Gap Sheaves and Vogel Cycles

Throughout our work on the Lê cycles of an affine hypersurface singularity (see [M2-5]), our primary algebraic tool consisted of a method for taking the Jacobian ideal of a complex analytic function and decomposing it into pure-dimensional “pieces”. These pieces were obtained by considering the relative polar varieties of Lê and Teissier (see, for example, [L-T], [T1], [T2]) as gap sheaves in the sense of [S-T]. A gap sheaf is a formal device which gives a scheme-theoretic meaning to the analytic closure of the difference of an initial scheme and an analytic set. We would like to extend our results on Lê cycles to functions on an arbitrary complex analytic space, and so we need to generalize this algebraic approach. We begin with an ordered set of generators for an ideal, and produce a collection of pure-dimensional analytic cycles, the Vogel cycles (see [G1], [G2], and [V]), which seem to contain a great deal of “geometric” data related to the original ideal. The Vogel cycles are defined using gap sheaves, together with the associated analytic cycles which they define, the gap cycles. If the underlying analytic space is not Cohen-Macaulay, the main technical problem is that there are, at least, three different reasonable definitions of the gap sheaves and cycles; we select as “the” definition the one that works most nicely in inductive proofs. We show, however, that if one re-chooses the functions defining the ideal in a suitably “generic” way, then all competing definitions for the gap cycles and Vogel cycles agree. In Section 3, we prove some extremely general Lê-Iomdine-Vogel formulas; these formulas generalize the Lê-Iomdine formulas that we used so profitably in [M2-5].