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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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References:
 [1] Anderson, M.T., Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. math., 102, 429-445, (1990) · Zbl 0711.53038 [2] Anderson, M.T., The $$L^2$$ structure of moduli spaces of Einstein metrics on 4-manifolds, Geom. funct. anal., 231-251, (1991) [3] Aspinwall, P.S.; Green, B.R.; Morrison, D.R., Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nuclear phys. B, 416, 414-480, (1994) · Zbl 0899.32006 [4] Baily, W.L., On the imbeddings of V-manifolds in projective space, Amer. J. math., 79, 2, 403-430, (1957) · Zbl 0173.22706 [5] Bando, S.; Kobayashi, R., Ricci-flat Kähler metrics on affine algebraic manifolds II, Math. ann., 287, 175-180, (1990) · Zbl 0701.53083 [6] Bedford, E.; Taylor, B.A., A new capacity for plurisubharmonic functions, Acta math., 194, 1-40, (1982) · Zbl 0547.32012 [7] Borzellino, J., Orbifolds of maximal diameter, Indiana univ. math. J., 42, 37-53, (1993) · Zbl 0801.53031 [8] Candelas, P.; de la Ossa, X.C., Comments on conifolds, Nuclear phys. B, 342, 1, 246-268, (1990) [9] Candelas, P.; Green, P.S.; Hübsch, T., Rolling among Calabi-Yau vacua, Nuclear phys. B, 330, 49-102, (1990) · Zbl 0985.32502 [10] Cheeger, J., Degeneration of Einstein metrics and metrics with special holonomy, (), 29-73 · Zbl 1053.53028 [11] Cheeger, J.; Colding, T.H., On the structure of space with Ricci curvature bounded below I, J. differential geom., 46, 406-480, (1997) · Zbl 0902.53034 [12] Cheeger, J.; Colding, T.H., On the structure of space with Ricci curvature bounded below II, J. differential geom., 52, 13-35, (1999) · Zbl 1027.53042 [13] Cheeger, J.; Tian, G., Anti-self-duality of curvature and degeneration of metrics with special holonomy, Comm. math. phys., 255, 391-417, (2005) · Zbl 1081.53038 [14] Cheeger, J.; Colding, T.H.; Tian, G., On the singularities of spaces with bounded Ricci curvature, Geom. funct. anal., 12, 873-914, (2002) · Zbl 1030.53046 [15] Cheng, S.Y.; Li, P., Heat kernel estimates and lower bound of eigenvalues, Comment. math. helv., 56, 327-338, (1981) · Zbl 0484.53034 [16] Colding, T.H., Ricci curvature and volume convergence, Ann. of math., 145, 477-501, (1997) · Zbl 0879.53030 [17] Cox, D.A.; Katz, S., Mirror symmetry and algebraic geometry, Math. surveys monogr., vol. 68, (1999), American Mathematical Society · Zbl 0951.14026 [18] Davis, E.B., Heat kernels and spectral theory, (1989), Cambridge Univ. Press Cambridge · Zbl 0699.35006 [19] J.P. Demailly, Complex Analytic and Differential Geometry, online book. [20] Eyssidieux, P.; Guedj, V.; Zeriahi, A., Singular Kähler-Einstein metrics, J. amer. math. soc., 22, 607-639, (2009) · Zbl 1215.32017 [21] Fornaess, J.E.; Narasimhan, R., The Levi problem on complex space with singularities, Math. ann., 248, 47-72, (1980) · Zbl 0411.32011 [22] Fukaya, K., Hausdorff convergence of Riemannian manifolds and its application, Adv. stud. pure math., 18, 143-234, (1990) · Zbl 0754.53004 [23] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1983), Springer · Zbl 0691.35001 [24] Green, P.S.; Hübsch, T., Connecting moduli spaces of Calabi-Yau threefolds, Comm. math. phys., 119, 431-441, (1988) · Zbl 0684.53077 [25] Green, R.E.; Wu, H., Lipschitz converges of Riemannian manifolds, Pacific J. math., 131, 119-141, (1988) · Zbl 0646.53038 [26] Griffiths, H.; Harris, J., Principles of algebraic geometry, (1978), John Wiley and Sons New York · Zbl 0408.14001 [27] Gromov, M., Metric structures for riemannian and non-Riemannian spaces, (1999), Birkhäuser [28] Gross, M.; Wilson, P.M.H., Mirror