東工大 幾何セミナー： Nicholas Lindsay 氏
- Nicholas Lindsay 氏（上海科技大学）
- Hamiltonian group actions and non-Kahler symplectic structures.
- There are many examples of symplectic manifolds which cannot have a Kahler structure. However, in the presence of a Hamiltonian torus action, such examples are more scarce. For example Delzant proved that if there is a Hamiltonian action of a half dimensional torus then the symplectic manifold is necessarily Kahler. Also, Karshon proved that the existence of a Hamiltonian circle action on a closed symplectic 4-manifold, implies the manifold is Kahler. For a long time, it was unknown whether a closed symplectic manifold with a Hamiltonian torus action with finite fixed point set has an invariant Kahler metric. Then, Tolman constructed a closed symplectic 6-manifold with a Hamiltonian T^2-action with 6 fixed points not having a T^2-invariant Kahler metric.
Recently, Goertsches, Konstantis and Zoller shows that Tolmans manifold is diffeomorphic to a projective bundle over CP^2, hence has some Kahler metric. I will discuss a joint work with Dmitri Panov, in which we showed that the symplectic form constructed by Tolman has no compatible Kahler metric. The settles the problem of finding a compact symplectic manifold with a Hamiltonian circle action with finite fixed point set not having a compatible Kahler metric.