Quantum spaces associated to mixed polarizations and their limiting behavior on toric varieties

Let $(X, \omega, J)$ be a toric variety of dimension $2n$ determined by a Delzant polytope $P$. As indicated in [40], $X$ admits a natural mixed polarization $\mathcal{P}_{k}$, induced by the action of a subtorus $T^{k}$. In this paper, we first establish the quantum space $\mathcal{H}_{k}$ for $\mathcal{P}_{k}$, identifying a basis parameterized by the integer lattice points of $P$. This confirms that the dimension of $\mathcal{H}_{k}$ aligns with those derived from K\"ahler and real polarizations. Secondly, we examine a one-parameter family of K\"ahler polarizations $\mathcal{P}_{k,t}$, defined via symplectic potentials, and demonstrate their convergence to $\mathcal{P}_{k}$. Thirdly, we verify that these polarizations $\mathcal{P}_{k,t}$ coincide with those induced by imaginary-time flow. Finally, we explore the relationship between the quantum space $\mathcal{H}_{k,0}$ and $\mathcal{H}_{k}$, establishing that ``$\lim_{t \rightarrow \infty} \mathcal{H}_{k,t} = \mathcal{H}_{k}$."