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Abstract We construct a family of examples of Legendrian subvarieties in some projective spaces. Although most of them are singular, a new example of smooth Legendrian varieties in dimension 8 is in this family. This 8-fold has interesting properties: it is a compactification of the special linear group, a Fano manifold of index 5 and Picard number 1.
Jarosław Buczyński (Fri,) studied this question.