Compact monotone tall complexity one $T$-spaces

Abstract: In this paper we study compact monotone tall complexity one $T$-spaces. We use the classification of Karshon and Tolman, and the monotone condition, to prove that any two such spaces are isomorphic if and only if they have equal Duistermaat-Heckman measures. Moreover, we show that the moment polytope is Delzant and reflexive, and provide a complete description of the possible Duistermaat-Heckman measures. Whence we obtain a finiteness result that is analogous to that for compact monotone symplectic toric manifolds. Furthermore, we show that any such $T$-action can be extended to a toric $(T \times S^1)$-action. Motivated by a conjecture of Fine and Panov, we prove that any compact monotone tall complexity one $T$-space is equivariantly symplectomorphic to a Fano manifold endowed with a suitable symplectic form and a complexity one $T$-action.