The even Clifford structure of the fourth Severi variety

Maurizio Parton; Paolo Piccinni

Complex Manifolds (2015)

  • Volume: 2, Issue: 1, page 89-104, electronic only
  • ISSN: 2300-7443

Abstract

top
TheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl0(E) → End(TM) mapping Ʌ2E into skew-symmetric endomorphisms, and the existence of a metric connection on E compatible with φ. We give an explicit description of such a vector bundle E as a sub-bundle of End(TM). From this we construct a canonical differential 8-form on EIII, associated with its holonomy Spin(10) · U(1) ⊂ U(16), that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at EIII as the smooth projective variety V(4) ⊂ CP26 known as the fourth Severi variety.

How to cite

top

Maurizio Parton, and Paolo Piccinni. "The even Clifford structure of the fourth Severi variety." Complex Manifolds 2.1 (2015): 89-104, electronic only. <http://eudml.org/doc/275869>.

@article{MaurizioParton2015,
abstract = {TheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl0(E) → End(TM) mapping Ʌ2E into skew-symmetric endomorphisms, and the existence of a metric connection on E compatible with φ. We give an explicit description of such a vector bundle E as a sub-bundle of End(TM). From this we construct a canonical differential 8-form on EIII, associated with its holonomy Spin(10) · U(1) ⊂ U(16), that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at EIII as the smooth projective variety V(4) ⊂ CP26 known as the fourth Severi variety.},
author = {Maurizio Parton, Paolo Piccinni},
journal = {Complex Manifolds},
keywords = {Clifford structure; exceptional symmetric space; octonions; canonical differential form},
language = {eng},
number = {1},
pages = {89-104, electronic only},
title = {The even Clifford structure of the fourth Severi variety},
url = {http://eudml.org/doc/275869},
volume = {2},
year = {2015},
}

TY - JOUR
AU - Maurizio Parton
AU - Paolo Piccinni
TI - The even Clifford structure of the fourth Severi variety
JO - Complex Manifolds
PY - 2015
VL - 2
IS - 1
SP - 89
EP - 104, electronic only
AB - TheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl0(E) → End(TM) mapping Ʌ2E into skew-symmetric endomorphisms, and the existence of a metric connection on E compatible with φ. We give an explicit description of such a vector bundle E as a sub-bundle of End(TM). From this we construct a canonical differential 8-form on EIII, associated with its holonomy Spin(10) · U(1) ⊂ U(16), that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at EIII as the smooth projective variety V(4) ⊂ CP26 known as the fourth Severi variety.
LA - eng
KW - Clifford structure; exceptional symmetric space; octonions; canonical differential form
UR - http://eudml.org/doc/275869
ER -

