Skip to main content
SpringerLink
Log in

Geometric Structures and Substructures on Uniruled Projective Manifolds

  • Conference paper
Foliation Theory in Algebraic Geometry

Part of the book series: Simons Symposia ((SISY))

Abstract

In a series of works on uniruled projective manifolds starting in the late 1990’s, Jun-Muk Hwang and the author have developed the basics of a geometric theory of uniruled projective manifolds arising from the study of varieties of minimal rational tangents (VMRTs), i.e., the collection at a general point of tangents to minimal rational curves passing through the point. From its onset, our theory is a cross-over between algebraic geometry and differential geometry. While we deal with problems in algebraic geometry, the heart of our perspective is differential-geometric in nature, revolving around foliations, G-structures, differential systems, etc. and dealing with various issues relating to connections, curvature and integrability.The current article is written with the aim of highlighting certain aspects in the geometric theory of VMRTs revolving around the theme of analytic continuation of geometric structures and substructures. For the parts of the article where adequate exposition already exists, we recall fundamental elements and results in the theory essential for the understanding of more recent development and provide occasional examples for illustration. The presentation will be more systematic on sub-VMRT structures since the latter topic is relatively new. We will discuss various perspectives concerning sub-VMRT structures, and indicate how the subject has intimate links with other areas of mathematics including several complex variables, local differential geometry and Kähler geometry.

An erratum to this chapter can be found at 10.1007/978-3-319-24460-0_8

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Campana F, Peternell T (1991) Projective manifolds whose tangent bundles are numerically effective. Math Ann 289:169–187

    Article  MathSciNet  MATH  Google Scholar 

  2. Casagrande C, Druel S (2012) Locally unsplit families of rational curves of large anticanonical degree on Fano manifolds. arXiv:1212.5083v2

    Google Scholar 

  3. Cho K, Miyaoka Y, Shepherd-Barron NI (2002) Characterizations of projective space and applications to complex symplectic manifolds, in Higher dimensional birational geometry (Kyoto, 1997), pp 1–88. Adv. Stud. Pure Math., vol 35. Math. Soc. Japan, Tokyo

    Google Scholar 

  4. Choe I, Hong J (2004) Integral varieties of the canonical cone structure on G∕P. Math Ann 329:629–652

    Article  MathSciNet  MATH  Google Scholar 

  5. Grauert H (1962) Über Modifikationen und exzeptionelle analytische Mengen. Math Ann 146:331–368

    Article  MathSciNet  MATH  Google Scholar 

  6. Grothendieck A (1957) Sur la classification des fibrés holomorphes sur la sphère de Riemann. Am J Math 79:121–138

    Article  MathSciNet  MATH  Google Scholar 

  7. Guillemin V (1965) The integrability problem for G-structures. Trans AMS 116:544–560

    MathSciNet  MATH  Google Scholar 

  8. Hong J, Hwang J-M (2008) Characterization of the rational homogeneous manifold associated to a long simple root by its variety of minimal rational tangents, in Algebraic Geometry in East Asia, Hanoi 2005. Advanced Studies in Pure Mathematics vol 50, pp 217–236

    MathSciNet  Google Scholar 

  9. Hwang J-M, Kim H (2013) Varieties of minimal rational tangents on double covers of projective space. Math Zeit 275:109–125

    Article  MathSciNet  MATH  Google Scholar 

  10. Hwang J-M, Kim H (2013) Varieties of minimal rational tangents on Veronese double cones. Preprint

    MATH  Google Scholar 

  11. Hong J (2000) Fano manifolds with geometric structures modeled after homogeneous contact manifolds. Int J Math 11:1203–1230

    MathSciNet  MATH  Google Scholar 

  12. Hong J, Mok N (2010) Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds. J Differ Geom 86:539–567

    MathSciNet  MATH  Google Scholar 

  13. Hong J, Mok N (2013) Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1. J Alg Geom 22:333–362

    Article  MathSciNet  MATH  Google Scholar 

  14. Hong J, Park K-D (2011) Characterization of standard embeddings between rational homogeneous manifolds of Picard number 1. Int Math Res Not 2011:2351–2373

    MathSciNet  MATH  Google Scholar 

  15. Hwang J-M (1997) Rigidity of homogeneous contact manifolds under Fano deformation. J Reine Angew Math 486:153–164

    MathSciNet  MATH  Google Scholar 

  16. Hwang J-M (2001) Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste 2000), ICTP Lect. Notes, vol 6. Abdus Salam Int. Cent. Theoret. Phys., Trieste, pp 335–393

    Google Scholar 

  17. Hwang J-M (2007) Rigidity of rational homogeneous spaces. Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol II. European Math. Soc. Publishing House, Zürich, pp 613–626

    Google Scholar 

  18. Hwang J-M (2012) Geometry of varieties of minimal rational tangents, in Current Developments in Algebraic Geometry, MSRI Publications, vol 59. Cambridge University Press, New York, pp 197–222

    Google Scholar 

  19. Hwang J-M (2014) Deformation of the space of lines on the 5-dimensional hyperquadric. Preprint

    Google Scholar 

  20. Hwang J-M (2014) Cartan-Fubini type extension of holomorphic maps respecting varieties of minimal rational tangents. Preprint

    MATH  Google Scholar 

  21. Hwang J-M (2015) Mori geometry meets Cartan geometry: varieties of minimal rational tangents. arXiv:1501.04720v1

    Google Scholar 

  22. Hwang J-M, Mok N (1997) Uniruled projective manifolds with irreducible reductive G-structures. J Reine Angew Math 490:55–64

    MathSciNet  MATH  Google Scholar 

  23. Hwang J-M, Mok N (1998) Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation. Invent Math 131:393–418

    Article  MathSciNet  MATH  Google Scholar 

  24. Hwang J-M, Mok N ( 1999) Varieties of minimal rational tangents on uniruled manifolds. In: Schneider M, Siu Y-T (eds) Several complex variables. MSRI publications, vol 37. Cambridge University Press, Cambridge, pp 351–389

    Google Scholar 

  25. Hwang J-M, Mok N (1999) Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds. Invent Math 136:209–231

    Article  MathSciNet  MATH  Google Scholar 

  26. Hwang J-M, Mok N (2001) Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1. J Math Pure Appl 80:563–575

    Article  MathSciNet  MATH  Google Scholar 

  27. Hwang J-M, Mok N (2002) Deformation rigidity of the rational homogeneous space associated to a long simple root. Ann Scient Éc Norm Sup 4e série 35:173–184

    Google Scholar 

  28. Hwang J-M, Mok N (2004) Deformation rigidity of the 20-dimensional F 4-homogeneous space associated to a short root. In: Popov VL (ed) Algebraic transformation groups and algebraic varieties, Encyclopaedia of Mathematical Sciences. Springer, Berlin-Heidelberg-New York, pp 37–58

    Chapter  Google Scholar 

  29. Hwang J-M, Mok N (2004) Birationality of the tangent map for minimal rational curves, Special Volume dedicated to Professor Yum-Tong Siu on his sixtieth birthday. Asian J Math 8:51–64

    Article  MathSciNet  Google Scholar 

  30. Hwang J-M, Mok N (2005) Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation. Invent Math 160:591–645

    Article  MathSciNet  MATH  Google Scholar 

  31. Kebekus S (2002) Families of singular rational curves. J Alg Geom 11:245–256

    Article  MathSciNet  MATH  Google Scholar 

  32. Kebekus S, Sola Conde L (2006) Existence of rational curves on algebraic varieties, minimal rational tangents, and applications. In: Global aspects of complex geometry. Springer, Berlin, pp 359–416

    Chapter  Google Scholar 

  33. Kollár J (1996) Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge) Band, vol 32. Springer, Berlin

    Google Scholar 

  34. Landsberg JM, Manivel L (2003) On the projective geometry of rational homogeneous varieties. Comment Math Helv 78:65–100

    Article  MathSciNet  MATH  Google Scholar 

  35. Manin YI (1997) Gauge field theory and complex geometry. Grundlehren der mathematischen Wissenschaften, vol 289, Second edition. Springer, Berlin

    Google Scholar 

  36. Miyaoka Y, Mori S (1986) A numerical criterion for uniruledness. Ann Math 124:65–69

    Article  MathSciNet  MATH  Google Scholar 

  37. Mok N (1988) The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J Differ Geom 27:179–214

    MathSciNet  MATH  Google Scholar 

  38. Mok N (1999) G-structures on irreducible Hermitian symmetric spaces of rank ≥ 2 and deformation rigidity. Contemp Math 222:81–107

    Google Scholar 

  39. Mok N (2002) On Fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents. Trans AMS 354:2639–2658

    Article  MathSciNet  MATH  Google Scholar 

  40. Mok N (2008) Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents. Sci China Ser A Math 51:660–684

    Article  MathSciNet  MATH  Google Scholar 

  41. Mok N (2008) Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents, Proceedings of the Conference “Géométrie différentielle, Physique mathématique, Mathématique et Société”. Astérisque 322(Volume II):151–205. published by Société Mathématique de France, Paris

