On the automorphism group of a symplectic half-flat 6-manifold

We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat SU(3)-structure has Abelian Lie algebra with dimension bounded by min{5,b1(M)}. Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on TS3 which are invariant under a cohomogeneity one action of SO(4).