The Fourier–Mukai transform from algebraic geometry may be formulated in KK-theory as the map of composition with a certain topological correspondence in the sense of Connes and Skandalis. The goal of this note is to analyze this correspondence and to describe the induced map in terms of certain natural Baum–Douglas cycles and cocycles for tori. This leads to a purely geometric description of the Baum–Connes assembly map for free Abelian groups.
"Baum–Connes and the Fourier–Mukai transform." Illinois J. Math. 66 (1) 31 - 57, April 2022. https://doi.org/10.1215/00192082-9725548