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We confirm the quasi-projective case of Saito’s conjecture (Invent. Math. 207:597–695, 2017), namely that the cohomological characteristic classes defined byAbbes and Saito can be computed in terms of the characteristic cycles. We constructa cohomological characteristic class supported on the non-acyclicity locus of a separated morphism relatively to a constructible sheaf. As applications of the functorialproperties of this class, we prove cohomological analogs of the Milnor formula andthe conductor formula for constructible sheaves on (not necessarily smooth) varieties.Mathematics Subject Classification Primary 14F20 · Secondary 14C17 · 11S15
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