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

Abstract

Let $(X, ω, J)$ 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 ``$\lim_{t \rightarrow \infty} \mathcal{H}_{k,t} = \mathcal{H}_{k}$."

Theorems & Definitions (37)

• Definition 1.1: Definition \ref{['def3-1']}
• Theorem 1.2: Theorem \ref{['thm5-0']}
• Theorem 1.3: Theorem \ref{['thm4-3']}
• Theorem 1.4: Theorem \ref{['thm3-0-4']}
• Corollary 1.5: Corollary \ref{['com-family']}
• Theorem 1.6: Theorem \ref{['thm5-1']}
• Remark 1.7
• Theorem 2.1
• Definition 2.2
• Theorem 2.3