On nilpotent commuting varieties of r-tuples in the Witt algebra

Let gg be the Witt algebra over an algebraically closed field k   of characteristic p>3p>3, and G=Aut(g)G=Aut(g) be the automorphism group of gg with Lie(G)=g0Lie(G)=g0. A result [12, Theorem 5.2] of A. Suslin, E. Friedlander and C. Bendel implies that the spectrum of the cohomology ring for the r-th Frobenius kernel of G is homeomorphic to the commuting variety of r  -tuples of nilpotent elements in g0g0. As an analogue of Ngo's result [5, Theorem 1.2.1] in the case of the classical Lie algebra sl2sl2 and a generalization of our previous work [15], in this paper we show that the varieties of r  -tuples of nilpotent elements in gg, as well as certain subalgebras, are reducible. Irreducible components and their dimensions are precisely presented. Moreover, these nilpotent commuting varieties of r  -tuples are neither normal nor Cohen–Macaulay. These results are different from those in the case of the classical Lie algebra sl2sl2.