Support varieties for Frobenius kernels of classical groups

Let G be a classical simple algebraic group over an algebraically closed field k of characteristic p>0, and denote by G(r) the r-th Frobenius kernel of G. We show that for p large enough, the support variety of a simple G-module over G(r) can be described in terms of support varieties of simple G-modules over G(1). We use this, together with the computation of the varieties VG(1)(H0(λ)), given by Jantzen (1987) in [8], and by Nakano et al. (2002) in [10], to explicitly compute the support variety of a block of Dist(G(r)).