Compact monotone tall complexity one 𝑇-spaces

In this paper we study compact monotone tall complexity one <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -action can be extended to a toric <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T times upper S Superscript 1 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo> × </mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(T \times S^1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -action. Motivated by a conjecture of Fine and Panov, we prove that any compact monotone tall complexity one <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -space is equivariantly symplectomorphic to a Fano manifold endowed with a suitable symplectic form and a complexity one <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -action.