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Torus invariant transverse Kähler foliations. (English) Zbl 1367.53063

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.

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

References:

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