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Global properties of toric nearly Kähler manifolds. (English) Zbl 1469.53069

The author studies \(6\)-dimensional complete, toric, strict nearly Kähler manifolds.
We recall that a nearly Kähler manifold is an almost Hermitian manifold \((M,g,J)\) such that \(\nabla J\) is skew-symmetric. Strict means that they are not Kähler. Finally, toric is in the sense that the automorphism group contains a \(3\)-torus. The only known example is the homogeneous nearly Kähler structure on \(S^3 \times S^3\), that the author studied in [Geom. Dedicata 200, 351–362 (2019; Zbl 1415.53066)].
The author extends the local theory developed by A. Moroianu and P.-A. Nagy [Ann. Global Anal. Geom. 55, No. 4, 703–717 (2019; Zbl 1482.53088)] in terms of a Monge-Ampère type equation. A toric nearly Kähler manifold is equipped with two multi-moment maps in the sense of T. B. Madsen and A. Swann [Geom. Dedicata 165, 25–52 (2013; Zbl 1275.53077)]. Theorem 1 gives a nearly-Kähler analogue of a classical result in symplectic geometry, by showing that the multi-moment maps induce a homeomorphism from the space of orbits onto its image \(S^3\). Theorem 2 studies the case when a hypothetical solution to the toric nearly Kähler equation is polynomial in the natural multi-moment map coordinates.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53D20 Momentum maps; symplectic reduction
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