Co-rank $1$ Arithmetic Siegel--Weil III: Geometric local-to-global

TL;DR

This paper completes the reduction of global arithmetic intersection numbers of special cycles on unitary Shimura varieties to local geometric data, advancing the proof of the arithmetic Siegel--Weil formula in co-rank 1.

Contribution

It finalizes the reduction process and proposes a construction for arithmetic special cycle classes associated with singular matrices of arbitrary co-rank.

Findings

Reduction of global intersection numbers to local geometric quantities

Construction of arithmetic cycle classes for singular matrices

Advancement in the proof of the arithmetic Siegel--Weil formula

Abstract

This is the third in a sequence of four papers, where we prove the arithmetic Siegel--Weil formula in co-rank $1$ for Kudla--Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. Our arithmetic Siegel--Weil formula implies that degrees of Kudla--Rapoport arithmetic special $1$-cycles are encoded in the first derivatives of unitary Eisenstein series Fourier coefficients. In this paper, we finish the reduction process from global arithmetic intersection numbers for special cycles to the local geometric quantities in our companion papers. Building on our previous companion papers, we also propose a construction for arithmetic special cycle classes associated to possibly singular matrices of arbitrary co-rank.