Residue currents on singular varieties

Doktorsavhandling, 2012

This thesis concerns various aspects of the theory of residue currents. Particularly, we study residue currents on singular varieties and duality theorems for such currents. On a singular variety, there are various notions of holomorphic functions. In Paper I, we study how to extend the definition of Coleff-Herrera products and Bochner-Martinelli type residue currents from the case of strongly holomorphic functions to weakly holomorphic functions, and investigate how various properties known in the strongly holomorphic case transform into the weakly holomorphic case. The duality theorem for Coleff-Herrera products on a complex manifold is one of the key properties of the Coleff-Herrera product. On a singular variety, the duality theorem for Coleff-Herrera products is in general false. In Paper II, we discuss necessary and sufficient conditions for when the duality theorem holds, and in particular we show that on any singular variety, one can find examples where the duality principle fails. Another important property of the Coleff-Herrera product is the transformation law. In Paper III, we describe a comparison formula for Andersson-Wulcan currents, generalizing the transformation law. Applications of this formula include giving a proof by means of residue currents of a theorem of Hickel related to the Jacobian of a holomorphic mapping, and constructing a current on a singular variety satisfying the duality principle. The failure of the duality theorem for Coleff-Herrera products leads to the search for an alternative. In Paper IV, we elaborate on the construction in Paper III, of a current satisfying the duality principle for an arbitrary ideal. In particular, using the comparison formula, we explain how we can view this construction as an intrinsic construction on the variety, generalizing the construction of Andersson and Wulcan.