Chern classes for coherent sheaves
We present in this paper a construction of Chern classes for a coherent sheaf S on a complex manifold X. In fact we construct classes Cp(S) in H2p(X, C), depending only on the smooth equivalence class of the sheaf S. and analytic classes CAp(S) in Hp(X, Ωp) (Ωp denotes the sheaf of germs of holomorphic p-forms on X). We show that these invariants extend the classical and Atiyah constructions for locally free sheaves, and that the analytic class CAp(S) is the 'leading term' of type (p, p) in the expansion by type of Cp(S). In the case of a compact manifold, we are able to show that the classes Cp(S) correspond (up to a constant factor) with those defined by Atiyah and Hirzebruch in . Thus in particular the Cp(S) are integral cohomology classes.