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Reproducing kernels in and volume forms on toric varieties

Among all the known integral representations in complex analysis, those of Cauchy and Bochner–Martinelli are remarkable for their simplicity. In essence, their kernels are unique differential forms of maximal degree among those defined on the same sets. Thus, the kernel of the Bochner–Martinelli integral representation in Cd generates the de Rham cohomology group H2d−1 DR (C d \{0}) C, while the Cauchy kernel in Cd is a generating element of Hd DR(C \Z) C, where Z is the union of the coordinate hyperplanes, that is, the set of singularities of the kernel. This observation motivates the following definition.