Examples of Non-kähler Hamiltonian Torus Actions
نویسنده
چکیده
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].
منابع مشابه
Generalized Complex Hamiltonian Torus Actions: Examples and Constraints
Consider an effective Hamiltonian torus action T ×M → M on a topologically twisted, generalized complex manifold M of dimension 2n. We prove that the rank(T ) ≤ n − 2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying rank(T ) = n − 2, using a surgery procedure on toric manifolds.
متن کاملContinuous Families of Hamiltonian Torus Actions
We determine conditions under which two Hamiltonian torus actions on a symplectic manifold M are homotopic by a family of Hamiltonian torus actions, when M is a toric manifold and when M is a coadjoint orbit. MSC 2000: 53D05, 57S05
متن کاملExamples of Non-kähler Hamiltonian Circle Manifolds with the Strong Lefschetz Property
In this paperwe construct six-dimensional compact non-Kähler Hamiltonian circle manifolds which satisfy the strong Lefschetz property themselves but nevertheless have a non-Lefschetz symplectic quotient. This provides the first known counter examples to the question whether the strong Lefschetz property descends to the symplectic quotient. We also give examples of Hamiltonian strong Lefschetz c...
متن کاملOn the Existence of Star Products on Quotient Spaces of Linear Hamiltonian Torus Actions Hans-christian Herbig, Srikanth B. Iyengar and Markus J. Pflaum
We discuss BFV deformation quantization [5] in the special case of a linear Hamiltonian torus action. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of [2] for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.
متن کاملMultiplicity-free Hamiltonian Actions Need Not Be K Ahler
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 KK ahler structures, and are therefore toric varieties. In this note we show that Delzant's result does not generalize to the non-abelian case. Our examples are constructed by applying U (2)-equivariant symplectic surgery to ...
متن کامل