多复变与复几何研讨班：Toeplitz operators and zeros of square-integrable Gaussian holomorphic sections

摘  要：For a complete Kähler manifold endowed with a positive line bundle, we use the theory of abstract Wiener spaces to construct a probabilistic model for Berezin-Toeplitz quantization. We associate to a function with compact support (a classical observable) a sequence of square-integrable Gaussian holomorphic sections. Our focus then is on the asymptotic distributions of their zeros in the semi-classical limit, in particular, we prove equidistribution results, large deviation estimates, central limit theorem of the random zeros on the support of the given function. This talk is based on the joint work with Alexander Drewitz and George Marinescu.

摘  要：For a Hermitian manifold equipped with a positive line bundle, Tian's approximation theorem states that, by considering the high tensor powers of this line bundle, the sequence of the induced Fubini-Study metrics via the Kodaira maps converges to the first Chern form of the line bundle. In this talk, we study the corresponding extensions of Tian's approximation theorem by combining the Kodaira maps with the Berezin-Toeplitz quantization, in particular with the twisting by the Toeplitz operators. Moreover, if the manifold is assumed to be compact, such a question is closely related to the study of the lowest eigenvalues of the Toeplitz operators with a given symbol. Finally, some estimates and further questions concerning the lowest eigenvalues of Toeplitz operators will be discussed.