  In this paper we introduce the Chern minimal surface in Hermitian surfaces by
using the Chern connection, and we show that it only has isolated complex and
anticomplex points for a generic one (neither holomorphic nor antiholomorphic).
For a generic Chern minimal $f$ from compact Riemann surface $\Sigma$ in a
Hermitian surface $M$, we establish two identities which related to the sum of
the orders of all complex points, anticomplex points denoted by $P$, $Q$
respectively, the cap product of the pull-back of the first Chern class
$f^*c_1(M)$ and $[\Sigma]$, the Euler characteristic of tangent bundle
$\chi(T\Sigma)$ and the Euler characteristic of normal bundle
$\chi(T^\perp\Sigma)$. More precisely, we obtain the formulae
$P-Q=-f^*c_1(M)[\Sigma]$ and
$P+Q=-\big(\chi(T\Sigma)+\chi(T^\perp\Sigma)\big)$. We also give some
applications of these formulae.
