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Convergence of Calabi-Yau manifolds. (English) Zbl 1232.32012
A Calabi-Yau \(n\)-variety is a normal Gorenstein projective variety \(N\) of dimension \(n\) admitting only canonical singularities, such that the dualizing sheaf \(K_N\) of \(N\) is trivial (i.e. \(K_N \cong {\mathcal O}_N\)) and \(H^2(N, {\mathcal O}_N) = \{ 0 \}\). We recall that \((M, \pi)\) is a resolution of \(N\) if \(M\) is a compact complex \(n\)-manifold and \(\pi: M \rightarrow N\) is a bi-rational proper morphism such that \(\pi: M \setminus \pi^{-1}(S) \rightarrow N \setminus S\) is biholomorphic, where by \(S\) we denote the singular set of \(N\). The resolution is called crepant if \(\pi^* K_N = K_M\), or equivalently if \(M\) is a compact Calabi-Yau \(n\)-manifold. For a Calabi-Yau variety \(N\) there are the notions of Kähler metrics, (smooth) Kähler forms and holomorphic volume forms, which are anologous to the ones for a Calabi-Yau manifold. Any Kähler form \(\omega\) represents a class \([\omega] \in H^1(N, \mathcal{PH}_N)\), where \(\mathcal{PH}_N\) denotes the sheaf of pluri-harmonic functions on \(N\). In [P. Eyssidieux, V. Guedj and A. Zeriahi, J. Am. Math. Soc. 22, No. 3, 607–639 (2009; Zbl 1215.32017)] it was proved that, for any \(\alpha \in H^1(N, \mathcal{PH}_N)\) which can be represented by a smooth Kähler form, there exists a unique Ricci-flat Kähler metric \(g\) with Kähler form \(\omega \in \alpha\). If \(N\) has a crepant resolution \((M, \pi)\) and \(\alpha_k \in H^{1,1} (M, R)\) is a family of Kähler classes such that \(\lim_{k \to \infty} \alpha_k = \pi^* \alpha\), then, by a result in [V. Tosatti, J. Eur. Math. Soc. (JEMS) 11, No. 4, 755–776 (2009; Zbl 1177.32015)], \(g_k\) converges to \(\pi^* g\) in the \({\mathcal C}^{\infty}\)-sense on any compact subset of \(M \setminus \pi^{-1} (S)\) when \(k\) goes to \(\infty\), where \(g_k\) is the unique Ricci-flat Kähler metric with Kähler form \(\omega_k \in \alpha_k\).
In the present paper the authors study the convergence of \((M, g_k)\) in the Gromov-Haussdorff topology. As an application they show that if \(N\) is a compact Calabi-Yau \(n\)-orbifold, which admits a crepant resolution \((M, \pi)\), if \(g\) is a Ricci-flat Kähler metric on \(N\) with Kähler form \(\omega\), and if \(g_k\) is a family of Ricci-flat Kähler metrics on \(M\) with the Kähler forms \(\omega_k\) such that the Kähler classes \([\omega_k]\) converge to \(\pi^* [\omega]\) in \(H^{1,1} (M, R)\) as \(k\) goes to \(\infty\), then \(\lim_{k \to \infty} d_{GH} \big((M, g_k), (N, g)\big) =0\), where \( d_{GH} \big((M, g_k), (N, g)\big)\) denotes the Gromov-Hausdorff distance.
Moreover, the authors study the convergence of Calabi-Yau manifolds obtained from a smoothing of a Calabi-Yau variety and the collapsing of a family of Calabi-Yau manifolds.
Reviewer: Anna Fino (Torino)

MSC:
32Q25 Calabi-Yau theory (complex-analytic aspects)
32U20 Capacity theory and generalizations
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