On nilpotent commuting varieties and cohomology of Frobenius kernels

The paper studies the dimensions of irreducible components of commuting varieties of (restricted) nilpotent r  -tuples in a classical Lie algebra gg defined over an algebraically closed field k  . As applications, we obtain some new results on the structure of the (even) cohomology ring of Frobenius kernels GrGr for each r≥1r≥1, where G   is the simply connected, simple algebraic group such that Lie(G)=gLie(G)=g. Explicit calculations for rank two groups are also presented.