# Theorem 84 let t be a classical monodromy operator

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Unformatted text preview: es an isometry of the Milnor lattice which is called a classical monodromy operator. Example 8.1. Let 2 2 q (z1 , . . . , zn+1 ) = z1 + . . . + zn+1 . It has a unique critical point at the origin with critical value 0. An isolated critical point (f, x0 ) (resp. isolated hypersurface singularity (V, x0 )) locally analytically isomorphic to (q, 0) (resp. (q −1 (0), 0)) is called a nondegenerate critical point (resp. ordinary double point or ordinary node ). √ ∗ . For any t ∈ Dδ , the intersection Let be any positive number and δ &lt; −1 Ft = q (t) ∩ B is nonempty and is given by the equations n+1 2 zi = t, ||z || &lt; . i=1 Writing zi = xi + √ −1yi we ﬁnd the real equations ||x||2 − ||y ||2 = t, x · y = 0, ||x||2 + ||y ||2 &lt; 2 . After some smooth coordinate change, we get the equations ||x||2 = 1, x · y = 0, ||y ||2 &lt; 1. It is easy to see that these are the equations of the open unit ball subbundle of the tangent bundle of the n-sphere S n . Thus we may take Ft to be a Milnor ﬁbre of (q, 0). The sphere S n is contained in Ft as the zero section of the tangent bundle, and Ft can be obviously retracted to S n . Let α denote the fundamental class [S n ] n in Hc (Ft , Z). We have n Hc (Ft , Z) = Zα. Our orientation on Ft is equal to the orientation of the tangent bundle of the sphere taken with the sign (−1)n(n−1)/2 . This gives (α, α) = (−1)n(n−1)/2 χ(S n ) = (−1)n(n−1)/2 (1 + (−1)n ), REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 43 and hence in the case when n is even, (8.4) (α, α) = −2 2 if n ≡ 2 mod 4 . if n ≡ 0 mod 4. By collapsing the zero section S n to the point, we get a map Ft to B ∩ q −1 (0) which is a diﬀeomorphism outside S n . For this reason the homology class δ is called the vanishing cycle. The classical Picard-Lefschetz formula shows that the monodromy operator is a reﬂection transformation (8.5) T (x) = x − (−1)n(n−1)/2 (x, δ )δ. Let (V, 0) be an isolated n-dimensional hypersurface singularity. A deformation of (V, 0) is a holomorphic map-germ φ : (Cn+k , 0) → (Ck , 0) with ﬁbre φ−1 (0) isomorphic to (V, 0). By deﬁnition (V, 0) admits a deformation map f : (Cn+1 , 0) → (C, 0). Let Jf be the jacobian algebra of f (8.2) and (8.6) ¯ Jf =0 = Jf /(f ), ¯ where f is the coset of f in Jf . Its dimension τ (the Tjurina number ) is less than or equal to µ. The equality occurs if and only if (V, 0) is isomorphic to a weighted homogeneous singularity. Choose a basis z a1 , . . . , z aτ of the algebra (8.6) represented by monomials with exponent vectors a1 , . . . , aτ with aτ = 0 (this is always possible). Consider a deformation τ (8.7) Φ : (Cn+τ , 0) → (Cτ , 0), (z, u) → (f (z ) + ui z ai , u1 , . . . , uτ −1 ). i=1 This deformation represents a semi-universal (miniversal ) deformation of (V, 0). Roughly speaking this means that any deformation φ : (Cn+k , 0) → (Ck , 0) is obtained from (8.7) by mapping (Ck , 0) to (Cτ , 0) and taking the pull-back of the map φ (this explains the versal part). The map-germ (Ck , 0) → (Cτ , 0) is not unique, but its derivative at 0 is unique (whence the semi in semi-universal). Let ∆ = {(t, u) ∈ Cτ : Φ−1 (t, u) has a singular point at some (z, u)}....
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