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

Abstract

The prolongation $\mathfrak{g}^{(k)}$ of a linear Lie algebra $\mathfrak{g}\subset \mathfrak{gl}(V)$ 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 $\mathfrak{g}\subset \mathfrak{gl}(V)$ with non-zero prolongations. If $\mathfrak{g}$ is the Lie algebra $\mathfrak{aut}(\hat{S})$ of infinitesimal linear automorphisms of a projective variety S⊂ℙV, its prolongation $\mathfrak{g}^{(k)}$ 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 $\mathfrak{aut}(\hat{S})^{(k)}$ 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 S⊂ℙV with $\mathfrak{aut}(\hat {S})^{(k)}\neq0$ , 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)≠ℙV, the blow-up Bl S (ℙV) has the target rigidity property, i.e., any deformation of a surjective morphism f:Y→Bl S (ℙV) comes from the automorphisms of Bl S (ℙV).