Unformatted text preview: alls the invariant hypersurface of degree 5 the
gem of the universe.
8. Monodromy groups
8.1. PicardLefschetz transformations. Let f : X → S be a holomorphic map
of complex manifolds which is a locally trivial C ∞ ﬁbration. One can construct
a complex local coeﬃcient system whose ﬁbres are the cohomology with compact
n
support Hc (Xs , Λ) with some coeﬃcient group Λ of the ﬁbres Xs = f −1 (s). The
local coeﬃcent system deﬁnes the monodromy map
(8.1) n
ρs0 : π1 (S, s0 ) → Aut(Hc (Xs0 , Λ)). We will be interested in the cases when Λ = Z, R, or C that lead to integral, real
or complex monodromy representations.
The image of the monodromy representation is called the monodromy group of
the map f (integral, real, complex).
We refer to [7], [53], [76] for some of the material which follows.
Let f : Cn+1 → C be a holomorphic function with an isolated critical point at
x0 . We will be interested only in germs (f, x0 ) of f at x0 . Without loss of generality
we may assume that x0 is the origin and f (x0 ) = 0. The level set V = f −1 (0) is
an analytic subspace of Cn+1 with isolated singularity at 0. The germ of (V, 0)
is an ndimensional isolated hypersurface singularity. In general the isomorphism
type of the germ of (V, 0) does not determine the isomorphism type of the germ
(f, 0). However, it does in one important case when f is a weighted homogeneous
polynomial.12 In this case we say that the germ (V, 0) is a weighted homogeneous
isolated hypersurface singularity.
It was shown by J. Milnor [79] that for suﬃciently small and δ , the restriction
of f to
X = {z ∈ Cn+1 : z  < , 0 < f (z ) < δ }
is a locally trivial C ∞ ﬁbration whose ﬁbre is an open ndimensional complex
manifold. Moreover, each ﬁbre has the homotopy type of a bouquet of nspheres.
The number µ of the spheres is equal to the multiplicity of f at 0 computed as the
12 This means that f (z , . . . , z
1
n+1 ) is a linear combination of monomials of the same degree,
where each variable is weighted with some positive number. 42 IGOR V. DOLGACHEV dimension of the jacobian algebra
Jf = dimC C[[z1 , . . . , zn+1 ]]/( (8.2) ∂f
∂f
,...,
).
∂z1
∂zn+1 ∗
Let D∗ = {t ∈ C : 0 < t < } and f : X → Dδ be the above ﬁbration, a Milnor
ﬁbration of (V, 0). Let
(8.3)
Mt = H n (Ft , Z) ∼ Hn (Ft , Z).
=
c The bilinear pairing
n
n
2
Hc (Ft , Z) × Hc (Ft , Z) → Hc n (Ft , Z) ∼ Z
= is symmetric if n is even and skewsymmetric otherwise. To make the last isomorphism unique we ﬁx an orientation on Ft deﬁned by the complex structure on the
open ball B = {z ∈ Cn+1 : z  < }. Thus, if n is even, which we will assume
from now on, the bilinear pairing equips Mt with a structure of a lattice, called the
Milnor lattice. Its isometry class does not depend on the choice of the point t in
∗
π1 (Dδ ; t0 ). Fixing a point t0 ∈ D∗ we obtain the classical monodromy map
π1 (D∗ ; t0 ) ∼ Z → O(Mt ).
=
0 Choosing a generator of π1 (D∗ ; t0 ), one sees that the map deﬁn...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, Vector Space, The Land, Igor V. Dolgachev, Reflection group

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