2022/05/25 by Schefers, Kendric
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2205.12436
Let Z be an l.c.i. scheme over ℂ. In this paper, we introduce a Kashiwara--Schapira-style functor of derived microlocalization, which we use to define a perverse sheaf μZ on the -1-shifted cotangent bundle, T^*[-1]Z. The sheaf μZ is designed to be a refinement of the microlocal homology of Z: a family of invariants introduced by Nadler that interpolates between the singular cohomology and Borel--Moore homology of Z. Our main result is an equivalence between μZ and the DT sheaf φT^*[-1]Z on T^*[-1]Z. This provides an alternative construction for the DT sheaf in the case of a shifted cotangent bundle. The main step of our argument, which may be of independent interest, is a local computation -- closely related to one obtained recently by Kinjo using different methods -- providing a description of the classical microlocalization functor in terms of vanishing cycles.