Hirzebruch-Zagier cycles in $p$-adic families and adjoint $L$-values

Let $E\mathrm{/}F$ be a quadratic extension of totally real number fields. We show that the generalized Hirzebruch-Zagier cycles arising from the associated Hilbert modular varieties can be put in $p$-adic families. As an application, using the theory of base change, we give a geometric construction of the multivariable $p$-adic adjoint $L$-function twisted by the Hecke character of $E\mathrm{/}F$, attached to Hida families of Hilbert modular forms over $F$.