Quantization of Kähler manifolds (Q2662761)

Fix a Kähler manifold \((X, \omega , J)\). The noncommutative algebra \(C^{\infty }(X)[[\hbar ]]\) equipped with a star product \(\ast _{\hbar }\) is the framework of the deformation quantization considered as a formal deformation of the commutative algebra \(C^{\infty }(X)\). The flat case \(X=\mathbb{C}^n\) is completely solved and is therefore natural to ask the following question: Do we have an honest Toeplitz action of \(C^{\infty }[[\hbar ]]\) on a ``geometric quantization'' so that this automatically determines a star product as in the flat case? In a series of three arXiv preprints the present authors gave a positive answer to this question by suitably localizing the Hilbert spaces \(H^0(X, L^{\otimes k})\) using the peak sections technique; here \(L\) is a line bundle over \(X\). The paper under review is a survey of these three works and presents new interesting relationships between the Berezin-Toeplitz quantization and the one-loop exact BV quantization on \(X\).