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Convergence of Bergman geodesics on $$\text{CP}^1$$. (English) Zbl 1144.53089
Let $$L\rightarrow X$$ be an ample holomorphic line bundle over a Kähler manifold $$X$$. Fix an integer $$N$$. For any basis $$\{s_0,s_1,\dots,s_d\}$$ of the space of holomorphic sections of $$L^N$$, we get the metric $$(1/N)\Phi*\omega_{FS}$$, where $$\Phi(z) =[s_0(z), s_1(z), \dots, s_d(z)]\in {\mathbb P}^d$$ and $$\omega_{FS}$$ is the Fubini-Study metric on $${\mathbb P}^d$$. The set of such metrics is denoted by $${\mathcal H}_N$$, while the space of all $$C^\infty$$ Kähler metrics in a fixed Kähler class $$[\omega]$$ of positive curvature is denoted by $${\mathcal H}$$. Then, it is known that the union $$\bigcup_N {\mathcal H}_N$$ is $$C^\infty$$ dense in $${\mathcal H}$$ and that the space $${\mathcal H}$$ is an infinite dimensional negatively curved symmetric space relative to a canonically defined Riemannian metric.
This article shows that to any geodesic on $${\mathcal H}$$, there is canonically associated a geodesic on $${\mathcal H}_N$$ that converges $$C^2$$ smoothly to the given geodesic for the case $$X={\mathbb P}^1$$; the result is a refinement of $$C^0$$ convergence given by D. H. Phong and J. Sturm [Invent. Math. 166, No. 1, 125–149 (2006; Zbl 1120.32026)].

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32L05 Holomorphic bundles and generalizations 53C22 Geodesics in global differential geometry
##### Keywords:
Bergman metric; Monge-Ampère equation
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##### References:
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