On the Bergman metric of Cartan-Hartogs domains
Andrea Loi, Roberto Mossa, Fabio ZUddas
TL;DR
This work investigates the Bergman metric on Cartan–Hartogs domains and introduces the Bergman dual via $K^*_{D}(z,\bar z)=K_D(z,-\bar z)$. For CH domains $M_{\Omega,\mu}$, the authors establish the equivalence of four conditions: (i) $M_{\Omega,\mu}$ is biholomorphic to the unit ball, (ii) the Bergman metric is Kähler–Einstein, (iii) after a constant rescaling, the Bergman dual is finitely projectively induced, and (iv) the dual metric yields a particular projective-induced structure. The core technique combines an explicit Bergman kernel for CH domains with analysis of a nonlinear ODE for Kähler–Einstein potentials; a Nash-algebraic diastasis argument shows that, when $\mu\in\mathbb{Q}$ and the KRS condition holds, the soliton must be trivial, forcing KE. The paper also analyzes the maximal domain of definition of the Bergman dual, provides an explicit example where the dual is not globally defined, and compares Bergman duality with other canonical CH metrics, highlighting the distinctive rigidity properties of the Bergman metric.
Abstract
We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_{Omega, mu} we prove the equivalence of: (i) M_{Omega, 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 g_{Omega, mu} and hat g_{Omega, mu}, and discuss open problems about the maximal domain on which the Bergman dual is defined.