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Register a new userAnalytic Bergman operators in the semiclassical limit
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Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $mathbb{C}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.
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P.G. Grinevich L.D. Landau Institute forn Theoretical Physics
, Lomonosov Moscow State University
2015
We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.
In this paper, we study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed submanifold in a complex manifold, we consider the operator which associates to a given holomorphic section of a positive line bundle over the submanifold the holomorphic extension of it to the ambient manifold with the minimal $L^2$-norm. When the tensor power of the line bundle tends to infinity, we prove an exponential estimate for the Schwartz kernel of this extension operator, and show that it admits a full asymptotic expansion in powers of the line bundle. Similarly, we study the asymptotics of the orthogonal Bergman kernel associated to the projection onto the holomorphic sections orthogonal to those which vanish along the submanifold. All our results are stated in the setting of manifolds and embeddings of bounded geometry.
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