Euler Obstruction and Defects of Functions on Singular Varieties

Several authors have proved Lefschetz type formulae for the local Euler obstruction. In particular, a result of this type is proved in [BLS].The formula proved in that paper turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of this work is to understand what prevents the local Euler obstruction of satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ``defect'') of such functions that we define below. We give an interpretation of this defect in terms of vanishing cycles, which allows us to calculate it algebraically. When the function has an isolated singularity, our invariant can be defined geometrically, via obstruction theory. We notice that this invariant unifies the usual concepts of {\it the Milnor number} of a function and of the {\it local Euler obstruction} of an analytic set.Comment: 18 page

Profitable Cost Increases and the Shifting of Taxation : Equilibrium Response of Markets in Oligopoly

This paper considers the conjectural variations model of oligopoly and introduces a shift in its equilibrium solution : a cost-side shift, such as a change in technology or input prices, or the introduction of excise tax. The equilibrium effects of this cost-displacement are then found, deriving and examining explicit expressions for the resulting movements in individual outputs and hence in price, profits and market structure. The main motivation we offer for the exercise is methodological : to derive, for the model adopted, certain industrial-organization results of general interest and applicability, which we then put to work mostly in a more specific public finance context. The results we are referring to are, very simply, the comparative statics (of our model) of oligopoly, in response to changes in costs. It is indeed surprising that the problem is not one which has been treated systematically in the literature except for particular cases, such as special functional forms and / or symmetric industry

An overview on complex Kleinian groups

Classical Kleinian groups are discrete subgroups of PSL(2,\C) acting on the complex projective line $\P^1$, which actually coincides with the Riemann sphere, with non-empty region of discontinuity. These can also be regarded as the monodromy groups of certain differential equations. These groups have played a major role in many aspects of mathematics for decades, and also in physics. It is thus natural to study discrete subgroups of the projective group PSL(n,\C), $n > 2$. Surprisingly, this is a branch of mathematics which is in its childhood, and in this article we give an overview of it

Real map germs and higher open books

We present a general criterion for the existence of open book structures defined by real map germs (\bR^m, 0) \to (\bR^p, 0), where $m> p \ge 2$, with isolated critical point. We show that this is satisfied by weighted-homogeneous maps. We also derive sufficient conditions in case of map germs with isolated critical value.Comment: 12 page