2025/10/07 by Andrea Loi, Loi, Andrea, Roberto Mossa +3
Mathematics · #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2510.06405
openalex publication_date 2025/10/07 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in Cn with Bergman kernel KD, we define the Bergman dual of (D, gD) as (D*, gD*), where D* is the maximal domain on which the modified kernel KD*(z, zbar) = KD(z, -zbar) is positive, and gD* is the Kahler metric obtained from KD*. For a Cartan-Hartogs domain MOmega, mu we prove the equivalence of: (i) MOmega, mu is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely gOmega, mu and hat gOmega, mu, and discuss open problems about the maximal domain on which the Bergman dual is defined.