Le concept de singularité isolée de fonction analytique. (On the concept of isolated singularity of analytic functions).

*(French)*Zbl 0656.32005
Complex analytic singularities, Proc. Semin., Ibaraki/Jap. 1984, Adv. Stud. Pure Math. 8, 215-227 (1987).

[For the entire collection see Zbl 0607.00005.]

Introduction (translated from the French): “We introduce the notion of isolated singularity for a complex analytic function on a reduced complex analytic space. This definition generalizes in a natural way the notion of Morse function on a singular space stratified by a complex analytic Whitney stratification introduced by F. Lazzeri and R. Pignoni [cf. R. Pignoni, Ann. Sc. Norm. Super. Pisa Cl. Sci., IV. Ser. 6, 593-608 (1979; Zbl 0432.58003); Boll. Unione Mat. Ital., V. Ser., A 17, 307-312 (1980; Zbl 0444.58002)] and used by M. Goresky and R. MacPherson [Singularities, Summer Inst., Arcata, Calif., 1981, Proc. Symp. Pure Math. 40, Part 1, 517-533 (1983; Zbl 0526.57022)]. The definition introduced here was inspired by the theory of \({\mathcal D}\)- modules, and similar results were presented at Luminy in July 1983. Note also that in the comptes-rendus of the JSPS-NSC conference, M. Kashiwara also developed a Morse theory on singular spaces inspired by the theory of differential systems à la Sato.

Our main result, announced in Section 4.2, was motivated by a question of Goresky involving Morse theory for the intersection homology of a singular analytic space. The methods used here were introduced by M. Kato and the author [Sûrikaisekikenhyûsho Kôkyûroku 266, 299-318 (1976)] and will be developed in a later work. We give mostly sketches of proofs, and our main result is stated and not proved. We have emphasized the motivations that led to the concept of isolated singularity for a complex analytic function on an arbitrary reduced complex analytic space.”

Introduction (translated from the French): “We introduce the notion of isolated singularity for a complex analytic function on a reduced complex analytic space. This definition generalizes in a natural way the notion of Morse function on a singular space stratified by a complex analytic Whitney stratification introduced by F. Lazzeri and R. Pignoni [cf. R. Pignoni, Ann. Sc. Norm. Super. Pisa Cl. Sci., IV. Ser. 6, 593-608 (1979; Zbl 0432.58003); Boll. Unione Mat. Ital., V. Ser., A 17, 307-312 (1980; Zbl 0444.58002)] and used by M. Goresky and R. MacPherson [Singularities, Summer Inst., Arcata, Calif., 1981, Proc. Symp. Pure Math. 40, Part 1, 517-533 (1983; Zbl 0526.57022)]. The definition introduced here was inspired by the theory of \({\mathcal D}\)- modules, and similar results were presented at Luminy in July 1983. Note also that in the comptes-rendus of the JSPS-NSC conference, M. Kashiwara also developed a Morse theory on singular spaces inspired by the theory of differential systems à la Sato.

Our main result, announced in Section 4.2, was motivated by a question of Goresky involving Morse theory for the intersection homology of a singular analytic space. The methods used here were introduced by M. Kato and the author [Sûrikaisekikenhyûsho Kôkyûroku 266, 299-318 (1976)] and will be developed in a later work. We give mostly sketches of proofs, and our main result is stated and not proved. We have emphasized the motivations that led to the concept of isolated singularity for a complex analytic function on an arbitrary reduced complex analytic space.”