David B. Massey

Lê Cycles and Hypersurface Singularities

Definitions and basic properties.- Elementary examples.- A handle decomposition of the milnor fibre.- Generalized Le-Iomdine formulas.- Le numbers and hyperplane arrangements.- Thoms a f condition.- Aligned singularities.- Suspending singularities.- Constancy of the Milnor fibrations.- Other characterizations of the Le cycles.

The Sebastiani–Thom Isomorphism in the Derived Category

We prove that the Sebastiani–Thom isomorphism for Milnor fibres and their monodromies exists as a natural isomorphism between vanishing cycles in the derived category.

Hypercohomology of Milnor fibres

In this paper, we prove a number of results which help describe the hypercohomology of general Milnor fibres with coefficients in bounded, constructible complexes of sheaves. The principal goal of all of these results is to provide a means of algebraically calculating some pieces of data typically associated with a complex analytic singularity: the cohomology groups of the Milnor fibre of a function, the cohomology of the complex link of a space at a point, and the characteristic cycle of a complex of sheaves. Some of these results have appeared in substantially weaker forms in our earlier paper [Mas1].

Critical points of functions on singular spaces

Abstract 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 Le and Saitos result that constant Milnor number implies that Thoms a f condition is satisfied.

The Lê varieties, I.

For a complex polynomial,f:(ℂ n+1 ,0) → (ℂ, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Le varieties, Λ () , off at 0. The multiplicities of these varieties, λ () , generalize the Milnor number for an isolated singularity. In particular, we show that ifs≤n-2, the Milnor, fibre off is obtained fromB 2n by successively attaching λ () k-handles, wheren-s≦k≦n Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of λ () circles by successively attaching λ () k-handles, where 2≦k≦n.

Real and Complex Singularities: Real analytic Milnor fibrations and a strong Łojasiewicz inequality

We give a strong version of a classic inequality of \L ojasiewicz; one which collapses to the usual inequality in the complex analytic case. We show that this inequality for a pair, quadruple, or octuple of real analytic functions allows us to construct a real Milnor fibration inside a ball.

Natural commuting of vanishing cycles and the Verdier dual

We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a {\bf natural} isomorphism, even when the coefficients are not in a field.

Singularities II: Geometric and Topological Aspects

In this paper we study the Milnor fibrations associated to real analytic map germs ψ : (R, 0) → (R, 0) with isolated critical point at 0 ∈ R. The main result relates the existence of called Strong Milnor fibrations with a transversality condition of a convenient family of analytic varieties with isolated critical points at the origin 0 ∈ R, obtained by projecting the map germ ψ in the family L −θ of all lines through the origin in the plane R .

INTERSECTION COHOMOLOGY, MONODROMY AND THE MILNOR FIBER

We say that a complex analytic space, X, is an intersection cohomology manifold if and only if the shifted constant sheaf on X is isomorphic to intersection cohomology; with field coefficients, this is quickly seen to be equivalent to X being a homology manifold. Given an analytic function f on an intersection cohomology manifold, we describe a simple relation between V(f) being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of f. We then describe how the Sebastiani–Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Finally, as an easy application, we obtain restrictions on the cohomology of the Milnor fiber of a hypersurface with a special type of one-dimensional critical locus.

Perversity, duality, and arrangements in C3

Abstract In this paper, we use the perversity and self-duality of the sheaf of vanishing cycles to obtain previously unknown bounds on the Betti numbers of the Milnor fibre of a central hyperplane arrangement in C 3 . Moreover, we obtain restrictions on the monodromy action on cohomology which yield number-theoretic constraints on the Betti numbers of the Milnor fibre.