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Journals and Conferences
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 af condition is satisfied.
Let f : X → C and g : Y → C be analytic functions. Let π1 and π2 denote the projections of X×Y onto X and Y , respectively. In [S-T], Sebastiani and Thom prove that the cohomology of the Milnor fibre of f ◦ π1 + g ◦ π2 is isomorphic to the tensor product of the cohomologies of the Milnor fibres of f and g (with a shift in degrees); they prove this in the… (More)
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. §
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… (More)
Let (f, g) be a pair of complex analytic functions on a singular analytic space X. We give “the correct” definition of the relative polar curve of (f, g), and we give a very formal generalization of Lê’s attaching result, which relates the relative polar curve to the relative cohomology of the Milnor fiber modulo a hyperplane slice. We also give the… (More)
These notes are my continuing effort to provide a sort of working mathematician’s guide to the derived category and perverse sheaves. They began as handwritten notes for David Mond and his students, and then I decided to type them for possible inclusion as an appendix in a paper. Now, however, these notes represent more of a journal of my understanding of… (More)
Suppose that f defines a singular, complex affine hypersurface. If the critical locus of f is onedimensional at the origin, we obtain new general bounds on the ranks of the homology groups of the Milnor fiber, Ff,0, of f at the origin, with either integral or Z/pZ coefficients. If the critical locus of f has arbitrary dimension, we show that the smallest… (More)
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.
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. 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… (More)
We give a strong version of a classic inequality of Lojasiewicz; one which collapses to the usual inequality in the complex analytic case. We show that this inequality for real analytic functions allows us to construct a real Milnor fibration inside a ball.