Skip to main content
SpringerLink
Log in

Geometry of Twisted Kähler–Einstein Metrics and Collapsing

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We prove that the twisted Kähler–Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi–Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kähler metrics on Calabi–Yau manifolds, and of the Kähler–Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension. Our results allow us to understand their collapsed Gromov–Hausdorff limits when the base is smooth and the discriminant has simple normal crossings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aleyasin, S.A.: Calabi problem for manifolds with edge-cone singularities, preprint, arXiv:1810.07823

  2. Barlet, D.: Développement asymptotique des fonctions obtenues par intégration sur les fibres. Invent. Math. 68(1), 129–174 (1982)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Berman, R.J.: K-polystability of \({\mathbb{Q}}\)-Fano varieties admitting Kähler-Einstein metrics. Invent. Math. 203(3), 973–1025 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  4. Boucksom, S., Jonsson, M.: Tropical and non-Archimedean limits of degenerating families of volume forms. J. Éc. Polytech. Math. 4, 87–139 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Brendle, S.: Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. IMRN 24, 5727–5766 (2013)

    MATH  Google Scholar 

  6. Campana, F., Guenancia, H., Păun, M.: Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. Ann. Sci. Éc. Norm. Supér. (4) 46(6), 879–916 (2013)

  7. Cattani, E., Kaplan, A., Schmid, W.: Degeneration of Hodge structures. Ann. Math. (2) 123(3), 457–535 (1986)

  8. Cattani, E., Kaplan, A., Schmid, W.: Variations of polarized Hodge structure: asymptotics and monodromy. Hodge theory (Sant Cugat, 1985). Lecture Notes in Mathematics, vol. 1246, pp. 16–31. Springer, Berlin (1987)

  9. Chambert-Loir, A., Tschinkel, Y.: Igusa integrals and volume asymptotics in analytic and adelic geometry. Conflu. Math. 2(3), 351–429 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Chen, G., Viaclovsky, J., Zhang, R.: Collapsing Ricci-flat metrics on elliptic K3 surfaces, preprint, arXiv:1910.11321

  11. Coman, D., Guedj, V., Zeriahi, A.: Extension of plurisubharmonic functions with growth control. J. Reine Angew. Math. 676, 33–49 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Datar, V., Jacob, A., Zhang, Y.: Adiabatic limits of anti-self-dual connections on collapsed \(K3\) surfaces. J. Differ. Geom. (to appear)

  13. Datar, V., Song, J.: A remark on Kähler metrics with conical singularities along a simple normal crossing divisor. Bull. Lond. Math. Soc. 47(6), 1010–1013 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Demailly, J.-P.: Estimations \(L^2\) pour l’opérateur \({\overline{\partial }}\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. Ann. Sci. École Norm. Sup. (4) 15(3), 457–511 (1982)

  15. Eriksson, D., Montplet, GFreixas i, Mourougane, C.: Singularities of metrics on Hodge bundles and their topological invariants. Algebr. Geom. 5(6), 742–775 (2018)

  16. Eriksson, D., Montplet, G. Freixas i., Mourougane, C.: BCOV invariants of Calabi-Yau manifolds and degenerations of Hodge structures, preprint, arXiv:1809.05452

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

    MATH  Google Scholar 

  18. Eyssidieux, P., Guedj, V., Zeriahi, A.: Convergence of weak Kähler-Ricci flows on minimal models of positive Kodaira dimension. Commun. Math. Phys. 357(3), 1179–1214 (2018)

    ADS  MATH  Google Scholar 

  19. Fine, J.: Fibrations with constant scalar curvature Kähler metrics and the CM-line bundle. Math. Res. Lett. 14(2), 239–247 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Fong, F.T.-H., Zhang, Z.: The collapsing rate of the Kähler-Ricci flow with regular infinite time singularity. J. Reine Angew. Math. 703, 95–113 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Freed, D.: Special Kähler Manifolds. Commun. Math. Phys. 203(1), 31–52 (1999)

    ADS  MATH  Google Scholar 

  22. Fu, X., Guo, B., Song, J.: Geometric estimates for complex Monge-Ampère equations. J. Reine Angew. Math. 765, 69–99 (2020)

    MathSciNet  MATH  Google Scholar 

  23. Fujita, T.: On Kähler fiber spaces over curves. J. Math. Soc. Jpn. 30(4), 779–794 (1978)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Griffiths, P.: Topics in Transcendental Algebraic Geometry. Annals of Mathematics Studies, 106. Princeton University Press, Princeton, NJ (1984)

