About GKM- and non-abelian Hamiltonian actions
This thesis revolves around two different, but not entirely unrelated topics. The first is the realization problem in GKM theory, the second is the topic of multiplicity free manifolds. Regarding the realization problem, we first show that a large class of GKM graphs is in fact a restriction of a...
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|This thesis revolves around two different, but not entirely unrelated topics. The first is the realization problem in GKM theory, the second is the topic of multiplicity free manifolds.
Regarding the realization problem, we first show that a large class of GKM graphs is in fact a restriction of a torus graph. This involves realizable GKM_4-graphs, so in particular realizable graphs in general position with valence at least 5. The corresponding GKM manifolds were studied first by Ayzenberg and later also Masuda in [A18] and [AM19].
Then, we give a sufficient criterion for when a T^2-manifold of dimension 6 is equivariantly formal, and use this, building on [GKZ22], to show that every orientable, 3-valent GKM graph is realizable as an equivariantly formal T^2-manifold.
After that, we switch to the realization of certain GKM fiber bundles, as first studied in [GKZ20]. More precisely, we characterize which GKM fiber bundles Γ -> Γ' -> B are realizable. Here B is the GKM graph of a quasitoric manifold of dimension 4, and Γ is the GKM graph of a generalized flag manifold of the form G/T, where T ⊂ G is a maximal torus.
At the end, we also construct many non-trivial examples of such GKM fiber bundles.
The last chapter is essentially the article [GSW22], where we study multiplicity free U(2)-manifolds. Multiplicity free manifolds naturally generalize the class of toric manifolds as studied in [Del88] to non-abelian Lie groups. Friedrich Knop [Kno11] was able to classify those in terms of their principal isotropy type and their invariant momentum polytope, building directly on work of Losev [Los09].
We restrict ourselves to the group U(2) and explicitly give the equivariant diffeomorphism types as well as the symplectic form of certain multiplicity free U(2)-manifolds, including those whose momentum image is a triangle.
We also give an easy-to-check characterization of when a multiplicity free U(2)-manifold admits a compatible U(2)-invariant Kähler structure. This turns out to be the case if and only if the corrsponding action of T^2 ⊂ U(2) admits an invariant Kähler structure.