References

top
  1. [1] M. Atiyah and J. Berndt. Projective planes, Severi varieties and spheres. In Surveys in Differential Geometry, volume VIII, pages 1–27. Int. Press, Somerville, MA, 2003. Zbl1057.53040
  2. [2] J. C. Baez. The octonions. Bull. Amer. Math. Soc. (N.S.), 39(2):145–205, 2002. Zbl1026.17001
  3. [3] M. Berger. Du cˆot´e de chez Pu. Ann. Sci. ´ Ecole Norm. Sup. (4), 5:1–44, 1972. 
  4. [4] A. Borel. Sur la cohomologie des espaces fibr´es principaux et des espaces homog`enes de groupes de Lie compacts. Ann. of Math. (2), 57:115–207, 1953. Zbl0052.40001
  5. [5] R. L. Bryant. Remarks on Spinors in Low Dimension, April 1999. Available at http://www.math.duke.edu/~bryant/Spinors. pdf. 
  6. [6] H. Duan and X. Zhao. The Chow rings of generalized Grassmannians. Found. Comput. Math., 10(3):245–274, 2010. Zbl1193.14008
  7. [7] J.-H. Eschenburg. Riemannian Geometry and Linear Algebra, 2012. Available at http://www.math.uni-augsburg.de/ ~eschenbu/riemlin.pdf. 
  8. [8] J.-H. Eschenburg. Symmetric Spaces and Division Algebras, 2012. Available at http://www.math.uni-augsburg.de/ ~eschenbu/symdiv.pdf. 
  9. [9] D. Ferus, H. Karcher, and H. F. M¨unzner. Cliffordalgebren und neue isoparametrische Hyperfl¨achen. Math. Z., 177(4):479– 502, 1981. Zbl0443.53037
  10. [10] T. Friedrich. Weak Spin(9)-structures on 16-dimensional Riemannian manifolds. Asian J. Math., 5(1):129–160, 2001. Zbl1021.53028
  11. [11] C. Gorodski and M. Radeschi. On homogeneous composed Clifford foliations, 2015. arXiv:1503.09058 [math.DG]. 
  12. [12] F. R. Harvey. Spinors and Calibrations, volume 9 of Perspectives in Mathematics. Academic Press Inc., Boston, MA, 1990. Zbl0694.53002
  13. [13] A. Iliev and L. Manivel. The Chow ring of the Cayley plane. Compos. Math., 141(1):146–160, 2005. Zbl1071.14056
  14. [14] S. Ishihara. Quaternion K¨ahlerian manifolds. J. Differential Geometry, 9:483–500, 1974. Zbl0297.53014
  15. [15] J. M. Landsberg and L. Manivel. The projective geometry of Freudenthal’s magic square. J. Algebra, 239(2):477–512, 2001. Zbl1064.14053
  16. [16] R. Lazarsfeld and A. Van de Ven. Topics in the geometry of projective space, volume 4 of DMV Seminar. Birkh¨auser Verlag, Basel, 1984. 
  17. [17] E. Meinrenken. Clifford algebras and Lie theory, volume 58 of Erg. der Math. und Grenz. Springer, Heidelberg, 2013. Zbl1267.15021
  18. [18] A. Moroianu and M. Pilca. Higher rank homogeneous Clifford structures. J. Lond. Math. Soc. (2), 87(2):384–400, 2013. Zbl1269.53055
  19. [19] A. Moroianu and U. Semmelmann. Clifford structures on Riemannian manifolds. Adv. Math., 228(2):940–967, 2011. Zbl1231.53042
  20. [20] L. Ornea, M. Parton, P. Piccinni, and V. Vuletescu. Spin(9) geometry of the octonionic Hopf fibration. Transformation Groups, 18(3):845–864, 2013. Zbl1298.53040
  21. [21] M. Postnikov. Lie groups and Lie algebras, volume V of Lectures in geometry. “Mir”, Moscow, 1986. 
  22. [22] M. Parton and P. Piccinni. Spin(9) and almost complex structures on 16-dimensional manifolds. Ann. Global Anal. Geom., 41(3):321–345, 2012. Zbl1258.53047
  23. [23] M. Parton and P. Piccinni. Spheres with more than 7 vector fields: All the fault of Spin(9). Linear Algebra Appl., 438(3):1113– 1131, 2013. Zbl1266.15044
  24. [24] M. Radeschi. Clifford algebras and new singular Riemannian foliations in spheres. Geom. Funct. Anal., 24(5):1660–1682, 2014. [WoS][Crossref] 
  25. [25] F. Russo. Tangents and secants of algebraic varieties: notes of a course. Publica¸c˜oes Matem´aticas do IMPA. Instituto de Matem´atica Pura e Aplicada (IMPA), Rio de Janeiro, 2003. 24o Col´oquio Brasileiro de Matem´atica. Zbl1061.14058
  26. [26] B. Segre. Prodromi di geometria algebrica. Roma: Edizioni Cremonese, 1972. Zbl0281.14001
  27. [27] F. Severi. Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’ suoi punti tripli apparenti. Palermo Rend., 15:33–51, 1901. Zbl32.0648.04
  28. [28] H. Toda and T. Watanabe. The integral cohomology ring of F4/T and E6/T. J. Math. Kyoto Univ., 14:257–286, 1974. [WoS] Zbl0289.57025
  29. [29] I. Yokota. Exceptional Lie groups, 2009. arXiv:0902.0431 [math.DG]. 
  30. [30] F. L. Zak. Severi varieties. Mat. Sb. (N.S.), 126(168)(1):115–132, 144, 1985. [WoS] Zbl0577.14031

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.