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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 <
−1
Ft = q (t) ∩ B is nonempty and is given by the equations
n+1
2
zi = t, z  < . i=1 Writing zi = xi + √
−1yi we ﬁnd the real equations x2 − y 2 = t, x · y = 0, x2 + y 2 < 2 . After some smooth coordinate change, we get the equations
x2 = 1, x · y = 0, y 2 < 1. It is easy to see that these are the equations of the open unit ball subbundle of the
tangent bundle of the nsphere 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 PicardLefschetz 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 ndimensional hypersurface singularity. A deformation
of (V, 0) is a holomorphic mapgerm 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 semiuniversal (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 pullback of the
map φ (this explains the versal part). The mapgerm (Ck , 0) → (Cτ , 0) is not
unique, but its derivative at 0 is unique (whence the semi in semiuniversal).
Let
∆ = {(t, u) ∈ Cτ : Φ−1 (t, u) has a singular point at some (z, u)}....
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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