Pullback formulas for arithmetic cycles on orthogonal Shimura varieties

TL;DR

This paper studies how special arithmetic cycles on orthogonal Shimura varieties behave under pullback to lower-dimensional embedded varieties, especially addressing cases with improper intersections using logarithmic Green current expansions.

Contribution

It introduces a method to analyze pullback behavior of special cycles, including improper intersections, via logarithmic expansions of Green currents on deformation spaces.

Findings

Describes pullback formulas for special cycles

Constructs logarithmic Green current expansions for improper intersections

Provides tools for arithmetic intersection theory on Shimura varieties

Abstract

On an orthogonal Shimura variety, one has a collection of special cycles in the Gillet-Soule arithmetic Chow group. We describe how these cycles behave under pullback to an embedded orthogonal Shimura variety of lower dimension. The bulk of the paper is devoted to cases in which the special cycles intersect the embedded Shimura variety improperly, for which we construct logarithmic expansions of certain Green currents on the deformation to the normal bundle of the embedding.