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arXiv · 2307.04198

Compact monotone tall complexity one $T$-spaces

Isabelle Charton·Silvia Sabatini·Daniele Sepe
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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.

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Isabelle Charton, Silvia Sabatini, Daniele Sepe. 2023-07-09. Compact monotone tall complexity one $T$-spaces. https://arxiv.org/abs/2307.04198

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