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PreprintCC BY 4.0arXiv

Finiteness and boundedness of positive monotone Hamiltonian GKM3 spaces

  • Panagiotis Konstantis
  • Silvia Sabatini
  • Leopold Zoller
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Abstract

In this paper, we establish several finiteness and boundedness theorems for compact positive monotone symplectic manifolds endowed with special Hamiltonian 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. As a byproduct of our results, we also obtain that there are finitely many combinatorially maximal, labeled, reflexive GKM3 graphs of valency n and with χ vertices.

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Cite

BibTeXRIS
BibTeX
@misc{konstantis2026finiteness,
  title = {{Finiteness and boundedness of positive monotone Hamiltonian GKM$_3$ spaces}},
  author = {Panagiotis Konstantis and Silvia Sabatini and Leopold Zoller},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2606.28768},
  eprint = {2606.28768},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.28768},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finiteness and boundedness of positive monotone Hamiltonian GKM_3 spaces
AU  - Panagiotis Konstantis
AU  - Silvia Sabatini
AU  - Leopold Zoller
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2606.28768
UR  - https://arxiv.org/abs/2606.28768
AB  - In this paper, we establish several finiteness and boundedness theorems for compact positive monotone symplectic manifolds endowed with special Hamiltonian actions, called GKM_3, 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. As a byproduct of our results, we also obtain that there are finitely many combinatorially maximal, labeled, reflexive GKM_3 graphs of valency n and with χ vertices.
ER  -