arXiv subjects
Publications and source records attributed to Achim Hennings.
We consider integrals over vanishing cycles in the Milnor fibration of an isolated singularity defined by a Newton non-degenerate function. We single out a condition where the leading logarithmic term of the expansion of the integral into a logarithmic sum can be determined exactly.
A three-term complex of free modules over a local ring determines a mixed Koszul complex, whose Euler characteristic can be expressed by mixed multiplicities. As an application, we offer a simple formula for the index of a holomorphic vector field at an isolated complete intersection singularity.
In an unpublished note [H1] we have described a method to obtain a formula for the index of an analytic vector field with (complex) isolated zero on a real analytic hypersurface with (complex) isolated singularity. This formula, like the one of Eisenbud-Levine and Khimshiashvili [AGV] for smooth points, expresses the index by the signature of bilinear forms, which are defined by a local residue symbol (cf. [Ma]). In the complete intersection case, we use a generalized residue symbol, defined for free resolutions in [LJ], in the special case of generalized Koszul complexes to obtain a suitable calculus for the bilinear forms involved.
Let f be a nondegenerate power series in several variables. We describe a condition for a polynomial g which implies that the product of g by the kth power of f is not contained in the Jacobian ideal of f.
We show that for a normal isolated singularity the module of square-integrable regular differential forms of top degree is not integral over the module of Kähler differential forms. This is related to the fibre dimension of the Nash transform.
We determine, complementing a paper by Marcel Morales, the log-pluricanonical forms on a nondegenerate hypersurface. This description shows that they are extendable to 1-parameter deformations. For an equisingular deformation we thus obtain a simultaneous log-canonical model.
We give a description of the equisingularity of a family of normal surface singularities by numerical invariants belonging to them. By equisingularity we mean Whitney regularity or a more restrictive condition using the Nash modification.
We show that the dimension of the set of limits of tangent spaces along a subvariety at an irreducible isolated singularity is one less than the dimension of the subvariety itself.
It is proved that the special fibre of the Nash transform of an irreducible isolated singularity has maximal dimension.