Equivariant automorphisms of the Cox construction and applications

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.