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

TL;DR

This paper constructs quantum spaces for mixed polarizations on toric varieties, analyzes their convergence from K"ahler polarizations, and establishes their limiting behavior, linking different polarization types through geometric flows.

Contribution

It introduces the quantum space for mixed polarizations on toric varieties and studies their convergence from K"ahler polarizations via symplectic potentials and imaginary-time flow.

Findings

Quantum space $\

Convergence of polarizations $\

Abstract

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 ``$\lim_{t \rightarrow \infty} \mathcal{H}_{k,t} = \mathcal{H}_{k}$."