Residual Kernels with Singularities on Coordinate Planes

A finite collection of planes $\{E_\nu \}$ in $\mathbb C^d$ is called 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 residual kernel for the atomic family. We construct new residual kernels in the case when $E_\nu$ are coordinate planes such that the complement $\mathbb C^d\setminus \bigcup E_\nu$ 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 differential forms. The kernels obtained are used to establish a new formula of integral representations for functions holomorphic in Reinhardt polyhedra.