Exponentiation of commuting nilpotent varieties

Let H   be a linear algebraic group over an algebraically closed field of characteristic p>0p>0. We prove that any “exponential map” for H induces a bijection between the variety of r  -tuples of commuting [p][p]-nilpotent elements in Lie(H)Lie(H) and the variety of height r infinitesimal one-parameter subgroups of H. In particular, we show that for a connected reductive group G in pretty good characteristic, there is a canonical exponential map for G and hence a canonical bijection between the aforementioned varieties, answering in this case questions raised both implicitly and explicitly by Suslin, Friedlander, and Bendel.