Reduction of coK\"ahler and 3-cosymplectic manifolds

The notion of coK\"ahler manifolds (resp. 3-cosymplectic manifolds) is an odd-dimensional analogue of the one of K\"ahler manifolds (resp. hyperK\"ahler manifolds). In this paper, we obtain reduction theorems of coK\"ahler manifolds and 3-cosymplectic manifolds. We prove that K\"ahler and coK\"ahler reductions have a natural compatibility with respect to cone constructions, that is, the coK\"ahler quotient of the cone of a K\"ahler manifold (resp. the K\"ahler quotient of the cone of a coK\"ahler manifold) coincides with the cone of the K\"ahler quotient (resp. the cone of the coK\"ahler quotient). We also show that hyperK\"ahler and 3-cosymplectic reductions admit the compatibility with respect to cone constructions. We further prove that the compatibility of K\"ahler and coK\"ahler reductions with respect to mapping torus constructions also does hold.