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Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. II. (English) Zbl 1388.53074

Given a sequence \((X_{i},L_{i},\omega_{i}, p_{i})\) of \(n\)-dimensional polarized Kähler manifolds one can obtain a (pointed) Gromov-Hausdorff limit \((Z,p)\). This length space has a decomposition \(Z=R\cup \Sigma\). Considering the structure sheaf \(\mathcal{O}_{R}\) and the inclusion \(\iota:R\rightarrow Z\), the authors define \(\mathcal{O}_{Z}=\iota_{*}\mathcal{O}_{R}.\) Then, they prove that \((Z,\mathcal{O}_{Z})\) is a normal complex analytic space.
When the limit space \((Z,p)\) is non-compact, the authors study the algebraicity of \(Z\). If \(R(Z)\) is the ring of holomorphic functions on \(Z\) with polynomial growth at infinity they show that \(R(Z)\) is finitely generated. Furthermore, Spec\((R(Z))\) is an affine algebraic variety which is complex analytically isomorphic to \((Z,\mathcal{O}_{Z})\).
The authors include several other results and the main application of these results is to study Kähler-Einstein metrics with positive Ricci curvature.
For Part I see [the authors, Acta Math. 213, No. 1, 63–106 (2014; Zbl 1318.53037)].

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Citations:

Zbl 1318.53037
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