  The Chern numbers of the title are the first coefficients (after the
multiplicities) of the Hilbert functions of various filtrations of ideals of a
local ring $(R, \mathfrak{m})$. For a Noetherian (good) filtration
$\mathcal{A}$ of $\mathfrak{m}$-primary ideals, the positivity and bounds for
$e_1(\mathcal{A})$ are well-studied if $R$ is Cohen-Macaulay, or more broadly,
if $R$ is a Buchsbaum ring or mild generalizations thereof.
  For arbitrary geometric local domains, we introduce techniques based on the
theory of maximal Cohen-Macaulay modules and of extended multiplicity functions
to establish the meaning of the positivity of $e_1(\mathcal{A})$, and to derive
lower and upper bounds for $e_1(\mathcal{A})$.
