Towards the affine and geometric invariant theory quotients of the Borel moment map

Abstract

We study the Borel moment map $\mu_B:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, given by $(r,s,i,j)\mapsto [r,s]+ij$, and describe our algorithm to construct the geometric invariant theory (GIT) quotients $\mu_B^{-1}(0)/\!\!/_{\det}B$ and $\mu_B^{-1}(0)/\!\!/_{\det^{-1}}B$, and the affine quotient $\mu_B^{-1}(0)/\!\!/B$. We also provide an insight of the singular locus of $2^n$ irreducible components of $\mu_B$. Finally, analogous to the Hilbert--Chow morphism, we discuss that the GIT quotient for the Borel setting is a resolution of singularities.