A direct approach to the analytic Bergman projection

Deleporte, Alix; Hitrik, Michael; Sjöstrand, Johannes

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 1, pp. 153-176

We develop a direct approach to the semiclassical asymptotics for Bergman projections in exponentially weighted spaces of holomorphic functions, with real analytic strictly plurisubharmonic weights. In particular, the approach does not make any direct use of the Kuranishi trick and it allows us to shorten and simplify proofs of a result due to [7] and [23], stating that in the analytic case, the amplitude of the asymptotic Bergman projection is a realization of a classical analytic symbol.

Nous développons une approche directe pour l’asymptotique semiclassique du projecteur de Bergman sur des espaces de fonctions holomorphes à poids exponentiel, dont le poids est analytique et strictement pluri-sous-harmonique. En particulier, cette approche n’utilise jamais directement l’astuce de Kuranishi et nous permet de raccourcir et de simplifier les preuves du fait, établi dans [7] et [23], que dans le cas analytique, l’amplitude du projecteur de Bergman asymptotique est la réalisation d’un symbole analytique classique.

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