  26. Griffiths, P., Schmid, W.: Locally homogeneous complex manifolds. Acta Math. 123, 253–302 (1969)

    MathSciNet  MATH  Google Scholar 

  27. Gross, M., Tosatti, V., Zhang, Y.: Collapsing of abelian fibred Calabi-Yau manifolds. Duke Math. J. 162(3), 517–551 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Gross, M., Tosatti, V., Zhang, Y.: Gromov-Hausdorff collapsing of Calabi-Yau manifolds. Commun. Anal. Geom. 24(1), 93–113 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Gross, M., Wilson, P.M.H.: Large complex structure limits of \(K3\) surfaces. J. Differ. Geom. 55(3), 475–546 (2000)

    MathSciNet  MATH  Google Scholar 

  30. Guenancia, H., Păun, M.: Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors. J. Differ. Geom. 103(1), 15–57 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Hein, H.-J.: Gravitational instantons from rational elliptic surfaces. J. Am. Math. Soc. 25(2), 355–393 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Hein, H.-J., Tosatti, V.: Remarks on the collapsing of torus fibered Calabi-Yau manifolds. Bull. Lond. Math. Soc. 47(6), 1021–1027 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Hein, H.-J., Tosatti, V.: Higher-order estimates for collapsing Calabi-Yau metrics, Camb. J. Math (to appear)

  34. Hitchin, N.: The moduli space of special Lagrangian submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(3–4), 503–515 (1997)

  35. Hitchin, N.: The moduli space of complex Lagrangian submanifolds. Asian J. Math. 3(1), 77–91 (1999)

    MathSciNet  MATH  Google Scholar 

  36. Huybrechts, D.: Products of harmonic forms and rational curves. Doc. Math. 6, 227–239 (2001)

    MathSciNet  MATH  Google Scholar 

  37. Jeffres, T., Mazzeo, R., Rubinstein, Y.A.: Kähler-Einstein metrics with edge singularities. Ann. Math. (2) 183(1), 95–176 (2016)

  38. Kashiwara, M.: The asymptotic behavior of a variation of polarized Hodge structure. Publ. Res. Inst. Math. Sci. 21(4), 853–875 (1985)

    MathSciNet  MATH  Google Scholar 

  39. Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973)

  40. Kim, D.: Canonical bundle formula and degenerating families of volume forms, preprint, arXiv:1910.06917

  41. Kollár, J.: Subadditivity of the Kodaira dimension: fibers of general type. In: Bando S, Mabuchi T (ed)Algebraic geometry, Sendai, 1985, pp 361–398, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam (1987)

  42. Kołodziej, S.: The complex Monge-Ampère equation. Acta Math. 180(1), 69–117 (1998)

    MathSciNet  MATH  Google Scholar 

  43. Kołodziej, S.: The Monge-Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52(3), 667–686 (2003)

    MathSciNet  MATH  Google Scholar 

  44. Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Cox, D.A., Katz, S. (eds.) Symplectic Geometry and Mirror Symmetry, 203–263. World Scientific Publishing, Singapore (2001)

    MATH  Google Scholar 

  45. Kontsevich, M., Soibelman, Y.: Affine Structures and Non-Archimedean Analytic Spaces. In: Atiyah, M.F. (ed.) The Unity of Mathematics. Springer, Berlin (2006)

    MATH  Google Scholar 

  46. Landman, A.: On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities. Trans. Am. Math. Soc. 181, 89–126 (1973)

    MathSciNet  MATH  Google Scholar 

  47. Lazarsfeld, R.: Positivity in Algebraic Geometry I & II. Springer, Berlin (2004)

    MATH  Google Scholar 

  48. Li, Y.: On collapsing Calabi-Yau fibrations. J. Differ. Geom (to appear)

  49. Li, Y.: A gluing construction of collapsing Calabi-Yau metrics on \(K3\) fibred \(3\)-folds. Geom. Funct. Anal. 29(4), 1002–1047 (2019)

    MathSciNet  MATH  Google Scholar 

  50. Magnússon, G.: The geometry of Kähler cones, preprint, arXiv:1211.6934

  51. McLean, R.C.: Deformations of calibrated submanifolds. Commun. Anal. Geom. 6(4), 705–747 (1998)

    MathSciNet  MATH  Google Scholar 

  52. Moishezon, B.G.: Singular Kählerian spaces. Manifolds-Tokyo 1973 Proceedings International Conference. Tokyo, 1973, pp. 343–351. University of Tokyo Press, Tokyo (1975)

