Fundamental forms and infinitesimal symmetries of projective varieties

Source abstract

Abstract We give a bound on the dimension of the linear automorphism group of a projective variety Z ⊂ P V $Z \subset {\mathbb P} V$ upper Z subset of double struck upper P upper V in terms of its fundamental forms at a general point. Moreover, we show that the bound is achieved precisely when Z ⊂ P V $Z \subset {\mathbb P} V$ upper Z subset of double struck upper P upper 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.