Degenerations of negative Kähler-Einstein surfaces. (English) Zbl 1537.32055
Summary: Every compact Kähler manifold with negative first Chern class admits a unique metric \(g\) such that \(\mathrm{Ric}(g) = -g\). Understanding how families of these metrics degenerate gives insight into their geometry and is important for understanding the compactification of the moduli space of negative Kähler-Einstein metrics. I study a special class of such families in complex dimension two. Following the work of S. Sun and R. Zhang in the Calabi-Yau case [“Complex structure degenerations and collapsing of Calabi-Yau metrics”, Preprint, arXiv:1906.03368], I construct a Kähler-Einstein neck region interpolating between canonical metrics on components of the central fiber. This provides a model for the limiting geometry of metrics in the family.
MSC:
| 32J27 | Compact Kähler manifolds: generalizations, classification |
| 32Q20 | Kähler-Einstein manifolds |
References:
| [1] | M. T.Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math.102 (1990), 429-445. · Zbl 0711.53038 |
| [2] | T.Aubin, Équations du type Monge‐Ampère sur les variétés kählériennes compactes, Bull. Sci. Math. (2)102 (1978), no. 1, 63-95. · Zbl 0374.53022 |
| [3] | S.Bando, Einstein Kähler metrics of negative Ricci curvature on open Kähler manifolds, Kähler metrics and moduli spaces, Adv. Stud Pur. Math.18 (1990), no. 2, 105-136. · Zbl 0754.53048 |
| [4] | T. H.Colding, Ricci curvature and volume convergence, Ann. Math.145 (1997), no. 3, 447-501. · Zbl 0879.53030 |
| [5] | J.Cheeger and A.Naber, Regularity of Einstein manifolds and the codimension 4 conjecture, Ann. Math.182 (2015), 1093-1165. · Zbl 1335.53057 |
| [6] | X.Chen, S.Donaldson, and S.Sun, Kähler‐Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Am. Math. Soc.28 (2014), 183-197. · Zbl 1312.53096 |
| [7] | X.Chen, S.Donaldson, and S.Sun, Kähler‐Einstein metrics on Fano manifolds. II: Limits with cone angles less than \(2\pi \), J. Am. Math. Soc.28 (2014), 199-234. · Zbl 1312.53097 |
| [8] | X.Chen, S.Donaldson, and S.Sun, Kähler‐Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \(2\pi\) and completion of the main proof, J. Am. Math. Soc.28 (2014), 235-278. · Zbl 1311.53059 |
| [9] | S.‐Y.Chen and S.‐T.Yau, On the existence of a complete Kähler metric on non‐compact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math.33 (1980), 507-544. · Zbl 0506.53031 |
| [10] | A.Futaki, An obstruction to the existence of Einstein Kähler metrics, Invent. Math.73 (1983), no. 3, 437-443. · Zbl 0506.53030 |
| [11] | R.Kobayashi, Kähler‐Einstein metric on an open algebraic manifold, Osaka J. Math21 (1984), no. 2, 399-418. · Zbl 0582.32011 |
| [12] | P.Li, Geometric analysis, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012. · Zbl 1246.53002 |
| [13] | A.Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, 1958. · Zbl 0096.16001 |
| [14] | Y. L.Luke, The special functions and their approximations, Mathematics in Science and Engineering, vol. 1, Academic Press, New York, 1969. · Zbl 0193.01701 |
| [15] | Y.Matsushima, Sur la structure du groupe d’homéomorphismes analytiques d’une certaine veriété kählérienne, Nagoya Math. J.11 (1957), 145-150. · Zbl 0091.34803 |
| [16] | P.Petersen, Riemannian geometry, Graduate Texts in Mathematics, Springer, New York, 2000. |
| [17] | W.‐D.Ruan, Degeneration of Kähler‐Einstein manifolds I: the normal crossing case, arXiv:math/0303112, 2003. |
| [18] | J.Song, Degeneration of Kähler‐Einstein manifolds of negative scalar curvature, arXiv:1706.01518, 2017. |
| [19] | S.Sun and R.Zhang, Complex structure degenerations and collapsing of Calabi‐Yau metrics, arXiv:1906.03368, 2019. |
| [20] | G.Tian, Degeneration of Kähler‐Einstein manifolds I, Proc. Symp. Pure Math.54 (1993), no. 2, 101-172. |
| [21] | G.Tian and S.‐T.Yau, Existence of Kähler‐Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory, Adv. Ser. Math. Phys., World Scientific, Singapore, 1987, pp. 574-628. |
| [22] | G.Tian and S.‐T.Yau, Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc.3 (1990), no. 3, 579-609. · Zbl 0719.53041 |
| [23] | S.‐T.Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge‐Ampére equation I, Comm. Pure Appl. Math.31 (1978), 339-411. · Zbl 0369.53059 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
