Geodesic rays in the space of Kähler metrics with $T$-symmetry

Let $(M,\omega ,J)$ be a Kähler manifold, equipped with an effective Hamiltonian torus action $\rho :T\to \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(M,\omega ,J)$ by isometries with moment map $\mu :M\to {\mathfrak{t}}^{\ast }$. We first construct a singular mixed polarization ${\mathcal{P}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ on $M$. Second, we construct a one-parameter family of complex structures $J_t$ on $M$ which are compatible with $\omega$. Furthermore, the path of corresponding Kähler metrics $g_t$ is a complete geodesic ray in the space of Kähler metrics of $M$, when $M$ is compact. Finally, we show that the corresponding family of Kähler polarizations ${\mathcal{P}}_t$ associated to $J_t$ converges to ${\mathcal{P}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ as $t\to \mathrm{\infty }$.