Toric varieties and residues

Abstract: The multidimensional residue theory as well as the theory of integral representations for holomorphic functions is a very powerful tool in complex analysis. The computation of integrals, solving algebraic or differential equations is usually reduced to some residue integral. It is a notable feature of the theory that it is based on few model differential forms. These are the Cauchy kernel and the Bochner-Martinelli kernel. These two model kernels have been the source of other fundamental kernels and residue concepts by means of homological procedures.The Cauchy and Bochner-Martinelli forms possess two common properties: firstly, their singular sets are the unions of complex subspaces, and secondly, the top cohomology group of the complement to the singular set is generated by a single element. We shall call such a set an atomic family and the corresponding form the associated residue kernel.A large class of atomic families is provided by the construction of toric varieties. The extensively developed techniques of toric geometry have already produced many explicit results in complex analysis. In the thesis, we apply these methods to the following two questions of multidimensional residue theory: simplification of the proof of the Vidras-Yger generalisation of the Jacobi residue formula in the toric setting; and construction of a residue kernel associated with a toric variety and its applications in the theory of residues and integral representations. The central role in our construction is played by the theorem stating that under some assumptions a toric variety admits realisation as a complete intersection of toric hypersurfaces in an ambient toric variety.