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.
@article{bwmeta1.element.doi-10_1515_coma-2015-0008,
author = {Maurizio Parton and Paolo Piccinni},
title = {The even Clifford structure of the fourth Severi variety},
journal = {Complex Manifolds},
volume = {2},
year = {2015},
zbl = {1336.53058},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0008}
}
Maurizio Parton; Paolo Piccinni. The even Clifford structure of the fourth Severi variety. Complex Manifolds, Tome 2 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0008/
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