Skip to main content
SpringerLink
Log in

A Note On the Nonexistence of a Complex Threefold as a Conjugate Orbit of \(G_2\)

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The existence or nonexistence of a complex structure on \(\mathrm{S}^6\) was a long standing unsolved problem. There is a well-known orbit \(O(\varLambda )\) in \(G_2\) which is diffeomorphic to \(\mathrm{S}^6\) and used by Gábor Etesi in an effort to find a complex structure. Etesi suggested to give a complex structure in \(\mathrm{S}^6\) through this orbit. In Daniel Guan’s earlier paper, he proved that the orbit can not be a complex submanifold. Since there was not a clear description of the map from \(O(\varLambda )\) to \(\mathrm{S}^6\) in that paper, we give another clearer, simpler and explicit proof of that result in this paper.

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. Agricola, I., Bazzoni, G., Goertsches, O., Konstantis, P., Rollenske, S.: On the history of the Hopf problem. Differ. Geom. Appl. 57, 1–9 (2018)

    Article  MathSciNet  Google Scholar 

  2. Campana, F., Demailly, J.P., Peternell, T.: The algebraic dimension of compact complex threefolds with vanishing second betti numbers. Compos. Math. 112, 77–91 (1998)

    Article  MathSciNet  Google Scholar 

  3. Campana, F., Demailly, J.-P., Peternell, T.: The algebraic dimension of compact complex threefolds with vanishing second Betti number. Compos. Math. 156(4), 679–696 (2020)

    Article  MathSciNet  Google Scholar 

  4. Chern, S.S.: Complex Manifolds without Potential Theory. D. Van. Nostrand Company, Inc., Princeton, New Jersey (1976)

  5. Chaves, L.M., Rigas, A.: On a conjugate orbits of \(G_2\). Math. J. Okayama Univ. 33, 155–161 (1991)

    MathSciNet  MATH  Google Scholar 

  6. Etesi, G.: Explicit Construction of the Complex Structure on the Six Dimensional Sphere, ArXiv:1509.02300 [Math.DG]

  7. Guan, D.: Classification of compact complex homogeneous spaces with invariant volumes. Trans. Am. Math. Soc. 354, 4493–4504 (2002)

    Article  MathSciNet  Google Scholar 

  8. Guan, D.: Examples of Compact Holomorphic Symplectic Manifolds which Admit No Kähler Structure, Geometry and Analysis on Complex Manifolds (Professor S. Kobayashi’s 60-th birthday) pp. 63–74. World Scientific Publishing Co. (1994)

  9. Guan, D.: Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one–IV. Ann. Glob. Anal. Geom. 30, 139–167 (2006)

    Article  MathSciNet  Google Scholar 

  10. Guan, D.: On the nonexistence of \({\rm S}^6\) type complex threefolds in any compact homogeneous complex manifolds with the compact lie group \(G_2\) as the Base Manifold. Pac. J. Math. 305(2), 641–644 (2019)

    Article  Google Scholar 

  11. Hano, J., Kobayashi, S.: A fibering of a class of homogeneous complex manifolds. Trans. Am. Math. Soc. 94, 233–243 (1960)

    Article  MathSciNet  Google Scholar 

  12. Kodaira, K., Morrow, J.: Complex Manifolds, American Mathematical Society (2006)

  13. Lamont, P.: Arithmetics in Cayley’s algebra. Proc. Glasgow Math. Ass. 6, 99–106 (1963)

    Article  MathSciNet  Google Scholar 

  14. Miyasaka, T.: Adjoint orbit types of compact exceptional Lie group \(G_2\) in its Lie algebra. Math. J. Okayama Univ. 43, 17–23 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Moufang, R.: Zur Struktur von Alternativkörpern. Math. Ann. 110, 416–430 (1934)

    Article  MathSciNet  Google Scholar 

  16. Tits, J.: Espaces homogènes complexes compacts. Commentarii Mathematici Helvetici 37(1), 111–120 (1962)

    Article  MathSciNet  Google Scholar 

  17. Wang, H.C.: Closed manifolds with homogeneous complex structure. Am. J. Math. 76, 1–32 (1954)

    Article  MathSciNet  Google Scholar 

  18. Yau, S.T., Schoen, R.: Differential Geometry, (in Chinese). The Science Press, Beijing (1983)

Download references

Acknowledgements

We would like to take this chance to thank S. Feng and the School of Mathematics and Statistics, Henan University for their supports when we are cooperating on this topic. Thanks also go to Josef Dorfmeister for showing the first author [11, 16, 17] when he was still a graduate.

Funding

The funding was provided by National Nature Science Foundation of China (Grant Nos. 11901157, 11971353, 12171140), China Postdoctoral Science Foundation (Grant No. 2020M680096).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Daniel Guan & Zhonghua Wang

  2. Department of Mathematics University of California at Riverside, Riverside, CA, 92521, USA

    Daniel Guan

Authors
  1. Daniel Guan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhonghua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhonghua Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

This research was supported by National Nature Science Foundation of China (Grant No.12171140), and partially supported by National Nature Science Foundation of China (Grant No.11901157, 11971353), and China Postdoctoral Science Foundation (2020M680096)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guan, D., Wang, Z. A Note On the Nonexistence of a Complex Threefold as a Conjugate Orbit of \(G_2\). Results Math 77, 139 (2022). https://doi.org/10.1007/s00025-022-01678-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-022-01678-5

Keywords

Mathematics Subject Classification

Search

Navigation