\(\hbox {G}_2\) Manifolds with Nodal Singularities Along Circles

Abstract

This paper finds matching building blocks for the construction of a compact manifold with \(\hbox {G}_2\) holonomy and nodal singularities along circles using twisted connected sum method by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further gluing is performed in order to get a compact manifold with \(\hbox {G}_2\) holonomy and isolated conical singularities with link \({\mathbb {S}}^3\times {\mathbb {S}}^3\).

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

References

  1. 1.

    Agmon, S., Nirenberg, L.: Properties of solutions of ordinary differential equations in Banach space. Commun. Pure Appl. Math. 16, 121–239 (1963)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Arezzo, C., Spotti, C.: On cscK resolutions of conically singular cscK varieties. J. Funct. Anal. 271, 474–494 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Atiyah, M., Witten, E.: \(M\)-theory dynamics on a manifold of \(G_2\) holonomy. Adv. Theor. Math. Phys. 6(1), 1–106 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Besse, A.L.: Einstein Manifolds, 10. Springer, Berlin (1987)

    Google Scholar 

  5. 5.

    Burns Jr., D., Rapoport, M.: On the Torelli problem for Kählerian K3 surfaces. Ann. Sci. Ecole Norm. Sup. 8(2), 235–273 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Candelas, P., de la Ossa, X.: Comments on conifolds. Nucl. Phys. B 342, 246–268 (1990)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cheeger, J.: On the spectral geometry of spaces with cone-like singularities. Proc. Natl. Acad. Sci. USA 76(5), 2103–2106 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Cheeger, J.: On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator. In: Proceedings of Symposia in Pure Mathematics, Univ. Hawaii, Honolulu, Hawaii, 1979), pp. 91–146, Proceedings of Symposia in Pure Mathematics, XXXVI, American Mathematical Society, Providence, RI (1980)

  9. 9.

    Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Differ. Geom. 18(4), 575–657 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

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

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

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

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

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

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Cheeger, J., Tian, G.: On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118, 493–571 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Chen, X., Wang, Y.: \(C^{2,\alpha }\)-estimate for Monge–Ampère equations with Hölder-continuous right hand side. Ann. Glob. Anal. Geom. 49(2), 195–204 (2016)

    MATH  Article  Google Scholar 

  15. 15.

    Chen, X., Donaldson, S.K., Sun, S.: Kähler–Einstein metrics on Fano manifolds. I: approximation of metrics with cone singularities. J. Am. Math. Soc. 28(1), 183–197 (2015)

    MATH  Article  Google Scholar 

  16. 16.

    Chen, X., Donaldson, S.K., Sun, S.: Kähler–Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2\(\pi \). J. Am. Math. Soc. 28(1), 199–234 (2015)

    MATH  Article  Google Scholar 

  17. 17.

    Chen, X., Donaldson, S.K., Sun, S.: 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(1), 235–278 (2015)

    MATH  Article  Google Scholar 

  18. 18.

    Chern, S.-S.: On holomorphic mappings of hermitian manifolds of the same dimension, 1968 Entire Functions and Related Parts of Analysis. In: Proceedings of Symposia in Pure Mathematics, La Jolla, California, pp. 157–170. American Mathematical Society, Providence, RI (1966)

  19. 19.

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

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

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

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Corti, A., Haskins, M., Nordström, J., Pacini, T.: \(\text{ G }_2\)-manifolds and associative submanifolds via semi-Fano 3-folds. Duke Math. J. 164(10), 1971–2092 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Crowley, D., Nordström, J.: Exotic \(\text{ G }_2\)-manifolds, preprint, arXiv:1411.0656

  23. 23.

    Demailly, J.P.: Complex analytic and differential geometry. https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

  24. 24.

    Doi, M., Yotsutani, N.: Doubling construction of Calabi–Yau threefolds. N. Y. J. Math. 20, 1203–1235 (2014)

    MathSciNet  MATH  Google Scholar 

  25. 25.

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

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Donaldson, S.K., Sun, S.: Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II. J. Differ. Geom. 107(2), 327–371 (2017)

    MATH  Article  Google Scholar 

  27. 27.

    Eells Jr., J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

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

    MATH  Article  Google Scholar 

  29. 29.

    Fernández, M., Gray, A.: Riemannian manifolds with structure group \(\text{ G }_2\). Ann. Mat. Pura Appl. 132(1982), 19–45 (1983)

    Google Scholar 

  30. 30.

    Foscolo, L., Haskins, M., Nordström, J.: Complete non-compact \(\text{ G }_2\)-manifolds from asymptotically conical Calabi–Yau 3-folds. arXiv:1709.04904

  31. 31.

    Fukuoka, T.: On the existence of almost Fano threefolds with del Pezzo fibrations. Math. Nachr. 290(8–9), 1281–1302 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 1998th edn. Springer, Berlin (2001)

    Google Scholar 

  33. 33.

    Hartshorne, R.: Deformation Theory, Graduate Texts in Mathematics, 257. Springer, New York (2010)

    Google Scholar 

  34. 34.

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

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Hein, H.-J.: On gravitational instantons, Thesis (Ph.D.)-Princeton University, 2010, 129 pp, ISBN: 978-1124-34891-9, ProQuest LLC

  36. 36.

