Various Numerical Invariants for Isolated Singularities

Abstract

Definition 0. Let V be a Stein analytic space with x as its only singular point. Let 7r: M -V be a resolution of the singularity of V. We shall denote dim H1(M, 9), 1 < i < n 1 by h(i), and dim Hq(M, UP) for 1 ? p < n, 1 < q < n by hP,q(M). So far as the classification problem is concerned, h (n-i) is one of the most important invariants. In this paper, we shall introduce a bunch of invariants (cf. Definition 2.6, Definition 4.1, and Definition 5.1) which are naturally attached to isolated singularities. These invariants are used to characterize the different notions of sheaves of germs of holomorphic differential forms on analytic spaces. Various formulae which relate all these invariants are proved. We also show how to calculate these invariants explicitly. Our paper is organized as follows. In section 2 we discuss the relationship between three different kinds of sheaves of germs of holomorphic forms introduced by Grauert-Grothendieck, Noether and Ferrari respectively, and the dualizing sheaf on a complex analytic space which admits only isolated singularities. We remark that the torsion sheaf of the sheaf of germs of holomorphic p-forms in the sense of Grauert-Grothendieck was studied by Brieskorn [9], Greuel [48], Kantor [19], Suzuki [38], Vetter [39] and the author [43]. In section 3 we relate the invariant g(n-l1) of a hypersurface isolated singularity with analytic invariants and topological invariants of any resolution of the singularity. The Noether's formula for the rank two bundle