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

Abstract

We study the Borel moment map ${\mu }_B:T^{\ast }(\mathfrak{b}\times {\mathbb{C}}^n)\to {\mathfrak{b}}^{\ast }$, 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.

12 pages