Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Institutions: Philipps University of Marburg, University of Cologne
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166648963