Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. Bloch's conductor conjecture expresses the dimension of the vanishing cohomology in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. This is joint work with Massimo Pippi.