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Complex Manifolds

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The even Clifford structure of the fourth Severi variety

Maurizio Parton
  • Corresponding author
  • Universit`a di Chieti-Pescara, Dipartimento di Economia, viale della Pineta 4, I-65129 Pescara, Italy
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Paolo Piccinni
  • Corresponding author
  • Sapienza-Universit`a di Roma, Dipartimento di Matematica, piazzale Aldo Moro 2, I-00185, Roma, Italy
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-08-04 | DOI: https://doi.org/10.1515/coma-2015-0008


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.

Keywords: Clifford structure; exceptional symmetric space; octonions; canonical differential form

MSC: Primary 53C26; 53C27; 53C38


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About the article

Received: 2015-06-17

Accepted: 2015-07-22

Published Online: 2015-08-04

Citation Information: Complex Manifolds, Volume 2, Issue 1, ISSN (Online) 2300-7443, DOI: https://doi.org/10.1515/coma-2015-0008.

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© 2015 Maurizio Parton and Paolo Piccinni. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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