Complements of Hypersurfaces, Variation Maps, and Minimal Models of Arrangements

Abstract

Abstract. We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in C n by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants. IntroductionTo study the topology of the complement C n \ V of an affine hypersurface V ⊂ C n one employs Morse theory, see for instance Randell [Ra], or the Lefschetz method of scanning by pencils of hyperplanes, as done e.g. by Dimca and Papadima in [DP]. Both methods yield in particular a CW-complex model of the complement C n \ V . It was proved in the above two papers that whenever V is a union of hyperplanes, then there exists a CWcomplex model which is minimal, in the sense that the number of q-cells equals the Betti number b q (C n \ V ), for any q. This notion of minimality was introduced by Papadima and Suciu in [PS] for studying the higher homotopy groups of complements of hyperplane arrangements. We give here a new proof of the minimality by using another method. We first prove the following result: Theorem 1.1. Let A be an affine arrangement of hyperplanes, not necessarily central. Let V A ⊂ C n denote the union of hyperplanes in A and let H be a generic hyperplane with respect to A. Then, one has the isomorphisms of Z-modules:Our alternate proof uses the behaviour of the variation maps within a pencil of hyperplanes on C n \ V A . It is based on a particular case (see Theorem 3.1) of a general result on vanishing cycles of pencils, which involves variation maps, proved in [Ti3,Ti6]. We discuss in §2 some aspects of the topology of pencils on complements of affine hypersurfaces, extracted from a general theory of non-generic Lefschetz pencils of hypersurfaces, which we have developped in a series of papers [Ti3,Ti4,Ti6,Ti5,Ti7]. In this context, we also give a method to compute inductively the betti numbers of complements of arrangements by using global polar invariants [Ti2].2000 Mathematics Subject Classification. 32S22, 14N20.