The big Chern classes and the Chern character

摘要

Let X be a smooth scheme over a field of characteristic 0. The Atiyah class of the tangent bundle T-X of X equips T-X[-1] with the structure of a Lie algebra object in the derived category D+(X) of bounded below complexes of O-X modules with coherent cohomology [6]. We lift this structure to that of a Lie algebra object L(D-poly(1)(X)) in the category of bounded below complexes of O-X modules in Theorem 2. The "almost free" Lie algebra L(D-poly(1)(X)) is equipped with Hochschild coboundary. There is a symmetrization map I : Sym(center dot)(L(D-poly(1)(X))) -> D-poly(center dot)(X) where D-poly(center dot)(X) is the complex of polydifferential operators with Hochschild coboundary. We prove a theorem (Theorem 1) that measures how I fails to commute with multiplication. Further, we show that D-poly(center dot)(X) is the universal enveloping algebra of L(D-poly(1)(X)) in D+(X). This is used to interpret the Chern character of a vector bundle E on X as the "character of a representation" (Theorem 4). Theorems 4 and 1 are then exploited to give a formula for the big Chern classes in terms of the components of the Chern character.