Higher Siegel--Weil formula for unitary groups II: corank one terms

Abstract

We prove the higher Siegel--Weil formula for \emph{corank one} terms, relating (1) the $r^{\rm th}$ central derivatives of corank one Fourier coefficients of Siegel--Eisenstein series, and (2) the degrees of special cycles of virtual dimension 0 on the moduli stack of Hermitian shtukas with $r$ legs. Notably, the formula holds for all $r$, regardless of the order of vanishing of the Eisenstein series. This extends earlier work of Feng--Yun--Zhang, who proved the higher Siegel--Weil formula for the non-singular (corank zero) terms.