Heights of Kudla-Rapoport divisors and derivatives of $$L$$ -functions.

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature $$(n-1,1)$$ . We construct an arithmetic theta lift from harmonic Maass forms of weight $$2-n$$ to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form $$f$$ a linear combination of Kudla-Rapoport divisors, equipped with the Green function given by the regularized theta lift of $$f$$ . Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of $$f$$ against a CM cycle, and (2) the central derivative of the convolution $$L$$ -function of a weight $$n$$ cusp form (depending on $$f$$ ) and the theta function of a positive definite hermitian lattice of rank $$n-1$$ . When specialized to the case $$n=2$$ , this result can be viewed as a variant of the Gross-Zagier formula for Shimura curves associated to unitary groups of signature $$(1,1)$$ . The proof relies on, among other things, a new method for computing improper arithmetic intersections. [ABSTRACT FROM AUTHOR]