Relative q-Cycles, Hilbert Exponent and Barlet’s Multiplicity

Abstract

The purpose of this note is to show how the mapping multiplicity can be used to define relative analytic cycles on a complex space Y, which yields, for a system of divisors in general position (in a suitable sense) on Y, an intersection degree which coincides with the “Barlet’s degree of an analytic cycle”, and serves as a global Hilbert exponent for the system. Consequently, conditions are given under which there is a continuous mapping of Y into the Barlet space \(C_{q}^\mathrm{loc}(Y)\) of all q-dimensional analytic cycles in Y. This leads to a generalization of a theorem of Siebert on the analyticity of a family of fiber-cycles on a complex space Y, to relative q-cycles on Y. Also, applications to the analyticity of a family of intersections of Schubert zeroes of an ample holomorphic vector bundle \(W\rightarrow Y\) associated to a multi-Schubert symbol are given.