Varieties with at most quadratic growth

Let V be a variety of non-necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions c _n (V), n = 1, 2, …, and here we study varieties of polynomial growth. Recently in [16], for any real number α, 3 < α < 4, a variety V was constructed satisfying C ₁ n ^α < c _n (V) < C ₂ n ^α , for some constants C ₁, C ₂. Motivated by this result here we try to classify all possible growth of varieties V such that c _n (V) < C _n ^α , with 0 < α < 2, for some constant C. We prove that if 0 < α < 1 then, for n large, c _n (V) ≤ 1, whereas if V is a commutative variety and 1 < α < 2, then lim _n→∞ log _n c _n (V) = 1 or c _n (V) ≤ 1 for n large enough.