Connectedness of level sets for non-degenerate integrable systems that extend complexity one torus actions

In this paper we study the connectedness of the level sets of integrable systems that have only non-degenerate singular points and extend complexity one $T$-spaces with proper moment maps. Our main result states that if there are no singular points with a hyperbolic block and connected $T$-stabilizer, then each level set is connected. Moreover, we prove that the above condition is necessary if either some reduced space is simply connected or the moment map for the integrable system is generic in a natural sense.

在本文中，我们研究了仅有非退化奇点的可积系统的水平集的连通性，并扩展了具有恰当矩映射的复杂度为一的$T$-空间。我们的主要结果表明，如果不存在带有双曲块且连通的$T$-稳定子的奇点，那么每个水平集都是连通的。此外，我们证明了如果某个约化空间是单连通的，或者可积系统的矩映射在自然意义下是一般的，那么上述条件是必要的。