Fundamental forms and infinitesimal symmetries of projective varieties

Abstract

We give a bound on the dimension of the linear automorphism group of a projective variety $Z \subset \mathbb{P} V$ in terms of its fundamental forms at a general point. Moreover, we show that the bound is achieved precisely when $Z \subset \mathbb{P} V$ is projectively equivalent to an Euler-symmetric variety. As a by-product, we determine the Lie algebra of infinitesimal automorphisms of an Euler-symmetric variety and also obtain a rigidity result on the specialization of an Euler-symmetric variety preserving the isomorphism type of the fundamental forms.