Unique toric structure on a Fano Bott manifold (Q6183441)

To each symplectic manifold \((M,\omega )\), one associates the Hamiltonian diffeomorphism group Ham\((M,\omega )\). It is a normal subgroup of the symplectomorphism group Symp\((M,\omega )\) and governs all possible Hamiltonian Lie group actions on \((M,\omega )\). The group Ham\((M,\omega )\) is infinite-dimensional and non-compact in general, and it might possess more than one maximal torus with distinct conjugacy classes. Due to \textit{T. Delzant}'s theorem [Bull. Soc. Math. Fr. 116, No. 3, 315--339 (1988; Zbl 0676.58029)], any closed symplectic toric manifold is equivariantly symplectomorphic to a smooth projective toric variety equipped with a torus invariant Kähler form. When a symplectic form is monotone, it is equivariantly symplectomorphic to a smooth Fano toric variety by Kleiman's ampleness criterion. A Bott manifold is the total space of Bott towers (introduced by \textit{M. Grossberg} and \textit{Y. Karshon}, [Duke Math. J. 76, No. 1, 23--58 (1994; Zbl 0826.22018)]). The authors of the present paper prove that if there exists a \(c_1\)-preserving graded ring isomorphism between integral cohomology rings of two Fano Bott manifolds (where \(c_1\) is the first Chern class), then they are isomorphic as toric varieties. As a consequence, they give an affirmative answer to \textit{D. McDuff}'s question ([Geom. Topol. 15, No. 1, 145--190 (2011; Zbl 1218.14045)]) on the uniqueness of a toric structure on a Fano Bott manifold.