Toric degenerations of weight varieties and applications

TL;DR

This paper demonstrates that weight varieties, including moduli spaces of polygons, can be degenerated into toric varieties, providing new insights into their geometric structure and applications in symplectic geometry.

Contribution

It introduces a method to degenerate weight varieties to toric varieties, extending to moduli spaces of polygons and general Hamiltonian quotients, with analysis of boundary divisors and real loci.

Findings

Weight varieties admit flat degenerations to toric varieties.

Moduli spaces of polygons degenerate to polarized toric varieties.

Results extend to Flaschka-Millson Hamiltonians on product quotients.

Abstract

We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric varieties with the moment polytopes defined by the lengths of their diagonals. We extend these results to more general Flaschka-Millson hamiltonians on the quotients of products of projective spaces. We also study boundary toric divisors and certain real loci.