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The even Clifford structure of the fourth Severi variety

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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.
Opis fizyczny
  • Universit`a di Chieti-Pescara, Dipartimento di Economia, viale della Pineta 4, I-65129 Pescara, Italy
  • Sapienza-Universit`a di Roma, Dipartimento di Matematica, piazzale Aldo Moro 2,
    I-00185, Roma, Italy
  • [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.
  • [2] J. C. Baez. The octonions. Bull. Amer. Math. Soc. (N.S.), 39(2):145–205, 2002.
  • [3] M. Berger. Du cˆot´e de chez Pu. Ann. Sci. ´ Ecole Norm. Sup. (4), 5:1–44, 1972.
  • [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.
  • [5] R. L. Bryant. Remarks on Spinors in Low Dimension, April 1999. Available at pdf.
  • [6] H. Duan and X. Zhao. The Chow rings of generalized Grassmannians. Found. Comput. Math., 10(3):245–274, 2010.
  • [7] J.-H. Eschenburg. Riemannian Geometry and Linear Algebra, 2012. Available at ~eschenbu/riemlin.pdf.
  • [8] J.-H. Eschenburg. Symmetric Spaces and Division Algebras, 2012. Available at ~eschenbu/symdiv.pdf.
  • [9] D. Ferus, H. Karcher, and H. F. M¨unzner. Cliffordalgebren und neue isoparametrische Hyperfl¨achen. Math. Z., 177(4):479– 502, 1981.
  • [10] T. Friedrich. Weak Spin(9)-structures on 16-dimensional Riemannian manifolds. Asian J. Math., 5(1):129–160, 2001.
  • [11] C. Gorodski and M. Radeschi. On homogeneous composed Clifford foliations, 2015. arXiv:1503.09058 [math.DG].
  • [12] F. R. Harvey. Spinors and Calibrations, volume 9 of Perspectives in Mathematics. Academic Press Inc., Boston, MA, 1990.
  • [13] A. Iliev and L. Manivel. The Chow ring of the Cayley plane. Compos. Math., 141(1):146–160, 2005.
  • [14] S. Ishihara. Quaternion K¨ahlerian manifolds. J. Differential Geometry, 9:483–500, 1974.
  • [15] J. M. Landsberg and L. Manivel. The projective geometry of Freudenthal’s magic square. J. Algebra, 239(2):477–512, 2001.
  • [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] E. Meinrenken. Clifford algebras and Lie theory, volume 58 of Erg. der Math. und Grenz. Springer, Heidelberg, 2013.
  • [18] A. Moroianu and M. Pilca. Higher rank homogeneous Clifford structures. J. Lond. Math. Soc. (2), 87(2):384–400, 2013.
  • [19] A. Moroianu and U. Semmelmann. Clifford structures on Riemannian manifolds. Adv. Math., 228(2):940–967, 2011.
  • [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.
  • [21] M. Postnikov. Lie groups and Lie algebras, volume V of Lectures in geometry. “Mir”, Moscow, 1986.
  • [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.
  • [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.
  • [24] M. Radeschi. Clifford algebras and new singular Riemannian foliations in spheres. Geom. Funct. Anal., 24(5):1660–1682, 2014. [WoS][Crossref]
  • [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.
  • [26] B. Segre. Prodromi di geometria algebrica. Roma: Edizioni Cremonese, 1972.
  • [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.
  • [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]
  • [29] I. Yokota. Exceptional Lie groups, 2009. arXiv:0902.0431 [math.DG].
  • [30] F. L. Zak. Severi varieties. Mat. Sb. (N.S.), 126(168)(1):115–132, 144, 1985. [WoS]
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