Compact monotone tall complexity one $T$-spaces

TL;DR

This paper classifies compact monotone tall complexity one $T$-spaces, showing they are uniquely determined by their Duistermaat-Heckman measures, and relates them to Fano manifolds with specific symplectic structures.

Contribution

It proves that such spaces are classified by their Duistermaat-Heckman measures and establishes their connection to Fano manifolds, extending the understanding of complexity one $T$-spaces.

Findings

Spaces are isomorphic iff their Duistermaat-Heckman measures are equal.

Moment polytopes are Delzant and reflexive.

Any such space can extend to a toric $(T imes S^1)$-action.

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.