Modularity in arithmetic Chow groups for higher codimension

Develop a canonical construction of arithmetic special cycles of codimension n>1 on regular integral models of unitary Shimura varieties—incorporating derived intersection to correct improper intersections, boundary and bad reduction corrections, and Green currents—and prove that their generating series defines a modular form valued in the arithmetic Chow group CĤ^n(𝒳).

Background

The author discusses Kudla’s modularity problem in arithmetic Chow groups for integral models of Shimura varieties. While the divisor case n=1 has been established in several settings (e.g., Bruinier–Howard–Kudla–Rapoport–Yang over Q, Qiu over totally real fields, Howard–Madapusi Pera for orthogonal groups), extending to higher codimension requires resolving technical issues: derived intersections to handle improper intersections in positive characteristic, defining correction terms at places of bad reduction and at the boundary, and constructing appropriate Green currents. Recent works begin addressing these issues in orthogonal/Hodge type settings, but the full unitary higher-codimension modularity remains open.