Heights of small CM cycles on Unitary Shimura Varieties

We compute the heights of small CM cycles on unitary Shimura varieties for general choices of the one-dimensional Hermitian subspace $W$ defining the embedding. The formula contains the logarithmic derivative of a quadratic Hecke $L$-function and explicit local terms depending on the choice of $W$. We prove this formula and an arithmetic Siegel–Weil identity by comparing the holomorphic projection of a mixed theta–Eisenstein derivative with arithmetic intersections of special divisors.