Poisson blow-ups and the adjoint quotient

Abstract

We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system $\tau :\mathfrak{X}⟶\mathfrak{B}$ to each affine Poisson scheme $\mathfrak{X}$ over $\mathbb{C}$. We prove that the blow-ups of $\mathfrak{X}$ along fibers of $\tau$ are Poisson schemes occurring in a family $\widetilde{\mathfrak{X}\times \mathfrak{B}}⟶\mathfrak{B}$, where $\widetilde{\mathfrak{X}\times \mathfrak{B}}$ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient $\tau :\mathfrak{g}⟶\mathfrak{g}//G=:\mathfrak{c}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with integrating algebraic group $G$. We show that the family $\widetilde{\mathfrak{g}\times \mathfrak{c}}⟶\mathfrak{c}$ is flat, conical, and equipped with a canonical Poisson Hamiltonian $G$-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of $\mathfrak{g}$ along regular adjoint orbit closures.