Table 3 gives the list of isomorphism classes of

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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 Picard-Lefschetz 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 1-dimensional deformation this point. The pre-image 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 semi-universal 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 Michigan-Dearborn.

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