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On the degeneration of asymptotically conical Calabi–Yau metrics

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Abstract

We study the degenerations of asymptotically conical Ricci-flat Kähler metrics as the Kähler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kähler metrics converge to a incomplete smooth Ricci-flat Kähler metric away from a compact subvariety. As a consequence, we construct singular Calabi–Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi–Yau manifolds.

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References

  1. Atiyah, M.F.: On analytic surfaces with double points. Proc. R. Soc. Lond. Ser. A 247, 237–244 (1958)

    Article  MathSciNet  Google Scholar 

  2. Bando, S., Kobayashi, R.: Ricci-flat Kähler metrics on affine algebraic manifolds. Geometry and analysis on manifolds (Katata/Kyoto, 1987), pp. 20–31. In: Lecture Notes in Mathematics, vol. 1339. Springer, Berlin (1988)

  3. Bando, S., Kobayashi, R.: Ricci-flat Kähler metrics on affine algebraic manifolds II. Math. Ann. 287(1), 175–180 (1990)

    Article  MathSciNet  Google Scholar 

  4. Bedford, E., Taylor, B.A.: The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37(1), 1–44 (1976)

    Article  MathSciNet  Google Scholar 

  5. Calabi, E.: Métriques kählériennes et fibrés holomorphes. Ann. Sci. École Norm. Sup. (4) 12(2), 269–294 (1979)

  6. Cheeger, J., Colding, T. H.: Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math. (2) 144(1), 189–237 (1996)

  7. Cheeger, J., Colding, T. H. On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46(3), 406–480 (1997)

  8. Cheeger, J., Colding, T. H.: On the structure of spaces with Ricci curvature bounded below. II, J. Differ. Geom. 54(1), 13–35 (2000)

  9. Cheeger, J., Colding, T. H.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54(1), 37–74 (2000)

  10. Cheeger, J., Colding, T.H., Tian, G.: On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12(5), 873–914 (2002)

    Article  MathSciNet  Google Scholar 

  11. Cheng, S.Y., Yau, S.T.: On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Comm. Pure Appl. Math. 33(4), 507–544 (1980)

  12. Collins, T.C., Székelyhidi, G.: Sasaki-Einstein metrics and K-stability. Geom. Topol. 23(3), 1339–1413 (2019)

    Article  MathSciNet  Google Scholar 

  13. Collins, T.C., Tosatti, V.: Kähler currents and null loci. Invent. Math. 202(3), 1167–1198 (2015)

    Article  MathSciNet  Google Scholar 

  14. Conlon, R.: On the construction of asymptotically conical Calabi-Yau manifolds. Ph.D. thesis, Imperial College London (2011)

  15. Conlon, R., Hein, H.: Asymptotically conical Calabi-Yau manifolds, I. Duke Math. J. 162(15), 2855–2902 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Conlon, R., Hein, H.: Asymptotically conical Calabi-Yau metrics on quasi-projective varieties. Geom. Funct. Anal. 25(2), 517–552 (2015)

    Article  MathSciNet  Google Scholar 

  17. Conlon, R., Hein, H.: Asymptotically Conical Calabi–Yau Manifolds III. arXiv.1405.7140

  18. Corti, A., Haskins, M., Nordström, J., Pacini, T.: Asymptotically cylindrical Calabi-Yau \(3\)-folds from weak Fano \(3\)-folds. Geom. Topol. 17(4), 1955–2059 (2013)

    Article  MathSciNet  Google Scholar 

  19. Croke, C.: Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13(4), 419–435 (1980)

  20. Demailly, J.-P.: Surveys of Modern Mathematics, 1. Analytic methods in algebraic geometry, Higher Education Press/International Press, Beijing/Somerville (2012)

    Google Scholar 

  21. Demailly, J.-P., Păun, M. Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. (2) 159(3) , 1247–1274 (2004)

  22. Donaldson, S. Kronheimer, P. The geometry of four-manifolds. In: Oxford Mathematical Monographs. (Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990)

  23. Donaldson, S., Sun, S.: Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. Acta Math. 213(1), 63–106 (2014)

    Article  MathSciNet  Google Scholar 

  24. Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. J. Am. Math. Soc. 22(3), 607–639 (2009)

    Article  Google Scholar 

  25. Faulk, M.: Asymptotically Conical Calabi-Yau Orbifolds, I. arxiv.1809.01556

  26. Futaki, A., Ono, H., Wang, G. Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differ. Geom. 83(3), 585–635 (2009)

  27. Gauntlett, J., Martelli, D., Sparks, J., Waldram, D.: Sasaki-Einstein metrics on \(S^2\times S^3\). Adv. Theor. Math. Phys. 8(4), 711–734 (2004)

    Article  MathSciNet  Google Scholar 

  28. Goto, R.: Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities. J. Math. Soc. Jpn. 64(3), 1005–1052 (2012)

    Article  MathSciNet  Google Scholar 

  29. Grauert, H.: Aber modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331–368 (1962)

    Article  MathSciNet  Google Scholar 

  30. Grauert, H., Remmert, R.: Plurisubharmonische Funktionen in komplexen Raumen (German). Math. Z. 65, 175–194 (1956)

    Article  MathSciNet  Google Scholar 

  31. Haskins, M., Hein, H,-J., Nordström, J.: Asymptotically cylindrical Calabi–Yau manifolds. J. Differ. Geom. 101(2), 213–265 (2015)

  32. Joyce, D. Compact manifolds with special holonomy. In: Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)

  33. Katz, S. Small resolutions of Gorenstein threefold singularities. Algebraic geometry: Sundance 1988. Contemp. Math. 116, 61–70 (American Mathematics Society, Providence) (1991)

