TAILIEUCHUNG - Đề tài " Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments "

The interplay between geometry and topology on complex algebraic varieties is a classical theme that goes back to Lefschetz [L] and Zariski [Z] and is always present on the scene; see for instance the work by Libgober [Li]. In this paper we study complements of hypersurfaces, with special attention to the case of hyperplane arrangements as discussed in Orlik-Terao’s book [OT1]. Theorem 1 expresses the degree of the gradient map associated to any homogeneous polynomial h as the number of n-cells that have to be added to a generic hyperplane section D(h) ∩ H to obtain the complement in. | Annals of Mathematics Hypersurface complements Milnor fibers and higher homotopy groups of arrangments By Alexandru Dimca and Stefan Papadima Annals of Mathematics 158 2003 473 507 Hypersurface complements Milnor fibers and higher homotopy groups of arrangments By Alexandru Dimca and Stefan Papadima Introduction The interplay between geometry and topology on complex algebraic varieties is a classical theme that goes back to Lefschetz L and Zariski Z and is always present on the scene see for instance the work by Libgober Li . In this paper we study complements of hypersurfaces with special attention to the case of hyperplane arrangements as discussed in Orlik-Terao s book OT1 . Theorem 1 expresses the degree of the gradient map associated to any homogeneous polynomial h as the number of n-cells that have to be added to a generic hyperplane section D h n H to obtain the complement in IP D h of the projective hypersurface V h . Alternatively by results of Le Le2 one knows that the affine piece V h a V h H of V h has the homotopy type of a bouquet of n 1 -spheres. Theorem 1 can then be restated by saying that the degree of the gradient map coincides with the number of these n 1 -spheres. In this form our result is reminiscent of Milnor s equality between the degree of the local gradient map and the number of spheres in the Milnor fiber associated to an isolated hypersurface singularity M . This topological description of the degree of the gradient map has as a direct consequence a positive answer to a conjecture by Dolgachev Do on polar Cremona transformations see Corollary 2. Corollary 4 and the end of Section 3 contain stronger versions of some of the results in Do and some related matters. Corollary 6 obtained independently by Randell R2 3 reveals a striking feature of complements of hyperplane arrangements. They possess a minimal CW-structure i. e. a CW-decomposition with exactly as many k-cells as the k-th Betti number for all k. Minimality may be viewed as an .

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