Degeneration of Kahler polarizations to mixed polarizations on toric varieties

Let $(X,\omega ,J)$ be a toric variety of dimension $2n$ determined by a Delzant polytope. In this paper, we first construct the polarizations ${\text{shP}}_k$ by the Hamiltonian $T^k$-action on $X$ (see Theorem 3.11). We will show that ${\text{shP}}_k$ is a singular mixed polarization for $1\le k<n$, and ${\text{shP}}_n$ is a singular real polarization which coincides with the real polarization studied in \cite{BFMN} on the open dense subset of $X$. Then for each $1\le k\le n$, we will find a one-parameter family of Kähler polarizations ${\text{shP}}_{k,t}$ on $X$ that converges to ${\text{shP}}_k$ (see Theorem 3.12). Finally, we will show that ${\text{shH}}_{k,t}^T$ the space of $T^k$-invariant $J_{k,t}$-holomorphic sections converges to ${\text{shH}}_k^0$ (see Theorem 3.18).