Limit of geometric quantizations on Kähler manifolds with T-symmetry (2307.07759v2)

Abstract: A compact K\"ahler manifold $\left( M,\omega ,J\right) $ with $T$-symmetry admits a natural mixed polarization $\mathcal{P}{\mathrm{mix}}$ whose real directions come from the $T$-action. In \cite{LW1}, we constructed a one-parameter family of K\"ahler structures $\left( \omega ,J{t}\right) $'s with the same underlying K\"a hler form $\omega $ and $J_{0}=J$, such that (i) there is a $T$-equivariant biholomorphism between $\left( M,J_{0}\right) $ and $\left( M,J_{t}\right) $ and (ii) K\"ahler polarizations $\mathcal{P} {t}$'s corresponding to $J{t}$'s converge to $\mathcal{P}{\mathrm{mix}}$ as $t$ goes to infinity. In this paper, we study the quantum analog of above results. Assume $L$ is a pre-quantum line bundle on $\left( M,\omega \right) $. Let $\mathcal{H}{t}$ and $ \mathcal{H}{\mathrm{mix}}$ be quantum spaces defined using polarizations $\mathcal{P}{t}$ and $\mathcal{P}{\mathrm{mix}}$ respectively. In particular, $\mathcal{H}{t}=H_{\bar{\partial}{t}}^{0}\left( M,L\right) $. They are both representations of $T$. We show that (i) there is a $T$-equivariant isomorphism between $\mathcal{H}{0}$ and $\mathcal{H}{\mathrm{mix}}$ and (ii) for regular $T$-weight $\lambda $, corresponding $\lambda $-weight spaces $ \mathcal{H}{t,\lambda }$'s converge to $\mathcal{H}_{\mathrm{mix},\lambda }$ as $t$ goes to infinity.