action. Now we define the G-equivariant monodromy group ΓG of (X, x0 ) following the definition in the case G = {1}. The group G acts obviously on the Milnor fibre REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 49 of f and hence on the Milnor lattice M , giving it a structure of a Z[G]-module. Since Cτ \ ∆G ⊂ Cτ \ ∆ we can choose a point s0 ∈ Cτ \ ∆G to define a homomorphism is0 : π1 (Cτ \ ∆G ; s0 ) → π1 (Cτ \ ∆; s0 ). This homomorphism induces a natural injective homomorphism of the monodromy groups i∗ : ΓG → Γ. This allows us to identify ΓG with a subgroup of Γ. Proposition 9.1 (P. Slodowy [107]). Let Γ be the monodr...
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