The big Chern classes and the Chern character

Let $X$ be a smooth scheme over a field of characteristic 0. Let ${\text{dd}}^{\bullet }(X)$ be the complex of polydifferential operators on $X$ equipped with Hochschild co-boundary. Let $L({\text{dd}}^1(X))$ be the free Lie algebra generated over $\text{strc}$ by ${\text{dd}}^1(X)$ concentrated in degree 1 equipped with Hochschild co-boundary. We have a symmetrization map $I: \oplus_k \sss^k(L(\dd^1(X))) \rar \dd^{\bullet}(X)$. Theorem 1 of this paper measures how the map $I$ fails to commute with multiplication. A consequence of Theorem 1 and Theorem 2 is Corollary 1, a result “dual” to Theorem 1 of Markarian [3] that measures how the Hochschild-Kostant-Rosenberg quasi-isomorphism fails to commute with multiplication. In order to understand Theorem 1 conceptually, we prove a theorem (Theorem 3) stating that ${\text{dd}}^{\bullet }(X)$ is the universal enveloping algebra of $T_X[-1]$ in $\text{dcat}$. An easy consequence of Theorem 3 is Theorem 4, which interprets the Chern character $E$ as the “character of the representation $E$ of $T_X[-1]$” and gives a description of the big Chern classes of $E$. Finally, Theorem 4 along with Theorem 1 is used to give an explicit formula (Theorem 5) expressing the big Chern classes of $E$ in terms of the components of the Chern character of $E$.