  53. Mourougane, C., Takayama, S.: Hodge metrics and positivity of direct images. J. Reine Angew. Math. 606, 167–178 (2007)

    MathSciNet  MATH  Google Scholar 

  54. Mourougane, C., Takayama, S.: Hodge metrics and the curvature of higher direct images. Ann. Sci. Éc. Norm. Supér. (4) 41(6), 905–924 (2008)

  55. Mourougane, C., Takayama, S.: Extension of twisted Hodge metrics for Kähler morphisms. J. Differ. Geom. 83(1), 131–161 (2009)

    MATH  Google Scholar 

  56. Oda, T.: Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 15. Springer, Berlin (1988)

  57. Odaka, Y., Oshima, Y.: Collapsing \(K3\) surfaces, tropical geometry and moduli compactifications of Satake, Morgan-Shalen type, preprint, arXiv:1810.07685

  58. Phong, D.H., Sturm, J.: On the singularities of the pluricomplex Green’s function. In: Fefferman, C., Ionescu, A.D., Phong, D.H., Wainger, S. (eds.) Advances in analysis. The legacy of Elias M. Stein, pp. 419–435. Princeton University Press, Princeton (2014)

  59. 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)

  60. Ruan, W.-D., Zhang, Y.: Convergence of Calabi-Yau manifolds. Adv. Math. 228(3), 1543–1589 (2011)

    MathSciNet  MATH  Google Scholar 

  61. Rubinstein, Y.A.: Smooth and singular Kähler-Einstein metrics. Geometric and spectral analysis. Contemporary Mathematics, Centre de Recherches Mathematiques Proceedings, vol. 630, pp. 45–138. American Mathematics Society, Providence, RI (2014)

  62. Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)

    ADS  MathSciNet  MATH  Google Scholar 

  63. Schoen, C.: On fiber products of rational elliptic surfaces with section. Math. Z. 197(2), 177–199 (1988)

    MathSciNet  MATH  Google Scholar 

  64. Song, J., Tian, G.: The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609–653 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  65. Song, J., Tian, G.: Canonical measures and Kähler-Ricci flow. J. Amer. Math. Soc. 25(2), 303–353 (2012)

    MathSciNet  MATH  Google Scholar 

  66. Song, J., Tian, G.: Bounding scalar curvature for global solutions of the Kähler-Ricci flow. Amer. J. Math. 138(3), 683–695 (2016)

    MathSciNet  MATH  Google Scholar 

  67. Song, J., Tian, G., Zhang, Z.: Collapsing behavior of Ricci-flat Kähler metrics and long time solutions of the Kähler-Ricci flow, preprint, arXiv:1904.08345

  68. Steenbrink, J.: Limits of Hodge structures. Invent. Math. 31(3), 229–257 (1975/76)

  69. Stoppa, J.: Twisted constant scalar curvature Kähler metrics and Kähler slope stability. J. Differ. Geom. 83(3), 663–691 (2009)

    MathSciNet  MATH  Google Scholar 

  70. Takayama, S.: On moderate degenerations of polarized Ricci-flat Kähler manifolds. J. Math. Sci. Univ. Tokyo 22(1), 469–489 (2015)

    MathSciNet  MATH  Google Scholar 

  71. Takayama, S.: A filling-in problem and moderate degenerations of minimal algebraic varieties. Algebr. Geom. 6(1), 26–49 (2019)

    MathSciNet  MATH  Google Scholar 

  72. Takayama, S.: Asymptotic expansions of fiber integrals over higher dimensional bases. J. Reine Angew. Math. https://doi.org/10.1515/crelle-2020-0027

  73. Temkin, M.: Functorial desingularization over \({\mathbf{Q}}\): boundaries and the embedded case. Israel J. Math. 224(1), 455–504 (2018)

    MathSciNet  MATH  Google Scholar 

  74. Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. In: Mathematical aspects of string theory (San Diego, California, 1986). Advance Series Mathematics Physics, 1, World Scientific Publishing, Singapore (1987)

