K\"ahler complexity one Hamiltonian $T$-manifolds have trivial paintings

Abstract

Let a torus $T$ act on a symplectic manifold $(M,\omega)$ with moment map $\phi$. We say that the Hamiltonian $T$-manifold $(M,\omega,\phi)$ has complexity one if $\frac{1}{2} \dim M - \dim T = 1$, and that it is K\"ahler if it admits an invariant compatible complex structure. In this paper, we show how the class of K\"ahler complexity one Hamiltonian $T$-manifolds sits inside the class of complexity one Hamiltonian $T$-manifolds by proving that every compact, connected K\"ahler complexity one Hamiltonian $T$-manifold has a trivial painting. As a corollary, we show that two tall compact, connected K\"ahler complexity one Hamiltonian $T$-manifolds are symplectomorphic exactly if they have the same genus, Duistermaat-Heckman measure, and skeleton. Here, $(M,\omega,\phi)$ is tall exactly if every non-empty fiber $\phi^{-1}(\alpha)$ contains more than one orbit.