The Chern character and the Todd class

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. Let $k$ be a field and let $X$ be a smooth equidimensional $k$-scheme of finite type (Smooth morphism of schemes), with Chow ring $A^{\ast }(X)$ (The intersection product and Chow ring of a smooth scheme) and vector-bundle group $K^0(X)$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme). When $X$ is additionally quasi-projective, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes identifies this group with coherent $K_0(X)$; that extra hypothesis is required for the coherent-group extension. Write $A^{\ast }(X{)}_{\mathbb{Q}}:=A^{\ast }(X){\otimes }_{\mathbb{Z}}\mathbb{Q}$. If $n=\dim X$, $A^i(X)=A_{n-i}(X)=0$ for $i>n$, since there are no negative-dimensional integral cycles. Every series below is evaluated only through degree $n$; in particular positive-codimension elements are nilpotent. For the rank-zero bundle the empty root list gives $\mathrm{ch}(0)=0$ and $\mathrm{td}(0)=1$.

Chern roots. The Chern roots of a finite locally free ${\mathcal{O}}_X$-module $\mathcal{E}$ of rank $r$ are formal symbols $x_1,\dots ,x_r$ with the property that the elementary symmetric functions in the $x_i$ are the Chern classes: $e_i(x_1,\dots ,x_r)=c_i(\mathcal{E})$ (Chern classes of a vector bundle on a smooth scheme); by the splitting principle every symmetric polynomial expression in the $x_i$ with rational coefficients defines a well-defined element of $A^{\ast }(X{)}_{\mathbb{Q}}$ (Additivity, naturality and the splitting principle for Chern classes).

Chern character. Define $\mathrm{ch}(\mathcal{E}):={\sum }_{j=1}^r{e}^{x_j}={\sum }_{m\ge 0}\frac{1}{m!}{p}_m(c_1,\dots ,c_m)\in A^{\ast }(X{)}_{\mathbb{Q}},$ where $p_0:=r=\mathrm{rank}\mathcal{E}$, while for $m\ge 1$ the power sum $p_m(c_1,\dots ,c_m)={\sum }_j{x}_j^m$ is a universal polynomial in the Chern classes. Thus ${\mathrm{ch}}_0(\mathcal{E})=r$; explicitly $\mathrm{ch}=r+c_1+\frac{1}{2}(c_1^2-2c_2)+\frac{1}{6}(c_1^3-3c_1{c}_2+3c_3)+\cdots$ (the denominators are why we tensor with $\mathbb{Q}$).

Todd class. Define $\mathrm{td}(\mathcal{E}):={\prod }_{j=1}^r\frac{x_j}{1-e^{-x_j}}\in A^{\ast }(X{)}_{\mathbb{Q}};$ explicitly $\mathrm{td}(\mathcal{E})=1+\frac{1}{2}{c}_1+\frac{1}{12}(c_1^2+c_2)+\frac{1}{24}{c}_1{c}_2+\cdots$. For a smooth $X$ the Todd class of $X$ is $\mathrm{td}(T_X)$ for the tangent bundle $T_X=\mathrm{Hom}({\mathrm{\Omega }}_{X\mathrm{/}k}^1,{\mathcal{O}}_X)$ (Sheaf of relative Kähler differentials, Differentials of a smooth morphism).

Line bundles. For an invertible sheaf $\mathcal{L}$ with $x=c_1(\mathcal{L})$: $\mathrm{ch}(\mathcal{L})=e^x$ and $\mathrm{td}(\mathcal{L})=x\mathrm{/}(1-e^{-x})=1+\frac{1}{2}x+\frac{1}{12}{x}^2-\cdots$.

Well-definedness. After truncation at degree $n$, the positive-degree components of the Chern character and all components of the Todd class are symmetric polynomials with rational coefficients in the Chern roots, hence universal polynomial expressions in their elementary symmetric functions, the Chern classes of Chern classes of a vector bundle on a smooth scheme. The degree-zero component of the Chern character is the rank $r$. Thus the Chern character depends only on the rank and Chern classes of $\mathcal{E}$, whereas the Todd class depends only on its Chern classes. For locally constant ranks these formulas are interpreted componentwise on the finite open-and-closed rank loci. By the splitting principle of Additivity, naturality and the splitting principle for Chern classes, these values are independent of the choice of flag bundle or filtration: any two filtrations have the same rank and elementary symmetric functions, and hence give the same polynomial values. Rational coefficients are needed for the exponential and geometric expansions, which is why the target is $A^{\ast }(X){\otimes }_{\mathbb{Z}}\mathbb{Q}$. The tangent bundle is finite locally free of rank $\dim X$ by Differentials of a smooth morphism, so its Chern roots and Todd class are defined; the Todd class is multiplicative in exact sequences, whereas the Chern character is additive, and the identification of $K^0(X)$ with $K_0(X)$ on a quasi-projective $X$ lets the character be evaluated on coherent classes through the finite locally free resolutions of Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes.