Berezin-Toeplitz Quantization of non-compact manifolds

Abstract

We develop Berezin-Toeplitz quantization in a non-compact complex geometric setting. Let $(X,\Theta)$ be a Hermitian manifold, $(L,h^L)$ a positive holomorphic line bundle, and $(E,h^E)$ a holomorphic Hermitian vector bundle. Assuming that the Kodaira Laplacian on $(0,1)$-forms with values in $L^p\!\otimes E$ has a spectral gap growing linearly in $p$, we prove that the Bergman projection onto the $L^2$-holomorphic space $H^0_{(2)}(X,L^p\!\otimes E)$ enjoys the usual off-diagonal decay and admits a full asymptotic expansion on compact subsets as $p\to\infty$. As a consequence, for every smooth symbol $f\in\mathcal{C}^\infty_{\mathrm{const}}(X,\operatorname{End}(E))$ (constant outside a compact set), the associated Toeplitz operators $T_{f,p}=P_p f P_p$ form a closed algebra and satisfy a complete composition expansion, yielding a star-product on $\mathcal C^\infty_{\mathrm{const}}(X,\operatorname{End}(E))$ and the expected semiclassical commutator formula. We also give intrinsic criteria characterizing Toeplitz families with compactly supported kernels. We then provide geometric conditions guaranteeing the spectral gap on large classes of non-compact manifolds, via fundamental $L^2$-estimates for $\bar\partial$ on complete Hermitian manifolds (including bounded-geometry complete K\"ahler manifolds, K\"ahler-Einstein manifolds, pseudoconvex/weakly $1$-complete, and quasi-projective manifolds). Finally, for compactly supported bounded symbols, we prove a Szeg\H{o}-type theorem describing the eigenvalue distribution of the compact Toeplitz operators $T_{f,p}$ as $p\to\infty$.