Conformally related Riemannian metrics with non-generic holonomy

In this paper we show that if a compact connected n-dimensional manifold M has a conformal class containing two non-homothetic metrics g and {\tilde{g}=e^{2\varphi}g} with non-generic holonomy, then after passing to a finite covering, either {n=4} and {(M,g,\tilde{g})} is an ambikähler manifold, or {n\geq 6} is even and {(M,g,\tilde{g})} is obtained by the Calabi Ansatz from a polarized Hodge manifold of dimension {n-2}, or both g and {\tilde{g}} have reducible holonomy, M is locally diffeomorphic to a product {M_{1}\times M_{2}\times M_{3}}, the metrics g and {\tilde{g}} can be written as{g=g_{1}+g_{2}+e^{-2\varphi}g_{3}}\quad\text{and}\quad{\tilde{g}=e^{2\varphi}(% g_{1}+g_{2})+g_{3}}for some Riemannian metrics {g_{i}} on {M_{i}}, and φ is the pull-back of a non-constant function on {M_{2}}.

A note on Berezin-Toeplitz quantization of the Laplace operator

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint, up to an error which tends to zero when taking higher powers of the polarization line bundle.