Group graded algebras and almost polynomial growth

Let F be a field of characteristic 0, G a finite abelian group and A a G-graded algebra. We prove that A generates a variety of G-graded algebras of almost polynomial growth if and only if A has the same graded identities as one of the following algebras: (1) FCp , the group algebra of a cyclic group of order p, where p is a prime number and p | |G|; (2) UTG 2 (F ), the algebra of 2×2 upper triangular matrices over F endowed with an elementary G-grading; (3) E, the infinite dimensional Grassmann algebra with trivial Ggrading; (4) in case 2 | |G|, EZ2 , the Grassmann algebra with canonical Z2- grading.