Tangent Currents, King's Residue Formula and Intersection Theory

Abstract: In this paper, we study the intersection of positive closed currents on domains using the theory of tangent currents in connection with King's residue formula. One of the main results provides a reasonable sufficient condition for the definition of the (proper) intersection of positive closed currents on domains. Another is an analytic description of the self-intersection of analytic subsets. The key idea is to connect tangent currents to complex Monge-Amp`ere type currents. We also investigate the existence, the $h$-dimension and the shadow of tangent currents. Additionally, we introduce local regularizations of positive closed currents, examine classical examples and look into the relationship between our approach and slicing theory. Our work extends to general complex manifolds including compact K\"ahler manifolds.