  In the present paper, we give a complete description of the group of
holomorphic automorphisms of the Cox construction of a simplicial fan
equivariant with respect to a large enough connected complex Lie subgroup of
the large torus acting on the Cox construction. We then apply this description
to find the groups of holomorphic automorphisms of rational moment-angle
manifolds, including Calabi--Eckmann manifolds and Hopf manifolds. We also
calculate the groups of equivariant automorphisms of complete simplicial toric
varieties. This paper is the first in a series of two papers dedicated to
studying the automorphism groups of toric stacks with applications to complex
geometry.
