arXiv subjects
Publications and source records attributed to Adam Parusinski.
Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield $2^{43}-43$ independent characteristic numbers mod 2, which generalize the Akbulut-King numbers in dimension three.
We give a new formula for the Chern-Schwartz-MacPherson class of a hypersurface in a nonsigular compact complex analytic variety. In particular this formula generalizes our previous result on the Euler characteristic of such a hypersurface. Two different approaches are presented. The first is based on the theory of characteristic cycle and the works of Sabbah, Briancon-Maisonobe-Merle, and Le-Mebkhout. In particular, this approach leads to a simple proof of a formula of Aluffi for the above mentioned class. The second approach uses Verdier's specialization property of the Chern-Schwartz-MacPherson classes. Some related new formulas are also given.
We show that every $μ$-constant family of isolated hypersurface singularities of type f(x) + tg(x), where t is a parameter, is topologically trivial. In the proof we construct explicitely a vector field trivializing the family. The proof uses only the curve selection lemma and hence, for an appropriately translated statement, also works over the reals. Some applications to the study of singularities at infinity of complex polynomials are given.
Let X be a complex analytic space and let f:X -> C be a proper complex analytic function with nonsingular generic fibres. By adapting the blowanalytic methods of Kuo we construct a retraction of a neighbourhood of the central fibre f^{-1}(0) onto f^{-1}(0). Our retraction is defined by the flow of a real analytic vector field on an oriented real analytic blow-up of X. Then we describe in terms of this blow-up the associated specialization map and local Milnor fibrations. The method also works in real analytic category.
We show that on real algebraic sets algebraically constructible functions coincide with the finite sums of signs of polynomials. Then we give some applications.
An algebraic version of Kashiwara and Schapira's calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and King's numerical conditions for a stratified set of dimension three to be algebraic. These properties, which include generalizations of the invariants modulo 4, 8, and 16 of Coste and Kurdyka, are defined using the link operator on the ring of constructible functions.
We study the Euler characteristic of the real Milnor fibres of a real analytic map, using a relation between complex monodromy and complex conjugation. We deduce the result of Coste and Kurdyka that the Euler characteristic of the link of an irreducible algebraic subset of a real algebraic set is generically constant modulo 4. We generalize this result to iterated links of ordered families of algebraic subsets.