Derivatives of Rankin-Selberg $L$-functions and heights of generalized Heegner cycles

Let $f$ be a newform of weight $2k$ and let $\chi$ be an unramified imaginary quadratic Hecke character of infinity type $(2t,0)$, for some integer $0<t\le k-1$. We show that the central derivative of the Rankin-Selberg $L$-function $L(f,\chi ,s)$ is, up to an explicit positive constant, equal to the Beilinson-Bloch height of a generalized Heegner cycle. This generalizes the Gross-Zagier formula (the case $k=1$) and Zhang’s higher weight formula (the case $t=0$).