The arithmetic geometry of Shimura varieties has been intensively studied since, about twenty years ago, Kudla made some conjectures relating their arithmetic Chow groups with derivatives of Eisenstein series and of Rankin-Selberg L-functions. The conjectures concern special cycles in orthogonal and unitary Shimura varieties and predict in particular that Green currents for these cycles should exist satisfying some additional properties, including an explicit expression for archimedean height pairings.
I will explain how to attach a natural superconnection to each special cycle and how results of Quillen and further developments by Bismut, Gillet and Soule allow to define natural Green forms for special cycles. For compact Shimura varieties with underlying group O(p,2) or U(p,1) I will explain how to compute the resulting archimedean heights and relate them to derivatives of Eisenstein series, essentially settling the archimedean aspect of Kudla's conjectures in this case. This is joint work with Siddarth Sankaran.