Dedicated to Professor Takuo Fukuda on his sixtieth birthday For a compact singular variety V, there are several definitions of Chern classes, the Mather class, the Schwartz-MacPherson class, the Fulton-Johnson class and so forth ([BrSc], [Fl, [FJ], [Ml, [Sell, see also [Al], [BLSS], [PP] and [Y] for recent developements). They are in the homology of V and, if Vis non-singular, they all reduce to the Poincare dual of the Chern class c*(TV) of the tangent bundle TV of V. On the other hand, for a

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... coherent sheaf F on V, the ( co homology) Chern character ch*(F) or the Chern class c*(F) makes sense if either V is non-singular or F is locally free. In this article, we propose a definition of the homology Chern character ch*(F) or the Chern class c*(F) for a coherent sheaf F on a possibly singular variety V. In this direction, the homology Chern character or the Chern class is defined in [Sc2] (see also [Kl) using the Nash type modification of V relative to the linear space associated to the coherent sheaf F. Also, the homology Todd class T(F) is introduced in [BFM] to describe their Riemann-Roch theorem. Our class is closely related to the latter. The variety V we consider in this article is a local complete intersection defined by a section of a holomorphic vector bundle over the ambient complex manifold M. If F is a locally free sheaf on V, then the class ch*(F) coincides with the image of ch*(F) by the Poincare homomorphism H*(V) ----, H*(V). This fact follows from the Riemann-Roch theorem for the embedding of V into M, which we prove at the level of Cech-de Rham cocycles. We also compute the Chern character and the Chern class of the tangent sheaf of V when V has only isolated singularities. In section 1, we discuss characteristic cocycles in the Cech-de Rham complex and define local Chern classes and characters in the Cech-de