Sheaf Singular Support

The singular support of a constructible sheaf is a closed conical subset of the cotangent bundle recording directions in which the sheaf fails local acyclicity. For a smooth variety over and constructible complexes with torsion coefficients prime to the field characteristic, it is the smallest such subset for which every -transversal test pair is locally acyclic relative to the pulled-back complex. Transversality requires that the pullback of contain no nonzero covector killed by the first differential, and that no nonzero covector from map into its image in . A nonzero locally constant sheaf has the zero section as singular support (Beilinson 2016). This cotangent-space invariant differs from the base-space singular support of a distribution.