Leafrians, Poisson structures, and Frobenius splittings on moduli spaces

Abstract

The Poisson $T$-Pfaffians of Bott--Samelson varieties, $G/U$, and $G/B$ produce Frobenius splittings, but only by a local, coordinate-bound argument. We isolate the local mechanism: a weighted trace pairing on the deformation space of a fat point, whose top coefficient factors over the blocks. At the members of a presentation, polyvector fields on the base $X$ transfer to the moduli space $M$. Applied to a Poisson bivector, this produces a Poisson bivector, extending Bottacin's structure from surfaces to arbitrary dimension. Applied to a top degree polyvector $s$ and wedged with a transverse Euler frame, it produces an anticanonical section on $M$, which we call the \emph{leafrian}. When $X$ is a Poisson surface, we prove that the leafrian splits the moduli of stable sheaves, of principal $G$-bundles, and of instantons if and only if $s$ Frobenius-splits the base. The mechanism itself is dimension-free. The construction also recovers and generalizes the splittings of the Lie-theoretic examples: the $r$-matrix need not satisfy the classical Yang--Baxter equation.