On some conjectures about the Chern numbers of filtrations

Let I be an m-primary ideal of a Noetherian local ring (R,m) of positive dimension. The coefficient e1(A) of the Hilbert polynomial of an I-admissible filtration A is called the Chern number of A. The Positivity Conjecture of Vasconcelos for the Chern number of the integral closure filtration is proved for a 2-dimensional complete local domain and more generally for any analytically unramified local ring R whose integral closure in its total ring of fractions is Cohen–Macaulay as an R-module. It is proved that if I is a parameter ideal then the Chern number of the I-adic filtration is non-negative. Several other results on the Chern number of the integral closure filtration are established, especially in the case when R is not necessarily Cohen–Macaulay.