Kähler quotients of compact Hamiltonian torus manifolds undergo explicit bimeromorphic transformations across walls, enabling comparison of Kähler classes and computation of Riemann–Roch numbers of singular quotients.
Constructibility of momentum maps and linear variation for singular symplectic reduced spaces
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abstract
In this paper we show that the transverse image of the momentum map of a Hamiltonian Lie group action admits a natural integral affine stratification with the property that over each stratum the momentum map is an equivariantly locally trivial fibration, provided the group is compact and the momentum map is proper. Using this we extend the linear variation theorem of Duistermaat and Heckman to singular values of the momentum map by showing that the cohomology classes of the symplectic forms on the reduced spaces at values within a stratum vary linearly. We also point out an instance of an invariant cycle theorem for momentum maps. Finally, we extend all of the above to Hamiltonian actions of proper quasi-symplectic groupoids.
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Comparison of K\"{a}hler quotients of torus actions
Kähler quotients of compact Hamiltonian torus manifolds undergo explicit bimeromorphic transformations across walls, enabling comparison of Kähler classes and computation of Riemann–Roch numbers of singular quotients.