TY - JOUR
T1 - A compact symmetric symplectic non-kaehler manifold
AU - Lerman, Eugene
PY - 1996
Y1 - 1996
N2 - In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of classification of Hamiltonian group actions. The Audin-Ahara-Hattori-Karshon classification of Hamiltonian circle actions on compact symplectic 4-manifolds showed that all of such manifolds are Kaehler. Delzant's classification of 2n-dimensional symplectic manifolds with Hamiltonian action of n-dimensional tori showed that all such manifolds are projective toric varieties, hence Kaehler. An example in this paper shows that not all compact symplectic manifolds that admit non-trivial Hamiltonian torus actions are Kaehler. Similar techniques allow us to construct a compact symplectic manifold with a Hamiltonian circle action that admits no invariant complex structures, no invariant polarizations, etc.
AB - In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an isolated fixed point. The motivation for this work comes from the program of classification of Hamiltonian group actions. The Audin-Ahara-Hattori-Karshon classification of Hamiltonian circle actions on compact symplectic 4-manifolds showed that all of such manifolds are Kaehler. Delzant's classification of 2n-dimensional symplectic manifolds with Hamiltonian action of n-dimensional tori showed that all such manifolds are projective toric varieties, hence Kaehler. An example in this paper shows that not all compact symplectic manifolds that admit non-trivial Hamiltonian torus actions are Kaehler. Similar techniques allow us to construct a compact symplectic manifold with a Hamiltonian circle action that admits no invariant complex structures, no invariant polarizations, etc.
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U2 - 10.4310/MRL.1996.v3.n5.a2
DO - 10.4310/MRL.1996.v3.n5.a2
M3 - Article
AN - SCOPUS:0030300552
SN - 1073-2780
VL - 3
SP - 587
EP - 590
JO - Mathematical Research Letters
JF - Mathematical Research Letters
IS - 5
ER -