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Let X be a scheme over a trait S. Under suitable assumptions, Spencer Bloch conjectured a formula (the Bloch conductor formula) which expresses the total dimension of vanishing cohomology in terms of algebraic differential forms. The case where the special fiber has an isolated singularity appeared before as a conjecture due to Pierre Deligne and goes under the name of Deligne-Milnor formula. I'll describe a proof of several new cases of (a generalization of) the Bloch conductor formula, which include the case of isolated singularities. The content of this talk is based on a joint work with Dario Beraldo.