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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)].

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32L05 Holomorphic bundles and generalizations
53C22 Geodesics in global differential geometry
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