Kendric Schefers, UC Berkeley

Abstract: The microlocal homology is a family of chain theories that interpolates between the Borel–Moore homology of a complex variety and its singular cohomology in the case when the variety is singular and Poincaré duality fails. Such a device allows one to speak of the singular support of classes in Borel–Moore homology, which we show decategorifies the Arinkin–Gaitsgory singular support of coherent sheaves in a precise sense.

The connection between these two singular support theories leverages a description of the microlocal homology in terms of the canonical perverse sheaf of vanishing cycles on shifted cotangent bundles established in our previous work, as well as the known relation between vanishing cycles and categories of matrix factorizations.