Studies on Characteristic Classes of Singular Varieties and Invariants of Singularities

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(1). We showed that the proof of Parusinski-Pragacz's Theorem on the Parusinski's generalized Milnor number also works for the Milnor class.(2). By introducing a "q-deformed" local Euler obstruction, we constructed a "q-deformed" Chern-Schwartz-MacPhjerson class which generalize both the Chern-Mather class and the Chern-Schwartz-MacPherson class. (A purpose of this paper is to point out that the Chern-Mather class can be also captured as a natural transformation).(3). We obtained some results on bivariant constructible functions, such as the observation that the Chern-Schwartz-MacPherson class of a bivariant constructible function restricted to each fiber of a morphism is locally constant, etc.(4). Using the bivariant theoru of Fulton-MacPherson, we obtained some reasonable pull-back formulas for the Milnor class for certain morphisms.(5). By obtaining product formulas for the Milnor class, we obtained a Parusinski-Pragacz-type formula for the product of hypersuraces and also we could give some kind of geometric meaning to a certain cohomology class appearing in the Bruselet-Lehmann-Seade-Suwa's formula for the Milnor class, and also we obtained a Thom-Sebastiani-type formula for the Milnor class.(6). We showed that the existence of bivariant Chern class with values in the bivariant Chow groups is equivalent to the existence of the commutaive "Verdier-Riemann-Roch" diagram for the Chern-Schwartz-MacPherson class.