Co-rank $1$ Arithmetic Siegel--Weil IV: Analytic local-to-global

Abstract

This is the fourth 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 pin down precise normalizations for some $U(m,m)$ Siegel Eisenstein series, give local Siegel--Weil special value formulas with explicit constants, and record a geometric Siegel--Weil result for degrees of complex $0$-cycles. Using this, we complete the proof of our arithmetic Siegel--Weil results by patching together the local main theorems from our companion papers.