Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity

• 成果类型: Article
• 署名作者: Fu, Baohua; Hwang, Jun-Muk
• 署名单位: Korea Institute for Advanced Study (KIAS); Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-011-0369-9
• 发表日期: 2012
• 页码: 457-513
• 关键词: rational homogeneous spaces; holomorphic maps; fano manifolds; CURVES; geometry

摘要

The prolongation of a linear Lie algebra plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras with non-zero prolongations. If is the Lie algebra of infinitesimal linear automorphisms of a projective variety SaS,a(TM) V, its prolongation is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety SaS,a(TM) V with , which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when S is linearly normal and Sec (S)not equal a(TM) V, the blow-up Bl (S) (a(TM) V) has the target rigidity property, i.e., any deformation of a surjective morphism f:Y -> Bl (S) (a(TM) V) comes from the automorphisms of Bl (S) (a(TM) V).