Locally conformally symplectic convexity (Q1757074)

The convexity theorem in symplectic geometry states that, for a Hamiltonian action of a compact torus \(T\) on a symplectic manifold \((M, \omega)\) with moment map \(\mu : M \to \mathfrak{t}^*\), the image \(\Delta = \mu(M)\) is a convex polytope given by the convex hull of the points in \(M\) fixed by \(T\), and \(\mu^{-1}(\alpha)\) is connected for every \(\alpha \in \mu(M)\) (see [\textit{M. F. Atiyah}, Bull. Lond. Math. Soc. 14, 1--15 (1982; Zbl 0482.58013); \textit{V. Guillemin} and \textit{S. Sternberg}, Symplectic techniques in physics. Cambridge: Cambridge University Press (1984, Zbl 0576.58012)]). In the paper under review the authors obtain results similar to the convexity theorem but in the locally conformally symplectic context. A \textit{locally conformally symplectic} (lcs) manifold is a compact manifold endowed with a non-degenerate \(2\)-form \(\omega\) that, around every point, is conformal to a symplectic form [\textit{I. Vaisman}, Int. J. Math. Math. Sci. 8, 521--536 (1985; Zbl 0585.53030)]. For a lcs manifold \((M, \omega)\) a 1-form \(\theta\) exists such that \(d\omega = \theta \wedge \omega\) (\(\theta\) is called the \textit{Lee form}), and the \textit{twisted differential} \(d_\theta = d - \theta \wedge { }\) is defined. The vector field \(V\) such that \(\omega(V, \cdot) ) = -\theta\) is called the \textit{s-Lee vector field}. A vector field \(X\) on an lcs manifold \((M, \omega)\) is called \textit{twisted Hamiltonian vector field} if there exists a smooth function \(h_X\) on \(M\) such that \(i_X \omega = d_\theta h_X\). An action of a Lie group \(G\) on \((M, \omega)\) is twisted \textit{Hamiltonian} if each fundamental field of the action is twisted Hamiltonian, and the \textit{action is of Lee type}, if the s-Lee field is a fundamental vector field of the action. The main result of the paper is: Theorem 1.1. Consider a twisted Hamiltonian action of Lee type of a compact torus \(T\) on a compact lcs manifold \((M, \omega, \theta)\) with twist moment map \(\mu : M \to \mathfrak{t}^*\). Then \(\mu : M \to \mu(M)\) is an open map with connected level sets, whose image is a convex polytope. The paper contains many other interesting results: The authors explain elements of the theory of locally conformally symplectic manifolds, conformally Kähler manifolds, and Vaisman manifolds, including group actions on these manifolds, prove a structure theorem for toric Vaisman manifolds, give examples of twisted Hamiltonian actions and computations of moment map images.