  34. Kolodziej, S.: The complex Monge-Ampere equation. Acta Math. 180(1), 69–117 (1998)

    Article  MathSciNet  Google Scholar 

  35. Li, C.: On sharp rates and analytic compactifications of asymptotically conical Kähler metrics. arxiv.1405.2433

  36. Li, P., Tam, L.F.: Green’s functions, harmonic functions, and volume comparison. J. Differ. Geom. 41(2), 277–318 (1995)

  37. Li, P., Yau, S.-T.: On the parabolic kernel of the Schrodinger operator. Acta Math. 156(3–4), 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  38. Lockhart, R., McOwen, R.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(3), 409–447 (1985)

  39. Marshall, S.: Deformations of special Lagrangian submanifolds. Ph.D. thesis, University of Oxford (2002)

  40. Martelli, D., Sparks, J., Yau, S.-T.: Sasaki-Einstein manifolds and volume minimization. Commun. Math. Phys. 280(3), 611–673 (2008)

    Article  Google Scholar 

  41. Matsuki, K.: Universitext. Introduction to the Mori program, Springer, New York (2002)

    Google Scholar 

  42. Matsuki, K.: Weyl groups and birational transformations among minimal models. Mem. Am. Math. Soc. 116(557) (1995)

  43. Matsushima, Y.: Sur la structure du groupe d’homéomorphismes analytiques d’ube certain variíeté Kählerienne. Nagoya Math. J. 11, 145–150 (1957)

  44. Mok, N., Siu, Y.-T., Yau, S.-T.: The Poincaré-Lelong equation on complete Kähler manifolds. Composit. Math. 44(1–3), 183–218 (1981)

    MATH  Google Scholar 

  45. Phong, D.H., Sesum, N., Sturm, J.: Multiplier ideal sheaves and the Kähler-Ricci flow. Commun. Anal. Geom. 15(3), 613–632 (2007)

    Article  Google Scholar 

  46. Reider, I.: Vector bundles of rank \(2\) and linear systems on algebraic surfaces. Ann. Math. (2) 127(2), 309–316 (1988)

  47. Rong, X., Zhang, Y.: Continuity of extremal transitions and flops for Calabi-Yau manifolds. Appendix B by Mark Gross. J. Differ. Geom. 89(2), 233–269 (2011)

  48. Rong, X., Zhang, Y.: Degenerations of Ricci-flat Calabi-Yau manifolds. Commun. Contemp. Math. 15(4), 1250057 (2013)

  49. Schoen, R.: Analytic aspects of the harmonic map problem. Math. Sci. Res. Inst. Publ. 2, 321–358 (1984)

    MathSciNet  MATH  Google Scholar 

  50. Sherman, M., Weinkove, B.: Local Calabi and curvature estimates for the Chern-Ricci flow. N. Y. J. Math. 19, 565–582 (2013)

    MathSciNet  MATH  Google Scholar 

  51. Song, J.: Riemannian geometry of Kähler-Einstein currents. arXiv.1404.0445

  52. Tian, G.: On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math. 101(1), 101–172 (1990)

  53. Tian, G., Yau, S.-T.: Complete Kähler manifolds with zero Ricci curvature. J. Am. Math. Soc. 3(3), 579–609 (1990)

    MATH  Google Scholar 

  54. Tian, G., Yau, S.-T.: Complete Kähler manifolds with zero Ricci curvature II. Invent. Math. 106(1), 27–60 (1991)

    Article  MathSciNet  Google Scholar 

  55. Tian, G., Yau, S.-T.: Kähler-Einstein metrics on complex surfaces with \(C_1>0\). Commun. Math. Phys. 112(1), 175–203 (1987)

    Article  Google Scholar 

  56. Tosatti, V.: Calabi-Yau manifolds and their degenerations, Ann. N.Y. Acad. Sci. 1260, 8–13 (2012)

  57. Tosatti, V.: Degenerations of Calabi-Yau metrics, in geometry and physics in Cracow. Acta Phys. Polon. B Proc. Suppl. 4(3), 495–505 (2011)

    Article  Google Scholar 

  58. Tosatti, V.: Collapsing Calabi-Yau manifolds. arXiv:2003.00673

  59. Tosatti, V.: Limits of Calabi-Yau metrics when the Kähler class degenerates. J. Eur. Math. Soc. (JEMS) 11(4), 755–776 (2009)

    Article  MathSciNet  Google Scholar 

  60. Tsuji, H.: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281(1), 123–133 (1988)

    Article  MathSciNet  Google Scholar 

  61. van Coevering, C.: Ricci-flat Kähler metrics on crepant resolutions of Kähler cones. Math. Ann. 347(3), 581–611 (2010)

    Article  MathSciNet  Google Scholar 

  62. van Coevering, C.: A Construction of Complete Ricci-flat Kähler Manifolds. arXiv:0803.0112

  63. Yau, S.-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation. I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)

  64. Yau, S.-T.: Survey on partial differential equations in differential geometry. Seminar on differential geometry, pp. 3–71. Annals of Mathematics Studies, Vol. 102. Princeton University Press, Princeton (1982)

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Acknowledgements

The authors are grateful to D. H. Phong for his interest and encouragement. The authors are also grateful to R. Conlon and H.-J. Hein for explaining aspects of their papers [15,16,17].

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Correspondence to Freid Tong.

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Communicated by Ngaiming Mok.

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T. C. Collins is supported in part by NSF grant DMS-1944952, DMS-1810924 and an Alfred P. Sloan Fellowship. B. Guo is supported in part by NSF grant DMS-1945869. F. Tong is supported in part by NSF grant DMS-1855947.

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Collins, T.C., Guo, B. & Tong, F. On the degeneration of asymptotically conical Calabi–Yau metrics. Math. Ann. 383, 867–919 (2022). https://doi.org/10.1007/s00208-021-02229-z

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