  In this article we show that any finite cover of the moduli space of closed
Riemann surfaces of $g$ genus with $g\geq 2$ does not admit any complete
finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we
also show that the total mass of the scalar curvature of any almost Hermitian
metric, which is equivalent to the Teichm\"uller metric, on any finite cover of
the moduli space is negative provided that the scalar curvature is bounded from
below.
