  In this article, we study the Chow group of the motive associated to a
tempered global $L$-packet $\pi$ of unitary groups of even rank with respect to
a CM extension, whose global root number is $-1$. We show that, under some
restrictions on the ramification of $\pi$, if the central derivative
$L'(1/2,\pi)$ is nonvanishing, then the $\pi$-nearly isotypic localization of
the Chow group of a certain unitary Shimura variety over its reflex field does
not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups
and $L$-functions, which generalizes the Birch and Swinnerton-Dyer conjecture.
Moreover, assuming the modularity of Kudla's generating functions of special
cycles, we explicitly construct elements in a certain $\pi$-nearly isotypic
subspace of the Chow group by arithmetic theta lifting, and compute their
heights in terms of the central derivative $L'(1/2,\pi)$ and local doubling
zeta integrals. This confirms the conjectural arithmetic inner product formula
proposed by one of us, which generalizes the Gross--Zagier formula to higher
dimensional motives.
