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

Inclusion-exclusion and Segre classes

We propose a variation of the notion of Segre class, by forcing a naive `inclusion-exclusion' principle to hold. The resulting class is computationally tractable, and is closely related to Chern-Schwartz-MacPherson classes. We deduce several general properties of the new class from this relation, and obtain an expression for the Milnor class of a scheme in terms of this class.Comment: 8 page

Verdier specialization via weak factorization

Let X in V be a closed embedding, with V - X nonsingular. We define a constructible function on X, agreeing with Verdier's specialization of the constant function 1 when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence on the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich et al. The main property of the specialization function is a compatibility with the specialization of the Chern class of the complement V-X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier's result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart in a motivic group. The specialization function and the corresponding Chern class and motivic aspect all have natural `monodromy' decompositions, for for any X in V as above. The definition also yields an expression for Kai Behrend's constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.Comment: Minor revision. To appear in Arkiv f\"or Matemati

Structure of molecular packing probed by polarization-resolved nonlinear four-wave mixing and coherent anti-Stokes Raman-scattering microscopy

International audienceWe report a method that is able to provide refined structural information on molecular packing in biomolecular assemblies using polarization-resolved four-wavemixing and coherent anti-Stokes Raman-scattering microscopy. These third-order nonlinear processes allow quantifying high orders of symmetry which are exploited here to reveal a high level of detail in the angular disorder behavior at the molecular scale in lipid membranes

Ultimate use of two-photon fluorescence microscopy to map orientational behavior of fluorophores

International audienceThe orientational distribution of fluorophores is an important reporter of the structure and function of their molecular environment. Although this distribution affects the fluorescence signal under polarized-light excitation, its retrieval is limited to a small number of parameters. Because of this limitation, the need for a geometrical model (cone, Gaussian, etc.) to effect such retrieval is often invoked. In this work, using a symmetry decomposition of the distribution function of the fluorescent molecules, we show that polarized two-photon fluorescence based on tunable linear dichroism allows for the retrieval of this distribution with reasonable fidelity and without invoking either an a priori knowledge of the system to be investigated or a geometrical model. We establish the optimal level of detail to which any distribution can be retrieved using this technique. As applied to artificial lipid vesicles and cell membranes, the ability of this method to identify and quantify specific structural properties that complement the more traditional molecular-order information is demonstrated. In particular, we analyze situations that give access to the sharpness of the angular constraint, and to the evidence of an isotropic population of fluorophores within the focal volume encompassing the membrane. Moreover, this technique has the potential to address complex situations such as the distribution of a tethered membrane protein label in an ordered environment