Bloch's conductor formula conjecture

Let $S=\operatorname{Spec} A$ be a discrete valuation ring with perfect residue field $k$ and fraction field $K$. Let $p:X\to S$ be proper and flat of finite type and relative dimension $n$, assume that the generic fiber $X_K$ is smooth over $K$ and that $X$ is regular, and write $\bar K$ for a separable closure of $K$. For a prime $\ell$ invertible in $k$, let $\chi(Y,\ell)$ denote the $\mathbb{Q}_{\ell}$-adic Euler characteristic, let $\mathsf{Sw}(X_{\bar K})$ be the Swan conductor of the $\operatorname{Gal}(\bar K/K)$-representation $H^*(X_{\bar K},\mathbb{Q}_{\ell})$, and let $[\Delta_X,\Delta_X]_S$ be Bloch's number, the degree in $\mathsf{CH}_0(k)\simeq\mathbb{Z}$ of Bloch's localized self-intersection $(\Delta_X,\Delta_X)_S\in\mathsf{CH}_0(X_k)$ of the diagonal in $X$. Bloch's conductor formula conjecture. Under these hypotheses,

$$[\Delta_X,\Delta_X]_S=\chi(X_{\bar k},\ell)-\chi(X_{\bar K},\ell)-\mathsf{Sw}(X_{\bar K}).$$

The negative of the right-hand side is the Artin conductor of $X/S$, denoted by $\mathsf{Art}(X/S)$. This conjecture identifies an intersection-theoretic invariant of a regular model with an arithmetic conductor and is a broad arithmetic analogue of the Gauss–Bonnet formula. The paper proves a version under the additional hypothesis that inertia acts with unipotent monodromy; in mixed characteristic, the conjecture remains open in general outside the cases covered by the cited results.