Abelianization and moment maps
Abstract
When XG=ℂn/ /G is a GIT quotient of a vector space, replacing G with a maximal torus T ⊂ G produces a closely related toric variety XT:=ℂN/ /T, the \emph{Abelianization} of XG. Abelian/non-Abelian correspondences connect structures on XG with structures on XT. In this paper, we study the interaction between the toric moment map μ of XT and XG, when XG=Gr(r,n). We do this by looking at the restriction of μ to an intermediate space Z/T ⊂ XT, where Z/G=XG. We show that the generic fibres of this map are co-Lagrangians that project to Lagrangians on XG. We completely describe the image in the case where XG=Gr(2,n), and show that in the general Gr(r,n) case the image can be sliced so that each piece is determined by a finite set of points, related to matroid circuits.
Related
- Equivariant Moment Maps and the Ricci Cocycle in Weighted Fujiki--Donaldson Theory
- Generalized Gelfand-Cetlin data and abelianization
- Abelianization of Symmetric Mapping Class Groups
- Representation and abelianization growth, particularly along expanding sequences
- Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators
BibTeX
@misc{kalashnikov2026abelianization,
title = {{Abelianization and moment maps}},
author = {Elana Kalashnikov and Ruxandra Moraru and Francisco Villacis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.42123},
eprint = {2610.42123},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.42123},
}
RIS
TY - UNPB
TI - Abelianization and moment maps
AU - Elana Kalashnikov
AU - Ruxandra Moraru
AU - Francisco Villacis
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.42123
UR - https://arxiv.org/abs/2610.42123
AB - When X_G=ℂ^n/ /G is a GIT quotient of a vector space, replacing G with a maximal torus T ⊂ G produces a closely related toric variety X_T:=ℂ^N/ /T, the \emph{Abelianization} of X_G. Abelian/non-Abelian correspondences connect structures on X_G with structures on X_T. In this paper, we study the interaction between the toric moment map μ of X_T and X_G, when X_G=Gr(r,n). We do this by looking at the restriction of μ to an intermediate space Z/T ⊂ X_T, where Z/G=X_G. We show that the generic fibres of this map are co-Lagrangians that project to Lagrangians on X_G. We completely describe the image in the case where X_G=Gr(2,n), and show that in the general Gr(r,n) case the image can be sliced so that each piece is determined by a finite set of points, related to matroid circuits.
ER -