Isolated non-normal crossings

Abstract

We describe a multivariable polynomial invariant for certain class of non isolated hypersurface singularities generalizing the characteristic polynomial on monodromy. The starting point is an extension of a theorem due to Lê and K.Saito on commutativity of the local fundamental groups of certain hypersurfaces. The description of multivariable polynomial invariants is given in terms of the ideals and polytopes of quasiadjunction generalizing corresponding data used in the study of the homotopy groups of the complements to projective hypersurfaces and Alexander invariants of plane reducible curves.

On the Alexander invariants of hypersurface complements

“…The following consequence of Theorem 3.2, Remark 3.3, and of Example 2.8 is similar to some results in [22], [36], [37]. …”

“…The first part of the example below corresponds to the germ of a normal crossing divisor. The second part of the example below corresponds to isolated non-normal crossing divisors (for short INNC); see [22], [36], [37]. Similarly, instead of localizing at a point, one may localize along the hyperplane H at infinity, i.e.…”

“…(ii) Let (Y, 0) be an INNC singularity at the origin of ab of U(Y, 0) is (n − 1)-connected; see Libgober [36]. More precisely, it is a bouquet of n-spheres, see [22], and hence…”

“…Alexander invariants in the form of Alexander modules, characteristic varieties and Alexander polynomials have been recently intensively studied, in particular in relation to the twisted cohomology of hypersurface arrangement complements; see for instance [1], [4], [5], [6], [18], [20], [21], [22], [25], [32], [33], [36], [42], [47].…”

“…Proposition 2.5 expresses the relation between the characteristic varieties defined using the Fitting ideals and the characteristic varieties defined using the jumping loci of the cohomology with rank one local coefficients. Example 2.8 treats the simplest local situations: the normal crossing case and the case of isolated non-normal crossing singularities, whose study was initiated by A. Libgober in [36].…”