Generalized cycles and Bochner-Martinelli currents: intersection and division

I will first explain in this talk why one needs to enlarge the geometric notion of complex cycle in order to conciliate local and global intersection numbers in improper intersection theory, focusing in particular on the situation in $\mathbb P^n(\mathbb C)$. This part of my talk will summarize the results obtained jointly with M. Andersson, D. Eriksson, H. Samuelsson Kalm, E. Wulcan in the last five years. I will then discuss the recent result obtained by my former student R. Gualdi, leading to a closed formula (in terms of the so-called Ronkin function) for the logarithmic height of an hypersurface (defined over $\mathbb Q$) in a complete toric variety and explain why the difficulties that arise when one tries to extend this result to cycles with higher codimension motivate the necessity to transpose the notion of generalized cycle to the arithmetic setting. Thinking about division instead of intersection, the Bochner-Martinelli residue currents play in improper intersection theory a role similar to that played by representants of generalized cycles in intersection theory. I will emphazise in this lecture the central role played by Crofton's formula since such objects can be interpreted as averaged over a product of projective spaces (with respect to Fubini-Study metric on the coordinate spaces) of Coleff-Herrera currents. After recalling the effectivity results obtained with M. Sombra with respect to multivariate residue calculus in the complete intersection setting, it will be natural to conclude this lecture addressing questions (still open) about the effectiveness of the realization of Briançon-Skoda theorem within an arithmetic frame. My talk intends also to be a tribute to Jan Erik Björk, who unfortunately left us very recently; he indeed inspired many of the topics I will discuss here.