Fundamental concepts in algebraic geometry such as algebraic cycles (with respect to intersection problems) or residues (with respect to division or interpolation ones) gain in flexibility once they are interpreted within the frame of currents in complex geometry ; such interpretation may even be transposed to the frame of arithmetics thanks to recent developments in analytic geometry over algebraically closed fields equipped with an ultrametric (instead of archimedean) absolute value. This survey intends to revisit from such currential interpretation (transposed if possible to the arithmetic setting, for example on Berkovich analytic spaces in place of complex spaces) improper intersection theory, together with the notions of local or global multiplicity, as well as multivariate residue theory with its articulations with Cauchy integral representation theory and Lagrange interpolation formula.

一旦在复杂几何形状的电流框架内解释了代数几何的基本概念，例如代数循环（关于相交问题）或残差（关于除法或插值问题），便具有灵活性。由于解析几何的最新发展，在配有超度量（而不是阿基米德）绝对值的代数封闭领域上，这种解释甚至可以转换为算术框架。这项调查旨在从这种当前的解释（如果可能的话，转换为算术设置，例如在Berkovich分析空间上代替复杂空间），不正确的相交理论以及局部或全局多重性的概念，