The infinitesimal Hodge conjecture for algebraic K-theory classes

Let $\mathcal{X}/S$ be a smooth projective scheme, where $S=\operatorname{Spec}(k[[t]])$ for a field $k$ of characteristic zero. Set $S_n=\operatorname{Spec}(k[t]/t^{n+1})$ and $X_n=\mathcal{X}\times_S S_n$. Let $K_0(X_0)$ be the Grothendieck group, let $H^*_{dR}(X_0/k)$ and $H^*_{dR}(\mathcal{X}/S)$ denote de Rham cohomology, and let

$$\operatorname{ch}:K_0(X_0)\longrightarrow H^*_{dR}(X_0/k)$$

be the Chern character. Let $\nabla$ be the Gauss–Manin connection on $H^*_{dR}(\mathcal{X}/S)$, and let

$$\Phi:H^*_{dR}(\mathcal{X}/S)^\nabla\overset{\sim}{\longrightarrow}H^*_{dR}(X_0/k)$$

be the canonical isomorphism. For $\xi_0\in K_0(X_0)_{\mathbb{Q}}$, write $F^iH^{2i}_{dR}(\mathcal{X}/S)$ for the Hodge filtration. The infinitesimal Hodge conjecture. The following statements are equivalent:

1. $\Phi^{-1}\circ\operatorname{ch}(\xi_0)\in\bigoplus_i H^{2i}_{dR}(\mathcal{X}/S)^\nabla\cap F^iH^{2i}_{dR}(\mathcal{X}/S)$;
2. there exists $\xi\in K_0(\mathcal{X})_{\mathbb{Q}}$ such that

$$\operatorname{ch}(\xi|_{X_0})=\operatorname{ch}(\xi_0)\in H^*_{dR}(X_0/k).$$

This conjecture characterizes which de Rham Chern characters of special-fibre K-theory classes lift to the family. It is the infinitesimal form of the Hodge conjecture in this K-theoretic setting; the supplied text does not state whether it has been proved or remains open.