Characteristic Classes of Coherent Sheaves on Singular Varieties

Abstract

As we have seen along this book, for a singular variety V , there are several definitions of Chern classes, the Mather class, the Schwartz. MacPherson class, the Fulton.Johnson class and so forth. They are in the homology of V and, if V is nonsingular, they all reduce to the Poincaré dual of the Chern class c^*(TV ) of the tangent bundle TV of V . On the other hand, for a coherent sheaf F on V , the (cohomology) Chern character ch^*(F) or the Chern class c^*(F) makes sense if either V is nonsingular or F is locally free. In this chapter, 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 [140] (see also [100]) using the Nash type modification of V relative to the linear space associated to the coherent sheaf F. Also, the homology Todd class τ(F) is introduced in [15] to describe their Riemann-Roch theorem. Our class is closely related to the latter.

The variety V we consider in this chapter 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 Poincaré homomorphism $H_* (V) \to H*(V)$. This fact follows from the Riemann–Roch theorem for the embedding of V into M, which we prove at the level of Čech–de Rham cocycles. We also compute the Chern character and the Chern class of the tangent sheaf of V , in the case V has only isolated singularities.