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Edit. The odd Bernoulli numbers are zero, of course! So the first attempt below is wrong. The revised examples use complete subvarietes of moduli spaces of curves.

Edit. The odd Bernoulli numbers are zero, of course! So the first attempt below is wrong. The revised examples use complete subvarietes of moduli spaces of curves. These revised examples use nonvanishing of the even Bernoulli numbers.

Theorem [Mumford, Formula (5.2), p. 304, Towards an enumerative geometry ...] For every integer $s\geq 1$, $\text{ch}_{2s}(\mathbf{E}_g)$ is torsion, and $$\text{ch}_{2s-1}(\mathbf{E}_g) =(-1)^{s+1}\frac{|B_{2s}|}{(2s)!}\kappa_{2s-1}\in \text{CH}^{2s-1}(\mathcal{M}_g)\otimes \mathbb{Q}.$$

Theorem [Mumford, Formula (5.2), p. 304, Towards an enumerative geometry ...] For every integer $s\geq 1$, $\text{ch}_{2s}(\mathbf{E}_g)$ is torsion, and $$\text{ch}_{2s-1}(\mathbf{E}_g) =(-1)^{s+1}\frac{|B_{2s}|}{(2s)!}\kappa_{2s-1}\in \text{CH}^{2s-1}(\mathcal{M}_g)\otimes \mathbb{Q},$$ where $B_{2s}$ is the Bernoulli number (nonzero since $2s$ is even).

Revised examples from complete subvarieties of moduli spaces of curves. Let $g\geq 2$ be an integer. Denote by $\mathcal{M}_g$ the Deligne-Mumford stack over $\text{Spec}\ \mathbb{C}$ parameterizing families of smooth, projective, geometrically connected curves of genus $g$. Denote the universal curve as $$\pi:\mathcal{C}_g \to \mathcal{M}_g.$$ Denote by $\omega_{\pi}$ the relative dualizing sheaf on $\mathcal{C}_g$, and denote the pushforward to $\mathcal{M}_g$ by $$\mathbf{E}_g := \pi_* \omega_{\pi}.$$ This is the Hodge bundle. It is a locally free $\mathcal{O}_{\mathcal{M}_g}$-module (for the étale site with the usual structure sheaf) of rank $g$. By relative duality, the K-theory class of $R\pi_*\mathcal{O}_{\mathcal{C}_g}$ equals $1-[\mathbf{E}_g^\vee]$. Thus, the Chern character of $R\pi_*\mathcal{O}_{\mathcal{C}_g}$ equals $$(1-g) + \sum_{r\geq 1}(-1)^{r+1}\text{ch}_r(\mathbf{E}_g).$$ Mumford computes the Chern character of the Hodge bundle over the compactified moduli stack $\overline{\mathcal{M}}_g$ in Formula (5.2), p. 304 of the following.

Theorem [Mumford, Formula (5.2), p. 304, Towards an enumerative geometry ...] For every integer $s\geq 1$, $\text{ch}_{2s}(\mathbf{E}_g)$ equals $0$, and $$\text{ch}_{2s-1}(\mathbf{E}_g) =(-1)^{s+1}\frac{|B_{2s}|}{(2s)!}\pi_*(c_1(\omega_\pi)^{2s}).$$

Revised examples from complete subvarieties of moduli spaces of curves. Let $g\geq 2$ be an integer. Denote by $\mathcal{M}_g$ the Deligne-Mumford stack over $\text{Spec}\ \mathbb{C}$ parameterizing families of smooth, projective, geometrically connected curves of genus $g$. Denote the universal curve as $$\pi:\mathcal{C}_g \to \mathcal{M}_g.$$ Denote by $\omega_{\pi}$ the relative dualizing sheaf on $\mathcal{C}_g$.

Definition. The Hodge bundle is the pushforward of the relative dualizing sheaf to $\mathcal{M}_g$, $$\mathbf{E}_g := \pi_* \omega_{\pi}.$$ For every integer $r\geq 0$, the kappa class of degree $r$ is the (cohomological) cycle class $$\kappa_r = \pi_*(c_1(\omega_{\pi})^{r+1})\in \text{CH}^{\ r}(\mathcal{M}_g).$$

The Hodge bundle is a rank $g$, locally free $\mathcal{O}_{\mathcal{M}_g}$-module (for the étale site with the usual structure sheaf). The kappa classes are elements in the Chow groups of the stack; the tensor product of these Chow groups with $\mathbb{Q}$ equal the associated $\mathbb{Q}$-Chow groups of the quasi-projective coarse moduli space $M_g$ by work of Angelo Vistoli. Using the Grothendieck-Riemann-Roch formula, Mumford computed the Chern character of the Hodge bundle on the compactified moduli space $\overline{\mathcal{M}}_g$.

Theorem [Mumford, Formula (5.2), p. 304, Towards an enumerative geometry ...] For every integer $s\geq 1$, $\text{ch}_{2s}(\mathbf{E}_g)$ is torsion, and $$\text{ch}_{2s-1}(\mathbf{E}_g) =(-1)^{s+1}\frac{|B_{2s}|}{(2s)!}\kappa_{2s-1}\in \text{CH}^{2s-1}(\mathcal{M}_g)\otimes \mathbb{Q}.$$

Corollary. The $\mathbb{Q}$-cycle class of the Chern character of $R\pi_*\mathcal{O}_{\mathcal{C}_g}$ equals $$(1-g) + \sum_{s\geq 1}(-1)^{s+1} \frac{|B_{2s}|}{(2s)!} \kappa_{2s-1}.$$

Proof. By relative duality, the K-theory class of $R\pi_*\mathcal{O}_{\mathcal{C}_g}$ equals $1-[\mathbf{E}_g^\vee]$. Thus, the Chern character of $R\pi_*\mathcal{O}_{\mathcal{C}_g}$ equals $$(1-g) + \sum_{r\geq 1}(-1)^{r+1}\text{ch}_r(\mathbf{E}_g).$$ QED