A finite collection of complex planes in an affine space is caled an atomic family if the top de Rham cohomology group of its complement is generated by a single element. A closed differential form generating this group is called a residue kernel for the atomic family. We construct new residual kernels in the case when the complement of this family admits a toric action with the orbit space being homeomorphic to a compact projective toric variety. They generalize the well known Bochner-Martinelli and Sorani kernels. The kernels obtained are used to establish a new formula of integral representation for function holomorphic in Reinhardt polyhedra. All proofs are based on the multidimensional perspective which allows us to glue a given toric variety to the affine space. As an application we consider a variation on the results of Chirka and Nemirovskij due to the Vitushkin's conjecture: if a two-dimensional sphere is embedded into the complex projective plane as a non-trivial homological cycle, then any holomorphic function in the neighbourhood of the sphere is constant.