  Let I be an m-primary ideal of a Noetherian local ring (R,m) of positive
dimension. The coefficient $e_1(A)$ 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 ${\bar{I^n}}$ 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.
