By Claire Voisin, Leila Schneps

The second one quantity of this contemporary account of Kaehlerian geometry and Hodge idea starts off with the topology of households of algebraic kinds. the most effects are the generalized Noether-Lefschetz theorems, the established triviality of the Abel-Jacobi maps, and most significantly, Nori's connectivity theorem, which generalizes the above. The final half bargains with the relationships among Hodge concept and algebraic cycles. The textual content is complemented through routines supplying worthy ends up in advanced algebraic geometry. additionally on hand: quantity I 0-521-80260-1 Hardback $60.00 C

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If x is contained in B. 10 If X ⊂ P N is a smooth projective complex variety, then a generic pencil (X t )t∈P1 of hyperplane sections of X is a Lefschetz pencil. 2 The holomorphic Morse lemma Let X be a complex variety, and let f : X → C be a holomorphic map. Let 0 ∈ X be a non-degenerate critical point of f . 8. 11 There exist holomorphic coordinates z 1 , . , z n on X , centred at 0, such that f can be written in these coordinates as n f (z) = f (0) + z i2 . i=1 Proof The proof is the same as in the differentiable case.

In other words, in the trivialisation C, K t is induced by the afﬁne retraction K t of the segment [− , ] onto the point − : K t (u) = (1 − t)(− ) + tu. 3) where the positive functions αt , βt for t ∈ [0, 1] are determined by the conditions αt2 f 1 + βt2 f 2 = 2 , −αt2 f 1 + βt2 f 2 = (1 − t)(− ) + t f. 12 We construct the homotopy Ht by setting Ht = Rt √ in the ball B η of radius , and Ht = K t on S[− , ] . e. 3) on η S[− , ] and with 1, t in B≤ . We set Ht = Id in B≤− . It is easy to check that we can construct such a pair (αt , βt ), which also satisﬁes the conditions Im H0 ⊂ B≤− ∪ B r and Ht |B− already satisﬁed in B η and S[− , ].

23. 24 Let X ⊂ Pn be a hypersurface. Then H k (X, Z) = 0 for k odd, k < dim X , and H 2k (X, Z) = Zh k for 2k < dim X . If moreover X is smooth, then the next result follows from Poincar´e duality. 25 Let X ⊂ Pn be a smooth hypersurface. Then H k (X, Z) = 0 for k odd, k > dim X , and H 2k (X, Z) = Zα for 2k > dim X , where the class α has intersection with h n−1−k equal to 1. 26 Let us take the case of a smooth hypersurface X in P4 . The preceding corollary shows that H2 (X, Z) = H 4 (X, Z) is generated by the unique class α such that α, h = 1.