Quantizations of Kähler metrics on blow-ups

The thesis consists of three main results related to Kähler metrics on blow-ups. In the first one, we prove that the blow-up C ̃^2 of C^2 at the origin endowed with the Burns–Simanca metric g_BS admits a regular quantization. We use this fact to prove that all coefficients in the Tian-Yau-Catlin-Zelditch expansion for the Burns–Simanca metric vanish and that a dense subset of (C ̃^2,g_BS) admits a Berezin quantization. In the second one, we prove that the generalized Simanca metric on the blow-up C ̃^n of C^n at the origin is projectively induced but not balanced for any integer n>=3. Finally, we prove as third result that any positive integer multiple of the Eguchi–Hanson metric, defined on a dense subset of C ̃^2/Z_2, is not balanced.