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 Kähler and real polarizations. Secondly, we examine a one-parameter family of Kähler 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 ``${\lim }_{t\to \mathrm{\infty }}{\mathcal{H}}_{k,t}={\mathcal{H}}_k$."