Nilpotent commuting varieties of the Witt algebra

Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$bethenilpotentvarietyof$\mathfrak{g}$,and$\mathscr{C}(\mathscr{N}):={(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0}$thenilpotentcommutingvarietyof$\mathfrak{g}$. As an analogue of Premet’s result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathcal{C}(\mathcal{N})$ is reducible and equidimensional. Irreducible components of $\mathcal{C}(\mathcal{N})$ and their dimension are precisely given. Furthermore, the nilpotent commuting varieties of Borel subalgebras are also determined.