  We discuss the Singer conjecture and Gromov-L\"uck inequality $\chi \geq
|\sigma|$ for aspherical complex surfaces. We give a proof of the Singer
conjecture for aspherical complex surface with residually finite fundamental
group that does not rely on Gromov's K\"ahler groups theory. Without the
residually finiteness assumption, we observe that this conjecture can be proven
for all aspherical complex surfaces except possibly those in Class
$\mathrm{VII}_0^+$ (a positive answer to the global spherical shell conjecture
would rule out the existence of aspherical surfaces in this class). We also
sharpen Gromov's inequality for aspherical complex surfaces that are not in
Class $\mathrm{VII}_0^+$. This is achieved by connecting the circle of ideas of
the Singer conjecture with the study of Reid's conjecture.
