The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and viceversa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes.

a b s t r a c tLet (R, m) be a d-dimensional Noetherian local ring. In this work we prove that the mixed Buchsbaum-Rim multiplicity for a finite family of R-submodules of R p of finite colength coincides with the Buchsbaum-Rim multiplicity of the module generated by a suitable superficial sequence, that is, we generalize for modules the well-known Risler-Teissier theorem. As a consequence, we give a new proof of a generalization for modules of the fundamental Rees' mixed multiplicity theorem, which was first proved by Kirby and Rees in (1994, [8]). We use the above result to give an upper bound for the minimal number of generators of a finite colength R-submodule of R p in terms of mixed multiplicities for modules, which generalize a similar bound obtained by Cruz and Verma in (2000, [5]) for m-primary ideals.

Let (R, m) be a Noetherian local ring. In this work we extend the notion of mixed multiplicities of modules, given in [12] and [9] (see also [3]), to an arbitrary family E, E 1 , . . . , E q of R-submodules of R p with E of finite colength. We prove that these mixed multiplicities coincide with the Buchsbaum-Rim multiplicity of some suitable R-module. In particular, we recover the fundamental Rees's mixed multiplicity theorem for modules, which was proved first by Kirby and Rees in [9] and recently also proved by the authors in [3]. Our work is based on, and extend to this new context, the results on mixed multiplicities of ideals obtained by Viêt in [25] and Manh and Viêt in [13]. We also extend to this new setting some of the main results of Trung in [20] and Trung and Verma in [21]. As in [12], [9] and [3], we actually work in the more general context of standard graded R-algebras.

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