Limit of geometric quantizations on Kähler manifolds with T‐symmetry.

A compact Kähler manifold M,ω,J$\left(M,\omega,J\right)$ with T$T$‐symmetry admits a natural mixed polarization Pmix$\mathcal {P}_{\mathrm{mix}}$ whose real directions come from the T$T$‐action. In Leung and Wang [Adv. Math. 450 (2024), 109756], we constructed a one‐parameter family of Kähler structures ω,Jt$\left(\omega,J_{t}\right)$'s with the same underlying Kähler form ω$\omega$ and J0=J$J_{0}=J$, such that (i) there is a T$T$‐equivariant biholomorphism between M,J0$\left(M,J_{0}\right)$ and M,Jt$\left(M,J_{t}\right)$ and (ii) Kähler polarizations Pt$\mathcal {P} _{t}$'s corresponding to Jt$J_{t}$'s converge to Pmix$\mathcal {P}_{\mathrm{mix}}$ as t$t$ goes to infinity. In this paper, we study the quantum analog of above results. Assume that L$L$ is a prequantum line bundle on M,ω$\left(M,\omega \right)$. Let Ht$\mathcal {H}_{t}$ and Hmix$\mathcal {H}_{\mathrm{mix}}$ be quantum spaces defined using polarizations Pt$\mathcal {P}_{t}$ and Pmix$\mathcal {P}_{\mathrm{mix}}$, respectively. In particular, Ht=H∂¯t0M,L$\mathcal {H}_{t}=H_{\bar{\partial }_{t}}^{0}\left(M,L\right)$. They are both representations of T$T$. We show that (i) there is a T$T$‐equivariant isomorphism between H0$\mathcal {H}_{0}$ and Hmix$\mathcal {H}_{\mathrm{mix}}$ and (ii) for regular T$T$‐weight λ$\lambda$, corresponding λ$\lambda$‐weight spaces Ht,λ$\mathcal {H}_{t,\lambda }$'s converge to Hmix,λ$\mathcal {H}_{\mathrm{mix},\lambda }$ as t$t$ goes to infinity. [ABSTRACT FROM AUTHOR]