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) → 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. © 2009 Springer-Verlag Berlin Heidelberg.