    Google Scholar 

  42. Mok N (2008) Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents. AMS/IP Stud Adv Math 42:41–61

    MathSciNet  MATH  Google Scholar 

  43. Mok N (2015) Holomorphic isometries of the complex unit ball into irreducible bounded symmetric domains, Preprint url: http://hkumath.hku.hk/~imr/IMRPrintSeries/2015/IMR2015-1.pdf

  44. Mori S (1979) Projective manifolds with ample tangent bundles. Ann Math 110:593–606

    Article  MathSciNet  MATH  Google Scholar 

  45. Mok N, Tsai I-H (1992) Rigidity of convex realizations of irreducible bounded symmetric domains of rank 2. J Reine Angew Math 431:91–122

    MathSciNet  MATH  Google Scholar 

  46. Mok N, Zhang Y (2014) Rigidity of pairs of rational homogeneous spaces of Picard number 1 and analytic continuation of geometric substructures on uniruled projective manifolds. http://hkumath.hku.hk/~imr/IMRPrintSeries/2015/IMR2015-8.pdf

  47. Neretin YuA (1999) Conformal geometry of symmetric spaces, and generalized linear-fractional Krein-Smul’yan mappings (Russian). Mat Sb 190:93–122. Translation in (1999) Sb Math 190:255–283

    Google Scholar 

  48. Ochiai T (1970) Geometry associated with semisimple flat homogeneous spaces. Trans AMS 152:159–193

    Article  MathSciNet  MATH  Google Scholar 

  49. Siu Y-T (1974) Techniques of extension of analytic objects. Lecture Notes in Pure and Applied Mathematics, vol 8. Marcel Dekker, New York

    Google Scholar 

  50. Pasquier B, Perrin N (2010) Local rigidity of quasi-regular varieties. Math Zeit 265: 589–600

    Article  MathSciNet  MATH  Google Scholar 

  51. Tsai I-H (1993) Rigidity of holomorphic maps between symmetric domains. J Diff Geom 37:123–160

    MathSciNet  MATH  Google Scholar 

  52. Tu, Z.-H: Rigidity of proper holomorphic mappings between non-equidimensional bounded symmetric domains Math. Z. 240 (2002), 13–35

    Google Scholar 

  53. Uhlenberg K, Yau S-T (1986) On the existence of hermitian-yang-mills connections in stable vector bundles. Comm Pure Appl Math 39(Issue Supplement S1):S257–S293

    Google Scholar 

  54. Wolf JA (1972) Fine structure of Hermitian symmetric spaces. In: Boothby-Weiss (ed) Geometry of symmetric spaces. Marcel-Dekker, New York, pp 271–357

    Google Scholar 

  55. Yamaguchi K (1993) Differential systems associated with simple graded Lie algebras. In: Progress in differential geometry, Adv. Study Pure Math. vol 22, pp 413–494

    Google Scholar 

  56. Zhang Y (2014) Admissible pairs of Hermitian symmetric spaces in the perspective of the theory of varieties of minimal rational tangents. arXiv:1412.7598

    Google Scholar 

Download references

Acknowledgements

In September 2013 a conference on Foliation Theory in Algebraic Geometry was held by the Simons Foundation in New York City organized by Paolo Cascini, Jorge Pereira, and James McKernan. The author would like to thank the organizers for their invitation and the Simons Foundation for the hospitality. For the Proceedings Volume the author has taken the opportunity to expound on a topic more ripe for an overview, viz., on foliation-theoretic aspects of the geometric theory of VMRTs. He would like to thank Jun-Muk Hwang for recent discussions on the topic some of which have been incorporated into the article, and the referee for helpful suggestions for the revision of the article.

This research was partially supported by the GRF grant 7039/12P of the Hong Kong Research Grants Council.

Author information

Authors and Affiliations

  1. The University of Hong Kong, Pokfulam Road, Pokfulam, Hong Kong

    Ngaiming Mok

Authors
  1. Ngaiming Mok
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ngaiming Mok .

Editor information

Editors and Affiliations

  1. Mathematics Department, Imperial College of London, London, UK

    Paolo Cascini

  2. Department of Mathematics, UC San Diego, La Jolla, CA, USA

    James McKernan

  3. IMPA, Rio de Janeiro, Brazil

    Jorge Vitório Pereira

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Mok, N. (2016). Geometric Structures and Substructures on Uniruled Projective Manifolds. In: Cascini, P., McKernan, J., Pereira, J.V. (eds) Foliation Theory in Algebraic Geometry. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-24460-0_5

Download citation

Publish with us

Policies and ethics

Search

Navigation