# Hard Lefschetz Property for Hamiltonian torus actions on 6-dimensional GKM manifolds

@article{Cho2013HardLP,
title={Hard Lefschetz Property for Hamiltonian torus actions on 6-dimensional GKM manifolds},
author={Yunhyung Cho and Min Kyu Kim},
journal={arXiv: Symplectic Geometry},
year={2013}
}
• Published 2013
• Mathematics
• arXiv: Symplectic Geometry
In this paper, we study the hard Lefschetz property of a symplectic manifold which admits a Hamiltonian torus action. More precisely, let $(M,\omega)$ be a 6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We will show that if the moment map image of $M$ is a GKM-graph and if the graph is index-increasing, then $(M,\omega)$ satisfies the hard Lefschetz property.
3 Citations

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#### References

SHOWING 1-10 OF 39 REFERENCES
Unimodality of the Betti numbers for Hamiltonian circle action with isolated fixed points
• Mathematics
• 2013
Let $(M,\omega)$ be an eight-dimensional closed symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points. In this article, we will show that the Betti numbers ofExpand
Multiplicity-free Hamiltonian actions need not be Kähler
Abstract. Multiplicity-free actions are symplectic manifolds with a very high degree of symmetry. Delzant [2] showed that all compact multiplicity-free torus actions admit compatible KählerExpand
On the cohomology rings of Hamiltonian T-spaces
• Mathematics
• 1998
Let $M$ be a symplectic manifold equipped with a Hamiltonian action of a torus $T$. Let $F$ denote the fixed point set of the $T$-action and let $i:F\hookrightarrow M$ denote the inclusion. By aExpand
Hard Lefschetz property of symplectic structures on compact Kähler manifolds
In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact K\"ahlerExpand
Examples of non-Kähler Hamiltonian torus actions
Abstract. An important question with a rich history is the extent to which the symplectic category is larger than the Kähler category. Many interesting examples of non-Kähler symplectic manifoldsExpand
Convexity and Commuting Hamiltonians
The converse was proved by A. Horn [5], so that all points in this convex hull occur as diagonals of some matrix A with the given eigenvalues. Kostant [7] generalized these results to any compact LieExpand
Frankel's Theorem in the Symplectic Category
We prove that if an (n − 1)-dimensional torus acts symplectically on a 2n-dimensional symplectic manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regardExpand
New Techniques for obtaining Schubert-type formulas for Hamiltonian manifolds
• Mathematics
• 2010
In [GT], Goldin and the second author extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points.Expand
Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31
These notes describe a general procedure for calculating the Betti numbers of the projective quotient varieties that geometric invariant theory associates to reductive group actions on nonsingularExpand
Towards Generalizing Schubert Calculus in the Symplectic Category
• Mathematics
• 2009
The main purpose of this article is to extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixedExpand