Varieties with at most quadratic growth

Let V be a variety of non necessarily associative algebras over afield of characteristic zero. The growth of V is determined by the asymptoticbehavior of the sequence of codimensions cn(V); n = 1; 2, … and here westudy varieties of polynomial growth. Recently, for any real number a, 3 < a < 4, a variety V was constructed satisfying C1n^a < cn(V) < C2n^a;for some constants C1;C2. Motivated by this result here we try to classifyall possible growth of varieties V such that cn(V) < Cn^a; with 0 < a <2, for some constant C. We prove that if 0 < a < 1 then, for n large,cn(V) ≤ 1, whereas if V is a commutative variety and 1 < a < 2, thenlim logn cn(V) = 1 or cn(V) ≤ 1 for n large enough.