    Hein, H.-J., Sun, S.: Calabi–Yau manifolds with isolated conical singularities. Publ. Math. Inst. Hautes Études Sci. 126, 73–130 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Hunsicker, E., Mazzeo, R.: Harmonic forms on manifolds with edges. Int. Math. Res. Not. 52, 3229–3272 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Joyce, D.: Compact Riemannian 7-manifolds with holonomy \(\text{ G }_2\). I. J. Differ. Geom. 43(2), 291–328 (1996)

    MATH  Article  Google Scholar 

  39. 39.

    Joyce, D.: Compact Manifolds with Special Holonomy, OUP Mathematical Monographs Series. Oxford University Press, Oxford (2000)

    Google Scholar 

  40. 40.

    Karigiannis, S., Lotay, J.: Deformation theory of \(\text{ G }_2\) conifolds. Commun. Anal. Geom (2017) (accepted)

  41. 41.

    Klimek, M.: Pluripotential Theory. The Clarendon Press, Oxford University Press, New York (1991)

    Google Scholar 

  42. 42.

    Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16, 209–292 (1967)

    MathSciNet  Google Scholar 

  43. 43.

    Kovalev, A.: Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 565, 125–160 (2003)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Kovalev, A., Singer, M.: Gluing theorems for complete anti-self-dual spaces. Geom. Funct. Anal. 11, 1229–1281 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Lee, N.-H.: Calabi–Yau construction by smoothing normal crossing varieties. Internat. J. Math. 21, 701–725 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Li, Y.: A new complete Calabi–Yau metric on \({\mathbb{C}}^3\). Invent. Math. 217, 1–34 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

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

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Looijenga, E., Peters, C.: Torelli theorems for Kähler K3 surfaces. Compos. Math. 42(2), 145–186 (1980)

    MATH  Google Scholar 

  49. 49.

    Lu, Y.-C.: Holomorphic mappings of complex manifolds. J. Differ. Geom. 2, 299–312 (1968)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Maz’ja, V.G., Plamenevskiǐ, B.A.: Estimates in L\(^p\) and Hölder classes and the Miranda–Agmon Maximum principle for solutions of elliptic boundary problems in domains with singular points on the boundary (in Russian). Math. Nachr. 81, 25–82 (1978)

    MathSciNet  Article  Google Scholar 

  51. 51.

    Mazzeo, R.: Elliptic theory of edge operators, I. Commun. Partial Differ. Equ. 16, 1615–1664 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Melrose, R.B.: The Atiyah–Patodi–Singer Index Theorem. Research Notes in Mathematics. A K Peters Ltd, Wellesley, MA (1993)

    Google Scholar 

  53. 53.

    Melrose, R.B., Mendoza, G.: Elliptic operators of totally characteristic type, unpublished preprint from MSRI, 047-83, (1983)

  54. 54.

    Moroianu, A., Semmelmann, U.: The Hermitian Laplacian operator on nearly Kähler manifolds. Commun. Math. Phys. 294(1), 251–272 (2010)

    MATH  Article  Google Scholar 

  55. 55.

    Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Pjateckiǐ-Šapiro, I.I., Šafarevič, I.R.: A Torelli theorem for algebraic surfaces of type K3. Math USSR Izvestiya. 35, 530–572 (1971)

    Google Scholar 

  57. 57.

    Reid, M.: Chapters on algebraic surfaces, Complex Algebraic Geometry (Park City,UT, 1993). In: IAS/Park City Math. Ser. 3, pp. 3–159, American Mathematical Society, Providence (1997)

    Google Scholar 

  58. 58.

    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)

    MATH  Article  Google Scholar 

  59. 59.

    Schulz, M.B., Tammaro, E.F.: M-theory/type IIA duality and K3 in the Gibbons–Hawking approximation. arXiv:1206.1070

  60. 60.

    Siu, Y.T.: A simple proof of the surjectivity of the period map of K3 surfaces. Manuscripta Math. 35(3), 311–321 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Stenzel, M.: Ricci-flat metrics on the complexification of a compact rank one symmetric space. Manuscripta Math. 80, 151–163 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  62. 62.

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

    MATH  Google Scholar 

  63. 63.

    Todorov, A.N.: Applications of the Kähler–Einstein–Calabi–Yau metric to moduli of K3 surfaces. Invent. Math. 61(3), 251–265 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  64. 64.

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

    MATH  Article  Google Scholar 

Download references

Acknowledgements

The author would like to thank Edward Witten for introducing him to this problem and for many fruitful discussions. The author is also grateful to the referees for detailed suggestions as well as Jeff Cheeger, Xiuxiong Chen, Sir Simon Donaldson, Lorenzo Foscolo, Mark Haskins, Hans-Joachim Hein, Helmut Hofer, Fanghua Lin, Rafe Mazzeo, Johannes Nordström, Song Sun, Akshay Venkatesh, Jeff Viaclovsky and Ruobing Zhang for helpful conversations. This material is based upon work supported by the National Science Foundation under Grant No. 1638352, as well as support from the S. S. Chern Foundation for Mathematics Research Fund.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Gao Chen.

Additional information

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

Verify currency and authenticity via CrossMark

Cite this article

Chen, G. \(\hbox {G}_2\) Manifolds with Nodal Singularities Along Circles. J Geom Anal (2019). https://doi.org/10.1007/s12220-019-00283-3

Download citation

Keywords

  • \(\hbox {G}_2\)
  • Holonomy
  • Calabi–Yau
  • Edge singularity

Mathematics Subject Classification

  • Primary 53C29
  • Secondary 32Q25