Let G be a connected semisimple group over an algebraically closed field K of characteristic p>0, and [gfr ]=Lie (G). Fix a linear function χ∈[gfr ]* and let [zfr ][gfr ](χ) denote the stabilizer of χ in [gfr ]. Set [Nscr]p([gfr ])={x∈[gfr ][mid ]x[p]=0}. Let [Cscr]χ([gfr ]) denote the category of finite-dimensional [gfr ]-modules with p-character χ. In [7], Friedlander and Parshall attached to each M∈Ob([Cscr]χ([gfr ])) a Zariski closed, conical subset [Vscr][gfr ](M)⊂[Nscr]p([gfr ]) called the support variety of M. Suppose that G is simply connected and p is not special for G, that is, p≠2 if G has a component of type Bn, Cn or F4, and p≠3 if G has a component of type G2. It is proved in this paper that, for any nonzero M∈Ob([Cscr]χ([gfr ])), the support variety [Vscr][gfr ](M) is contained in [Nscr]p([gfr ])∩[zfr ][gfr ](χ). This allows one to simplify the proof of the Kac–Weisfeiler conjecture given in [18].