Nonproper intersection theory and positive currents I, local aspects

Details Nonproper intersection theory and positive currents I, local aspects Nonproper intersection theory and positive currents I, local aspects

2010-09-13

arxiv.org/abs/1009.2458v1

Mathematics

Complex Variables

Scientific paper

We introduce a current calculus to deal with (local) non-proper intersection theory, especially construction of local cycles of St\"uckrad-Vogel type (Vogel cycles). Given a coherent ideal sheaf $\J$, generated by a tuple of functions $f$ semiglobally on a reduced analytic space $X$, we construct a current $M^f$, obtained as a limit of explicit expressions in $f$, whose Lelong numbers at each point of its components of various bidegrees are precisely the Segre numbers associated to $\J$ at the point. The precise statement is a generalization of the classical King formula. The current $M^f$ can be interpreted, at each point, as a mean value of various local Vogel cycles. Our current calculus also admits a convenient approach to Tworzewski's locally defined invariant intersection theory.

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