Varieties of superalgebras of almost polynomial growth

Let Vgr be a variety of superalgebras and let , n=1,2,… , be its sequence of graded codimensions. Such a sequence is polynomially bounded if and only if Vgr does not contain a list of five superalgebras consisting of a commutative superalgebra, the infinite dimensional Grassmann algebra and the algebra of 2×2 upper triangular matrices with trivial and natural Z2-gradings. In this paper we completely classify all subvarieties of the varieties generated by these five superalgebras, by giving a complete list of finite dimensional generating superalgebras.