Ishida, Hiroaki Torus invariant transverse Kähler foliations. (English) Zbl 1367.53063 Trans. Am. Math. Soc. 369, No. 7, 5137-5155 (2017). Let \(M\) be a manifold equipped with a foliation \(\mathcal F\). A transverse symplectic structure with respect to \(\mathcal F\) is a closed \(2\)-form \(\omega\) such that \(\ker \omega = T \mathcal F\). If \(M\) is a complex manifold, there is a similar notion of a transverse Kähler structure. The author proves convexity of a moment map for a torus action on a manifold equipped with a certain transverse symplectic structure. Specifically, he proves the following: Let \(M\) be a compact connected manifold acted on by a torus \(G\) with Lie algebra \(\mathfrak g\), let \(\mathcal F\) be the foliation by orbits of a subspace \(\mathfrak h \subset \mathfrak g\) acting locally freely on \(M\), and let \(\omega\) be a \(G\)-invariant transverse symplectic structure with respect to \(\mathcal F\). If \(v_1, \ldots, v_k \in \mathfrak g\) are such that there exists a smooth map \(h = (h_1, \ldots, h_k) : M \to \mathbb R^k\) such that \(dh_i = - i_{X_{v_i}} \omega\) for all \(i = 1, \ldots, k\), then: 1) for every \(c \in \mathbb R^k\), the fiber \(h^{-1}(c)\) is either empty or connected, 2) if \(Z_j\) are the connected components of the set of common critical points of the \(h_i\), then \(h(Z_j)\) is a point, say \(c_j\), for each \(j\), and \(h(M)\) is the convex hull of the \(c_j\), 3) in particular \(h(M)\) is convex. The author does also discuss properties of moment maps in the transverse Kähler case. Finally, he uses these results to prove a conjecture of Cupit-Foutou and Zaffran stating that an LVBM manifold \(M\) is LVM if and only if the foliation (that they themselves construct on \(M\)) admits a transverse Kähler structure. Reviewer: Luca Vitagliano (Fisciano) Cited in 7 Documents MSC: 53D20 Momentum maps; symplectic reduction 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 32M25 Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions 57S25 Groups acting on specific manifolds Keywords:torus action; complex manifold; toric variety; moment-angle manifold; LVM manifold; LVMB manifold; non-Kähler manifold; transverse Kähler form; moment map × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Atiyah, M. F., Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14, 1, 1-15 (1982) · Zbl 0482.58013 [2] Battaglia, Fiammetta; Zaffran, Dan, Foliations modeling nonrational simplicial toric varieties, Int. Math. Res. Not. IMRN, 22, 11785-11815 (2015) · Zbl 1351.14032 [3] Battisti, L., LVMB manifolds and quotients of toric varieties, Math. 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