Abstract
A guiding principle in Kähler geometry is that the infinite-dimensional symmetric space of Kähler metrics in a fixed Kähler class on a polarized projective Kähler manifold should be well approximated by finite-dimensional submanifolds of Bergman metrics of height (Yau, Tian, Donaldson). The Bergman metric spaces are symmetric spaces of type where for certain . This article establishes some basic estimates for Bergman approximations for geometric families of toric Kähler manifolds.
The approximation results are applied to the endpoint problem for geodesics of , which are solutions of a homogeneous complex Monge–Ampère equation in , where is an annulus. Donaldson, Arezzo and Tian, and Phong and Sturm raised the question whether -geodesics with fixed endpoints can be approximated by geodesics of . Phong and Sturm proved weak -convergence of Bergman to Monge–Ampère geodesics on a general Kähler manifold. Our approximation results show that one has convergence in the case of toric Kähler metrics, extending our earlier result on .
Citation
Jian Song. Steve Zelditch. "Bergman metrics and geodesics in the space of Kähler metrics on toric varieties." Anal. PDE 3 (3) 295 - 358, 2010. https://doi.org/10.2140/apde.2010.3.295
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