Real Structures on Horospherical Varieties

Abstract

We study the equivariant real structures on complex horospherical varieties, generalizing classical results known for toric varieties and flag varieties. In particular, we obtain a necessary and sufficient condition for the existence of such real structures and determine the number of equivalence classes. We then apply our results to classify the equivariant real structures on smooth projective horospherical varieties of Picard rank 1. Contents Introduction 1 1. Real group structures on complex algebraic groups 5 2. Equivariant real structures on quasi-homogeneous spaces 13 3. Equivariant real structures on horospherical varieties 16 Appendix A. Tits classes for the simply-connected simple algebraic groups (by Mikhail Borovoi) 26 References 29