Residue currents are multivariate generalizations of one complex variable residues. They have found various applications in algebra, geometry and analysis, including effective versions of Hilbert ́s Nullstellensatz, generalizations of the Jacobi vanishing theorem, and explicit versions of the Ehrenpreis-Palamodov Fundamental principle. These applications all use the central idea that residue currents can be thought of as analytic representations of varieties or ideals.
In this project we will use a construction of residue currents and apply techniques, recently developed by my collaborators and me, to problems in complex and algebraic geometry. More specifically, the aims of the proposed subprojects are: to find a new realization of Serre duality, to find a new representation of the fundamental cycle of an ideal, and to understand non-proper intersection theory in terms of currents.