New PDF release: Hodge Theory and Complex Algebraic Geometry

Algebraic Geometry

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 forms. the most effects are the generalized Noether-Lefschetz theorems, the prevalent 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 providing precious ends up in complicated algebraic geometry. additionally on hand: quantity I 0-521-80260-1 Hardback $60.00 C

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7 Let (x, H ) ∈ Z . Then X H has an ordinary double point at X if and only if pr2 : Z → (P N )∗ is a immersion at the point (x, H ). 18. Taking a local chart C N ⊂ P N containing x, and taking normalised affine representatives for the homogeneous forms of degree 1 on P N in the neighbourhood of H , we can replace P N in the statement by C N , and (P N )∗ by the set K of affine forms 1≤i≤N αi xi + β on C N . Then, in the neighbourhood of (x, H ), Z ⊂ X × K is defined by Z = {(y, k) ∈ X × K | k(y) = 0, dk(y) = 0}, where k is viewed as the restriction to X of the affine function k on C N .

We locally choose a complex hypersurface Y ⊂ X passing through 0, of equation t = 0, where t is a holomorphic function with non-zero differential at 0. We apply the induction hypothesis to Y , which gives holomorphic coordinates z 1 , . , z n−1 on Y satisfying the conclusion of the lemma. Extending these functions to holomorphic functions on X , we obtain n−1 f − f (0) − z i2 = 0 i=1 on Y . Then we have n−1 f − f (0) − z i2 = tg, i=1 where g is holomorphic and vanishes at 0, since d f (0) = 0 = g(0)dt.

When the Hessian is non-degenerate, the local inversion theorem shows that in a neighbourhood of 0, the set χ −1 (0) of critical points of f is reduced to {0}. 8 We proceed by induction on n. If n > 0, then clearly we can find a hypersurface Y ⊂ X passing through 0, defined in the neighbourhood of 0, and smooth at 0, such that f |Y admits 0 as a non-degenerate 22 1 The Lefschetz Theorem on Hyperplane Sections critical point. Indeed, for this last condition to be satisfied, it suffices that the non-degenerate quadratic form Hess0 f remain non-degenerate on the hyperplane TY,0 ⊂ TX,0 .

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