On divisorial filtrations on sheaves

In this paper, we generalize the notion of Poincaré series of a multi-index divisorial filtration corresponding to a collection of sigma-processes to the case of an arbitrary locally-free sheaf on the space of blow-ups of the complex plane $\mathbb C^2$. For an arbitrary sheaf, we establish a representation of the series in terms of topological invariants of the sheaf. In particular, for the sheaf of functions, this representation coincides with the Poincaré series obtained by Gusein-Zade and Delgado.