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Unformatted text preview: Let (∆, 0) be the germ of ∆ at 0. It is called the bifurcation diagram or discriminant
of f . Choose a representative of Φ deﬁned by restricting the map to some open ball
B in Cn+τ with center at 0 and let U be an open neighborhood of 0 in Cτ such
that π −1 (U ) ⊂ B . When B is small enough one shows that the restriction map
π : π −1 (U \ U ∩ ∆) → U \ ∆
is a locally trivial C ∞ ﬁbration with ﬁbre diﬀeomorphic to a Milnor ﬁbre F of (f, 0).
Fixing a point u0 ∈ U \ ∆, we get the global monodromy map of (X, 0),
(8.8) n
π1 (U \ ∆; u0 ) → O(Hc (π −1 (u0 ), Z)) ∼ O(M ),
= where M is a ﬁxed lattice in the isomorphism class of Milnor lattices of (f, 0).
One can show that the conjugacy class of the image of the global monodromy
map (the global monodromy group of (X, 0)) does not depend on the choices of
B, U, u0 . 44 IGOR V. DOLGACHEV Theorem 8.2. The global monodromy group Γ of an isolated hypersurface singularity of even dimension n = 2k is a subgroup of Ref−2 (M ) if k is odd and Ref2 (M )
otherwise.
Proof. This follows from the PicardLefschetz theory. Choose a point (t0 , u(0) ) ∈
(0)
Cτ \ ∆ close enough to the origin and pass a general line ui = ci (t − t0 )+ ui through
n+τ
is the 1dimensional deformation
this point. The preimage of this line in C
˜
Cn+1 → C given by the function t = f (z ) implicitly deﬁned by the equation
τ f (z ) + t(−1 + τ
(0) a
ci zi i ) +
i=1 ( ui a
− ci t0 )zi i = 0. i=1 By choosing the line general enough, we may assume that all critical points of this
function are nondegenerate and critical values are all distinct. Its ﬁbre over the
(0) a
τ
point t = t0 is equal to the ﬁbre of the function f (z ) + i=1 ui zi i over t0 . Since
the semiuniversal deformation is a locally trivial ﬁbration over the complement of
∆, all nonsingular ﬁbres are diﬀeomorphic. It follows from the Morse theory that
˜
the number of critical points of f is equal to the Milnor number µ. Let t1 , . . . , tµ be
the critical values. Let S (ti ) be a small circle around ti and ti ∈ Si . The function
˜
f deﬁnes a locally trivial C ∞ ﬁbration over C \ {t1 , . . . , tµ } so that we can choose
a diﬀeomorphism of the ﬁbres:
˜
φ i : f − 1 ( t 0 ) → π − 1 ( u0 ) .
˜
˜
The Milnor ﬁbre Fi of f at ti is an open subset of f −1 (ti ) and hence deﬁnes an
inclusion of the lattices:
˜
˜
Hn (Fi , Z) → Hn (f −1 (t ), Z) ∼ Hn (f −1 (t0 ), Z) ∼ Hn (π −1 (u0 ), Z) ∼ M.
=
=
=
i The image δi of the vanishing cycle δi generating Hn (Fi , Z) in M is called a vanishing cycle of the Milnor lattice M . The vanishing cycles (δ1 , . . . , δµ ) form a basis
in M .
For each ti choose a path γi in C \ {t1 , . . . , tµ } which connects t0 with ti and
when continues along the circle S (ti ) in counterclockwise fashion until returning to
ti and going back to t0 along the same path but in the opposite direction. The
homotopy classes [γi ] of the path...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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