Finiteness and boundedness of positive monotone Hamiltonian GKM₃ spaces

2026/06/27 by Panagiotis Konstantis, Silvia Sabatini, Leopold Zoller · 1 voice
Mathematics · #math.SG

paper · pdf

Abstract

In this paper, we establish three finiteness and boundedness theorems for compact positive monotone symplectic manifolds endowed with special actions, called GKM₃, which generalize smooth toric varieties. Specifically, we prove that, for fixed dimension and Euler characteristic, there are only finitely many complex cobordism classes of such spaces. Moreover, modulo lattice transformations, the moment map image can be embedded into a box of explicitly bounded size, and all Chern numbers satisfy quantitative bounds. In particular, this yields a bound on the volume of the underlying symplectic manifold, analogous to the one obtained by Kollár-Miyaoka-Mori for Fano varieties.

Citations

• On integral Chang-Skjelbred computations with disconnected isotropy groups
• On a symplectic generalization of a Hirzebruch problem
• Monotone Symplectic Six-Manifolds that admit a Hamiltonian GKM Action are diffeomorphic to Smooth Fano Threefolds
• 12, 24 and Beyond
• Circle-invariant fat bundles and symplectic Fano 6-manifolds
• New tools for classifying Hamiltonian circle actions with isolated fixed points
• Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle
• Displacing Lagrangian toric fibers via probes
• Rigid subsets of symplectic manifolds
• Exact cohomology sequences with integral coefficients for torus actions
• The reflexive dimension of a lattice polytope
• One-skeleta, Betti numbers and equivariant cohomology
• Equivariant de Rham Theory and Graphs
• Equivariant cohomology, Koszul duality, and the localization theorem
• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• The moment map and equivariant cohomology

Discussions

• Panagiotis Konstantis, Silvia Sabatini, Leopold Zoller: Finiteness and boundedness of positive monotone Hamiltonian GKM$_3$ spaces https://arxiv.org/abs/2606.28768 https://arxiv.org/pdf/2606.28768 htt [bsky, 0 points, 0 comments]

Related

• Monotone Symplectic Six-Manifolds that admit a Hamiltonian GKM Action are diffeomorphic to Smooth Fano Threefolds
• On a symplectic generalization of a Hirzebruch problem
• Moment polytopes for symplectic manifolds with monodromy
• GKM theory for torus actions with non-isolated fixed points
• 12, 24 and Beyond