Hamiltonian circle action, invariant hypersurface and the complex projective space

2025/10/22 by Ping Li, Li, Ping
Mathematics · #32Q60 #37B05 #53D05 #57R20 #58J20 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2510.19190

openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Let M be a 2n-dimensional closed symplectic manifold admitting a Hamiltonian circle action with isolated fixed points. We show that if M contains an S¹-invariant symplectic hypersurface D such that M∖ D is a homology cell, which is satisfied when M∖ D is contractible, then M and D are homotopy complex projective spaces with standard Chern classes and the S¹-representations on the fixed-point set of (M,D) are the same as those arising from the standard linear actions on (ℙⁿ,ℙ^n-1), provided that n \not ≡ 3 \pmod 4. This can be viewed as the transformation group analogue to a recent result obtained by Peternell and the author, where the latter was conjectured by Fujita more than four decades ago.

Related

• Hamiltonian circle actions with almost minimal isolated fixed points
• New tools for classifying Hamiltonian circle actions with isolated fixed points
• Hamiltonian S¹-manifolds which are like a coadjoint orbit of G₂
• Embedded surfaces for symplectic circle actions
• On semifree symplectic circle actions with isolated fixed points