Given a polarized complex compact projective manifold, one is led to study high tensor powers of line bundles on it. The orthogonal projection from the space of square integrable sections onto the subspace of holomorphic ones, is the Bergman projection. The investigation of its asymptotics properties turns out to have important consequences in geometry and quantization. The aim of this talk is to give an overview of these results, especially in the equivariant setting.

 

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