Milnor number

Abstract: The chromatic polynomial χG(q) of a graph G counts the num- ber of proper colorings of G. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. The proof is obtained by identifying χG(q) with a sequence of numerical invariants of a projective hypersurface analogous to the Milnor number of a local analytic hypersurface. As a by-product of our approach, we obtain an analogue of Kouchnirenko’s

Abstract: The Case of Manifolds.- The Schwartz Index.- The GSV Index.- Indices of Vector Fields on Real Analytic Varieties.- The Virtual Index.- The Case of Holomorphic Vector Fields.- The Homological Index and Algebraic Formulas.- The Local Euler Obstruction.- Indices for 1-Forms.- The Schwartz Classes.- The Virtual Classes.- Milnor Number and Milnor Classes.- Characteristic Classes of Coherent Sheaves on Singular Varieties.

Abstract: We study deformations of germs of reduced complex curve singularities and of singular projective curves in some Pn(ℂ). In both cases a deformation is topologically trivial iff the Milnor numbers of the singularities are constant during the deformation. The Milnor number also occurs naturally in the degree of the singular Todd class of Baum-Fulton-MacPherson and in a formula of Deligne concerning the dimension of the base space of the semiuniversal deformation. Some applications of this fact are given in particular to the non-smooth-ability of certain curves.