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Register a new userSemiclassical Ohsawa-Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel
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Authors
Siarhei Finski
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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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We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $frac{1}{sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kahler potential $kvarphi$ in a $frac{1}{sqrt{k}}$-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.
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.
We show that if a compact complex manifold admits a Kahler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.
We shall give an explicit estimate of the lower bound of the Bergman kernel associated to a positive line bundle. In the compact Riemann surface case, our result can be seen as an explicit version of Tians partial $C^0$-estimate.
We study the asymptotics of the Poisson kernel and Greens functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Greens functions of the conformal Laplacian near their singularities. Our expansions of the Greens functions answer the first part of the conjecture of Kim-Musso-Wei[22] in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Greens functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [30] to show solvability of the fractional Yamabe problem for conformal infinities of dimension $3$ and fractional parameter in $(frac{1}{2} , 1)$ to a global case left by previous works.
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