Todd class

Definition

Let $X$ be a smooth quasi-projective variety over a field, with rational Chow ring $CH^\bullet(X)_\Q = CH^\bullet(X) \otimes \Q$. Let $E$ be a vector bundle of rank $r$ on $X$ with Chern classes $c_1, \ldots, c_r$. Write formally $\sum_{i=0}^r c_i(E) t^i = \prod_{j=1}^r (1 + \alpha_j t)$, where $\alpha_1, \ldots, \alpha_r$ are the Chern roots of $E$. The Todd class of $E$ is $$\operatorname{td}(E) = \prod_{j=1}^r \frac{\alpha_j}{1 - e^{-\alpha_j}} \in CH^\bullet(X)_\Q,$$ where each factor is expanded as the power series $\frac{x}{1 - e^{-x}} = 1 + \frac{x}{2} + \frac{x^2}{12} - \frac{x^4}{720} + \cdots$. The product is symmetric in the $\alpha_j$, so it is a polynomial in $c_1, \ldots, c_r$ in each degree, and only finitely many degrees are nonzero. The Todd class of $X$ is $\operatorname{td}(X) = \operatorname{td}(T_X)$, for the tangent bundle $T_X$.