TY - JOUR
T1 - Examples of non-Kähler Hamiltonian torus actions
AU - Tolman, Susan
PY - 1998/2
Y1 - 1998/2
N2 - 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 manifolds have been constructed [T] [M] [G]. However, sufficiently large symmetries can force a symplectic manifolds to be Kähler [D] [Kn]. In this paper, we solve several outstanding problems by constructing the first symplectic manifold with large non-trivial symmetries which does not admit an invariant Kähler structure. The proof that it is not Kähler is based on the Atiyah-Guillemin-Sternberg convexity theorem [At] [GS]. Using the ideas of this paper, C. Woodward shows that even the symplectic analogue of spherical varieties need not be Kähler [W].
AB - 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 manifolds have been constructed [T] [M] [G]. However, sufficiently large symmetries can force a symplectic manifolds to be Kähler [D] [Kn]. In this paper, we solve several outstanding problems by constructing the first symplectic manifold with large non-trivial symmetries which does not admit an invariant Kähler structure. The proof that it is not Kähler is based on the Atiyah-Guillemin-Sternberg convexity theorem [At] [GS]. Using the ideas of this paper, C. Woodward shows that even the symplectic analogue of spherical varieties need not be Kähler [W].
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U2 - 10.1007/s002220050205
DO - 10.1007/s002220050205
M3 - Article
AN - SCOPUS:0032340512
SN - 0020-9910
VL - 131
SP - 299
EP - 310
JO - Inventiones Mathematicae
JF - Inventiones Mathematicae
IS - 2
ER -