arXiv subjects
Publications and source records attributed to David B. Massey.
We introduce graded, enriched characteristic cycles as a method for encoding Morse modules of strata with respect to a constructible complex of sheaves. Using this new device, we obtain results for arbitrary complex analytic functions on arbitrarily singular complex analytic spaces.
Given a complex analytic function with a one-dimensional critical locus at the origin, we examine the monodromy action on the integral cohomology of the Milnor fiber. We relate this monodromy to that of a generic hyperplane slice through the origin, and to that of a generic hyperplane slice near the origin. We thereby obtain number-theoretic restrictions on the monodromy and on the cohomology of the original Milnor fiber.
We show that there are obstructions to the existence of certain types of invariant subspaces of the Milnor monodromy; this places restrictions on the cohomology of Milnor fibres of non-isolated hypersurface singularities.
If a complex analytic function, $f$, has a stratified isolated critical point, then it is known that the cohomology of the Milnor fibre of $f$ has a direct sum decomposition in terms of the normal Morse data to the strata. We use microlocal Morse theory to obtain the same result under the weakened hypothesis that the vanishing cycles along $f$ have isolated support. We also investigate an index-theoretic proof of this fact.
The characteristic cycle of a complex of sheaves on a complex analytic space provides weak information about the complex; essentially, it yields the Euler characteristics of the hypercohomology of normal data to strata. We show how perverse cohomology actually allows one to extract the individual Betti numbers of the hypercohomology of normal data to strata, not merely the Euler characteristics. We apply this to the ``calculation'' of the vanishing cycles of a complex, and relate this to the work of Parusiński and Briançon, Maisonobe, and Merle on Thom's $a_f$ condition.
We prove a derived category version of the Sebastiani-Thom Theorem, which describes the vanishing cycles of the sum of two functions of disjoint variables.
We compare and contrast various notions of the "critical locus" of a complex analytic function on a singular space. After choosing a topological variant as our primary notion of the critical locus, we justify our choice by generalizing Lê and Saito's result that constant Milnor number implies that Thom's a_f condition is satisfied.
Throughout our work on the Lê cycles of an affine hypersurface singularity, 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 as gap sheaves. 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 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, which seem to contain a great deal of ``geometric'' data related to the original ideal. We prove a number of useful results, including some extremely general Lê-Iomdine-Vogel formulas; these formulas generalize the Lê Iomdine formulas that we used so profitably in previous work.