Torus manifolds with non-abelian symmetries

Abstract:

Let $G$ be a connected compact non-abelian Lie group and $T$ be a maximal torus of $G$. A torus manifold with $G$-action is defined to be a smooth connected closed oriented manifold of dimension $2\dim T$ with an almost effective action of $G$ such that $M^T\neq \emptyset$. We show that if there is a torus manifold $M$ with $G$-action, then the action of a finite covering group of $G$ factors through $\tilde {G}=\prod SU(l_i+1)\times \prod SO(2l_i+1)\times \prod SO(2l_i)\times T^{l_0}$. The action of $\tilde {G}$ on $M$ restricts to an action of $\tilde {G}’=\prod SU(l_i+1)\times \prod SO(2l_i+1)\times \prod U(l_i)\times T^{l_0}$ which has the same orbits as the $\tilde {G}$-action.

We define invariants of torus manifolds with $G$-action which determine their $\tilde {G}’$-equivariant diffeomorphism type. We call these invariants admissible 5-tuples. A simply connected torus manifold with $G$-action is determined by its admissible 5-tuple up to a $\tilde {G}$-equivariant diffeomorphism. Furthermore, we prove that all admissible 5-tuples may be realised by torus manifolds with $\tilde {G}''$-action, where $\tilde {G}''$ is a finite covering group of $\tilde {G}’$.