symmetry via 3-tori for a class of Calabi-Yau threefolds, Math. ann., 309, 505-531, (1997) · Zbl 0901.14024 [29] Gross, M.; Wilson, P.M.H., Large complex structure limits of K3 surfaces, J. differential geom., 55, 475-546, (2000) · Zbl 1027.32021 [30] Guedj, V.; Zeriahi, A., Intrinsic capacities on compact Kähler manifolds, J. geom. anal., 15, 607-639, (2005) · Zbl 1087.32020 [31] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Ann. of math., 79, 109-326, (1964) · Zbl 0122.38603 [32] Kobayashi, R., Moduli of Einstein metrics on K3 surface and degeneration of type I, Adv. stud. pure math., 18, II, 257-311, (1990) · Zbl 0755.32023 [33] Kobayashi, R.; Todorov, A.N., Polarized period map for generalized K3 surfaces and the moduli of Einstein metrics, Tohoku math. J., 39, 341-363, (1987) · Zbl 0646.14029 [34] Kodaira, K., On compact complex analytic surfaces I, Ann. of math., 71, 111-152, (1960) · Zbl 0098.13004 [35] Kolodziej, S., The complex Monge-Ampère equation, Acta math., 180, 69-117, (1998) · Zbl 0913.35043 [36] Kontsevich, M.; Soibelman, Y., Homological mirror symmetry and torus fibrations, (), 203-263 · Zbl 1072.14046 [37] Li, P.; Tian, G., On the heat kernel of the bergmann metric on algebraic varieties, J. amer. math. soc., 8, 857-877, (1995) · Zbl 0864.58058 [38] Lu, P., Kähler-Einstein metrics on Kummer threefold and special Lagrangian tori, Comm. anal. geom., 7, 787-806, (1999) · Zbl 1023.32015 [39] Miyaoka, Y.; Peternell, T., Geometry of higher-dimensional algebraic varieties, DMV seminar, vol. 26, (1997), Birkhäuser Verlag · Zbl 0865.14018 [40] Petersen, P., Riemannian geometry, (1997), Springer [41] Reid, M., The moduli space of 3-folds with $$K = 0$$ may nevertheless be irreducible, Math. ann., 287, 329-334, (1987) · Zbl 0649.14021 [42] Roan, S.S., Minimal resolution of Gorenstein orbifolds, Topology, 35, 487-508, (1996) [43] Rossi, M., Geometric transitions, J. geom. phys., 56, 9, 1940-1983, (2006) · Zbl 1106.32019 [44] Ruan, W.D., On the convergence and collapsing of Kähler metrics, J. differential geom., 52, 1-40, (1999) · Zbl 1039.53045 [45] Satake, I., The Gauss-Bonnet theorem for V-manifolds, J. math. soc. Japan, 9, 464-492, (1957) · Zbl 0080.37403 [46] Schoen, R.; Yau, S.T., Lectures on differential geometry, (1994), International Press · Zbl 0830.53001 [47] Song, J.; Tian, G., Canonical measures and Kähler-Ricci flow · Zbl 1239.53086 [48] Strominger, A.; Yau, S.T.; Zaslow, E., Mirror symmetry is T-duality, Mirror symmetry, vector bundles and Lagrangian submanifolds, Stud. adv. math., 23, 333-347, (2001) · Zbl 0998.81091 [49] Tian, G., Smoothing 3-folds with trivial canonical bundle and ordinary double points, (), 458-479 · Zbl 0829.32012 [50] Tosatti, V., Limits of Calabi-Yau metrics when the Kähler class degenerates, J. eur. math. soc. (JEMS), 11, 4, 755-776, (2009) · Zbl 1177.32015 [51] Yau, S.T., On the Ricci curvature of a compact Kähler manifold and complex Monge-Ampère equation I, Comm. pure appl. math., 31, 339-411, (1978) · Zbl 0369.53059 [52] Yau, S.T., A general Schwarz lemma for Kähler manifolds, Amer. J. math., 100, 197-204, (1978) · Zbl 0424.53040 [53] Yau, S.T., Survey on partial differential equations in differential geometry, (), 3-71 [54] Yau, S.T., Einstein manifolds with zero Ricci curvature, (), 1-14 · Zbl 1023.53027 [55] Yoshikawa, K., Degeneration of algebraic manifolds and the spectrum of Laplacian, Nagoya math. J., 146, 93-129, (1997) · Zbl 0880.58030 [56] Y.G. Zhang, The convergence of Kähler manifolds and calibrated fibrations, PhD thesis, Nankai Institute of Mathematics, 2006.
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