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

TL;DR

This paper classifies certain projective varieties with non-zero prolongations, extending classical results, and applies this to demonstrate target rigidity of blow-ups along these varieties under specific conditions.

Contribution

It provides a complete classification of irreducible non-degenerate nonsingular varieties with non-zero prolongations, generalizing Cartan and Kobayashi-Nagano's work, and applies it to target rigidity in algebraic geometry.

Findings

Classified irreducible non-degenerate varieties with non-zero prolongations.

Extended classical classification results to a broader class of varieties.

Proved target rigidity for blow-ups along these varieties under certain conditions.

Abstract

The prolongation g^{(k)} of a linear Lie algebra g \subset 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 g \subset gl(V) with non-zero prolongations. If g is the Lie algebra aut(\hat{S}) of infinitesimal linear automorphisms of a projective variety S \subset \BP V, its prolongation 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 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 with non zero prolongations, 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) \neq P(V), the blow-up of P(V) along S has the target rigidity property, i.e., any deformation of a surjective morphism Y \to Bl_S(PV) comes from the automorphisms of Bl_S(PV).