One can show that the conjugacy class of the image of

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Unformatted text preview: alls the invariant hypersurface of degree 5 the gem of the universe. 8. Monodromy groups 8.1. Picard-Lefschetz 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 n-dimensional 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 n-dimensional complex manifold. Moreover, each fibre has the homotopy type of a bouquet of n-spheres. 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 skew-symmetric 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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