Berezin–Toeplitz quantization in real polarizations with toric singularities

On a compact Kähler manifold $X$, Toeplitz operators determine a deformation quantization $(\mathrm{C}^\infty (X,\mathbb{C}) [\![\hbar]\!], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1,0} X, T^{0,1} X$ as $\hbar \to 0^{+}$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13].

In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $\hbar \to 0^{+}$.

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This research was substantially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK14301619 and CUHK14306720) and direct grants from the Chinese University of Hong Kong.

Received 23 May 2021

Accepted 22 September 2022

Published 21 October 2022