2025/07/07 by Victor Matveevich Buchstaber, Buchstaber, Victor M., Svjetlana Terzić +1
Mathematics · #Geometry and complex manifolds #Advanced Algebra and Geometry #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2507.04582
Plücker coordinates define the Tn-equivariant embedding p : Gn,2→ \C PN of a complex Grassmann manifold Gn,2 into the complex projective space \C PN, N=\binomn2-1 for the canonical Tn-action on Gn,2 and the Tn-action on \C PN given by the second exterior power representation Tn→ TN and the standard TN-action. Let μ: Gn,2→ Δn,2⊂ \R n and μ: \C PN→ Δn,2⊂ \Rn be the moment maps for the Tn-actions on Gn,2 and \C PN respectively, such that μ ∘ p=μ. The preimages μ-1(\bf x) and μ -1(\bf y) are smooth submanifolds in Gn, 2 and \C PN, for any regular values \bf x, \bf y ∈ Δn,2 for these maps, respectively. The orbit spaces μ-1(\bf x)/Tn and μ-1(\bf y)/Tn are symplectic manifolds, which are known as symplectic reduction. The regular values for μ and μ coincide for n=4 and we prove that μ-1(\bf x) and μ-1(\bf x) do not depend on a regular value \bf x∈ Δ4,2. We provide their explicit topological description, that is we prove μ-1(\bf x)≅ S3× T2 and μ -1(\bf x)≅ S5× T2. The Deligne - Mumford compactification M0, n is proved to be a symplectic reduction of Gn,2 by the canonical Tn-action if and only if n=4,5, while the Losev - Manin compactification is a such symplectic reduction if and only if n=5.