Aspherical complex surfaces, the Singer conjecture, and Gromov–Lück inequality χ ≥ |σ |

We discuss the Singer conjecture and Gromov–Lück inequality χ≥ |σ| for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group that does not rely on Gromov’s Kähler 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 VII₀⁺ (a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen the Gromov-Lück inequality for aspherical complex surfaces that are not in Class VII₀⁺ . This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid’s conjecture.