Locally conformally symplectic convexity

Abstract

We investigate special lcs and twisted Hamiltonian torus actions on strict lcs manifolds and characterize them geometrically in terms of the minimal presentation. We prove a convexity theorem for the corresponding twisted moment map, establishing thus an analog of the symplectic convexity theorem of Atiyah and Guillemin-Sternberg. We also prove similar results for the symplectic moment map (defined on the minimal presentation) whose image is then a convex cone. In the special case of a compact toric Vaisman manifold, we obtain a structure theorem. Preliminaries on lcs, lcK and Vaisman manifoldsLet (M, ω) be an almost symplectic manifold of real dimension greater than 2, where ω is a non-degenerate 2-form. Often ω := g(J·, ·) will be the fundamental form of an (almost) Hermitian metric g on (M, J), where J : T M → T M is an (almost) complex structure on M . We will usually consider the complex case, when J is integrable.If every point of M admits a neighborhood U and a smooth function f U : U → R such that the two-form e −fU ω| U is closed, we call (M, ω) a locally conformally symplectic manifold (lcs). If ω is the fundamental (or Kähler) form of a Hermitian manifold (M, g, J), then we call it a locally conformally Kähler manifold (lcK).From the definition, it follows that the local 1-forms df U glue together to a global 1-form θ, called the Lee form, satisfying on M

LcK structures with holomorphic Lee vector field on Vaisman-type manifolds

“…The above formula (3) shows that Vaisman metrics have potential, and by (1), if (Ω, θ) is an lcK structure with potential h, then (e f Ω, θ + df ) is an lcK structure with potential e f h. Remark 2.3. Every lcK metric with potential, in particular every Vaisman metric, on a compact complex manifold is strict.…”

“…These lcK structures, however, have the same Lee vector field as the Vaisman structures. More recently, F. Belgun [1] constructed examples of lcK manifolds with holomorphic Lee vector field on compact complex manifolds which are not of Vaisman type.…”

“…Every lcK metric with potential, in particular every Vaisman metric, on a compact complex manifold is strict. Indeed, if θ = −df and Ω = d θ d c θ h, then (1) shows that e f Ω = dd c (e f h) is a Kähler metric with global potential, which is not possible on compact manifolds.…”

LCK metrics on toric LCS manifolds

“…In [I17], we showed that any compact toric LCK manifold admits a toric Vaisman metric. Finally, in [BGP19], Belgun, Goertsches and Petrecca studied the moment map of a certain class of toric LCS manifolds and showed a corresponding convexity property.…”

The convexity package for Hamiltonian actions on conformal symplectic manifolds

“…Belgun-Goertsches-Petrecca [5] have recently extended the convexity theorem of Atiyah and Guillemin-Sternberg to Hamiltonian torus actions on conformal symplectic manifolds. The purpose of this paper is to generalize their result to the nonabelian case.…”