arXiv subjects
Publications and source records attributed to David B. Massey.
We investigate how the complex analytic Morse Splitting Lemma relates to the generic Lê numbers of a function, and what the relation tells us about the function. As special cases, we consider functions for which the 0-dimensional generic Lê number is at most 3 and functions for which the critical locus has dimension 1.
By using our previous results on Lê modules and an upper-bound on the betti numbers which we proved with Lê, we investigate the cohomology of Milnor fibers and the internal local systems given by the vanishing cycles of hypersurfaces with one-dimensional singular sets and small Lê numbers.
A working mathematician's summary of many results on the derived category, perverse sheaves, and vanishing cycles. This is the August 2025 version, with a completely revised section on vanishing cycles.
For a hypersurface defined by a complex analytic function, we obtain a chain complex of free abelian groups, with ranks given in terms of relative polar multiplicities, which has cohomology isomorphic to the reduced cohomology of the real link. This leads to Morse-type inequalities between the Betti numbers of the real link of the hypersurface and the relative polar multiplicities of the function.
We define and describe the properties of a class of perverse sheaves which is very useful when the base ring is not a field.
We derive a number of inequalities involving Lê numbers of non-isolated hypersurface singularities. In particular, we derive Lê-Iomdine formulas with inequalities and use these, together with Teissier's Minkowski inequalities for sectional Milnor numbers of isolated hypersurface singularities, to arrive at ``Minkowski inequalities'' for Lê numbers.
We prove a number of results involving the kernel of the identity minus the monodromy on the vanishing cycles.
We describe how the canonical and variation maps between the shifted nearby and vanishing cycles interact with Verdier dualizing.
For a complex analytic function $f$, the exceptional divisor of the jacobian blow-up is of great importance. In this paper, we show what a lemma from the thesis of Lazarsfeld tells one about the structure of this exceptional divisor.
We show that the Theorem of Bobadilla and Pełka implies a non-isolated version for aligned hyper surface singularities.
We give a characterization of the support and cosupport conditions for a perverse sheaf in terms of the Whitney filtration.
We consider a homogeneous polynomial of degree equal to a prime power and examine the cohomology of the Milnor fiber.
Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the stalk cohomology of the Milnor fiber in lowest degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant.
We describe the relationship between intersection cohomology with twisted coefficients and the perverse sheaves which play the role of the eigenspaces for the Milnor monodromy of an affine hypersurface.
We prove a result on the relationship between the relative conormal space of an analytic function $f$ on affine space and the relative conormal space of $f$ restricted to a hyperplane slice, at a point where the relative conormal space of $f$ is "microlocally trivial".
In this paper, we investigate the local Euler obstruction and the relative local Euler obstruction in terms of constructible complexes of sheaves, characteristic cycles, and vanishing cycles. The fundamental tool that we use is the notion of a characteristic complex for an analytic space embedded in affine space.
We discuss and prove a number of results for calculating characteristic cycles, or graded, enriched characteristic cycles. We concentrate particularly on results related to hypersurfaces.
We investigate deformations of functions on affine space, deformations in which the changes specialize to a distinguished point in the zero-locus of the original function. Such deformations enable us to obtain nice results on the cohomology of the Milnor fiber of the original function.