  75. Tian, G.: Some progresses on Kähler-Ricci flow. Boll. Unione Mat. Ital. 12(1–2), 251–263 (2019)

    MathSciNet  MATH  Google Scholar 

  76. Tian, G., Zhang, Z.: Relative volume comparison of Ricci Flow and its applications, preprint, arXiv:1802.09506

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

    MathSciNet  MATH  Google Scholar 

  78. Tosatti, V.: Adiabatic limits of Ricci-flat Kähler metrics. J. Differ. Geom. 84(2), 427–453 (2010)

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

  81. Tosatti, V.: Families of Calabi-Yau manifolds and canonical singularities. Int. Math. Res. Not. IMRN 20, 10586–10594 (2015)

    MathSciNet  MATH  Google Scholar 

  82. Tosatti, V.: KAWA lecture notes on the Kähler-Ricci flow. Ann. Fac. Sci. Toulouse Math. 27(2), 285–376 (2018)

    MathSciNet  MATH  Google Scholar 

  83. Tosatti, V., Weinkove, B., Yang, X.: The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits. Am. J. Math. 140(3), 653–698 (2018)

    MATH  Google Scholar 

  84. Tosatti, V., Zhang, Y.: Triviality of fibered Calabi-Yau manifolds without singular fibers. Math. Res. Lett. 21(4), 905–918 (2014)

    MathSciNet  MATH  Google Scholar 

  85. Tosatti, V., Zhang, Y.: Infinite time singularities of the Kähler-Ricci flow. Geom. Topol. 19(5), 2925–2948 (2015)

    MathSciNet  MATH  Google Scholar 

  86. Tosatti, V., Zhang, Y.: Collapsing hyperkähler manifolds. Ann. Sci. Éc. Norm. Supér. 53(3), 751–786 (2020)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  88. Wang, C.-L.: On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds. Math. Res. Lett. 4(1), 157–171 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

  89. Wang, C.-L.: Quasi-Hodge metrics and canonical singularities. Math. Res. Lett. 10(1), 57–70 (2003)

    MathSciNet  MATH  Google Scholar 

  90. Wilson, P.M.H.: Sectional curvatures of Kähler moduli. Math. Ann. 330(4), 631–664 (2004)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  92. Yau, S.-T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100(1), 197–203 (1978)

    MATH  Google Scholar 

  93. Yoshikawa, K.-I.: On the boundary behavior of the curvature of \(L^2\)-metrics, preprint, arXiv:1007.2836

  94. Zhang, Y.: Note on equivalences for degenerations of Calabi-Yau manifolds. Surveys in Geometric Analysis 2017, pp. 186–202. Science Press, Beijing (2018)

  95. Zhang, Y.S.: Collapsing limits of the Kähler-Ricci flow and the continuity method. Math. Ann. 374(1–2), 331–360 (2019)

    MathSciNet  MATH  Google Scholar 

  96. Zhang, Y.S.: Infinite-time singularity type of the Kähler-Ricci flow II, preprint, arXiv:1809.01305

Download references

Acknowledgements

We are grateful to H. Guenancia, H.-J. Hein, D. Kim, M. Popa and P.M.H. Wilson for discussions. This work was done during the second-named author’s visits to the Institut Henri Poincaré in Paris in 2018 (supported by a Chaire Poincaré at IHP funded by the Clay Mathematics Institute) and to the Center for Mathematical Sciences and Applications at Harvard University in 2018, and during the third-named author’s visit to Northwestern University in 2016, which we would like to thank for the hospitality and support. Mark Gross was supported by EPSRC Grant EP/N03189X/1 and a Royal Society Wolfson Research Merit Award. Valentino Tosatti was also partially supported by NSF Grants DMS-1610278 and DMS-1903147. Yuguang Zhang was supported by the Simons Foundation’s program Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics (Grant #488620).

Author information

Author notes

  1. Yuguang Zhang

    Present address: Riemann Center for Geometry and Physics, Leibniz Universität Hannover, 30167, Hannover, Germany

Authors and Affiliations

  1. DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Mark Gross

  2. Department of Mathematics and Statistics, McGill University, Montréal, QC, H3A 0B9, Canada

    Valentino Tosatti

  3. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA

    Valentino Tosatti

  4. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Yuguang Zhang

Authors
  1. Mark Gross
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valentino Tosatti
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuguang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valentino Tosatti.

Additional information

Communicated by H. T. Yau.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gross, M., Tosatti, V. & Zhang, Y. Geometry of Twisted Kähler–Einstein Metrics and Collapsing. Commun. Math. Phys. 380, 1401–1438 (2020). https://doi.org/10.1007/s00220-020-03911-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-020-03911-0

Search

Navigation