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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. Eigenmonodromy 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 χmonodromy
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
¯
skewsymmetric. 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 χmonodromy 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 Gequivariant deformation of (X, 0) and show that a semiuniversal Gequivariant
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 Gequivariant 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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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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